Eglash Ron African Fractals Modern Computing and Indigenous Design

— African---— Fractals---M O D E R N C O M P U T IN G
A N D IN D IG E N O U S D E S IG N
RO N
EG LA SH
C o n ten ts
A cknow ledgm ents
ix
PART I
I n tr o d u c tio n
CHAPTER J
I n t r o d u c t i o n t o f r a c ta l g e o m e t r y
CHAPTER 2
F r a c t a ls in A f r i c a n s e t t l e m e n t a r c h i t e c t u r e
CHAPTER 3
F ra c t a ls in c r o s s - c u l tu r a l c o m p a r i s o n
CHAPTER 4
I n t e n t i o n a n d i n v e n t i o n in d e s i g n
P A R T II
A fr ic a n fr a c ta l 7 n a t h e m a t i c s
CHAPTER 5
G e o m e tric a lg o rith m s
CHAPTER 6
S calin g
CHAPTER 7
N u m e r i c s y s te m s
CHAPTER 8
R ecursion
CHAPTER 9
I n f in it y
CHAPTER IO
C o m p lex ity
3
20
39
49
61
71
86
/
109
147
151
C ontents
v iii
PART III
Im plication s
a p t e r
11
c h a p t e r
12
c h a p t e r
13
F ra c t a ls in t h e E u r o p e a n h i s t o r y o f m a t h e m a t i c s
c h a p t e r
14
F u tu r e s fo r A f r i c a n f r a c t a l s
Ch
a p p e n d ix
T h e o r e t i c a l fr a m e w o r k s i n c u l t u r a l s t u d i e s o f k n o w l e d g e 1 79
T h e p o li ti c s o f A f r i c a n f r a c t a l s
192
203
216
M easu rin g th e fractal d im e n s io n o f A fric a n s e ttle m e n t a r c h ite c tu r e
N o te s
235
R eferences
Index
253
243
231
•Acknowledgments
T h a n k s to g o first t o m y wife, N a n c y C a m p b e l l , w h o h a s t o l e r a t e d m y o b se ssio n s
w i t h g ra c e , a n d to E v e ly n , A l b e r t , a n d J o a n n e E glash, w h o in s p ire d m a n y o f t h e m .
I a m g ra te fu l for t h e a ss ista n c e o f m y p rofesso rs a t U C S C : R a l p h A b r a h a m , S t e v e
C a t o n , J a m e s C lifford, D o n n a H a ra w a y , G o t t f r i e d M ayer-Kress, K e n N o rris, H ein zO u o P e ii g e n , C a r o ly n M a r t i n S h a w , a n d P a tr ic i a Z av ella. E q u ally i m p o r t a n t were
m y fello w U C S C g r a d u a t e s t u d e n t s , in p a r t i c u l a r D a v i d B a in , J u l i a n B le e c k e r,
P e t e r B r o a d w e l l , K irb y B u n a s , C l a u d i a C a s t e n a d a , G i o v a n n a D i C h i r o , J o e
D u m it , V i n c e n t e Diaz, P a u l E d w ard s, L i n d a G a r c i a , J e n n i fe r G o n z a le s , C h r i s G ray,
J o h n H a r ti g a n , S h a r o n H elsel, L aura K an g, L o rrain e Kenny, M a t t h e w K obbe, A n g ie
Rosga, W a r r e n S ack, M e g S a tt e r t h w a i te , S a n d y S to n e , M a r ita S tu r k e n , B e m t W a h l,
a n d S a r a h W il li a m s . T h a n k s a ls o t o B illie H a r r i s , M i r a n d a H a y s, R e b e c c a Lyle,
K e n M a r t i n , S h e i l a P e u s e , A d o l p h S m i t h , J o s h u a W e i n s t e i n , a n d P a u l Yi.
R e s e a r c h f u n d i n g fr o m t h e I n s t i t u t e for I n t e r c u l t u r a l S t u d i e s a n d t h e Fulb r i g h t p r o g r a m m a d e p o s s ib le m y f i e ld w o rk in w e st a n d c e n t r a l A f r i c a . A s c h a p ­
t e r 10 m a k e s clear, 1 o w e m u c h to m y S e n e g a le s e colleag ues, C h r i s t i n e S t n a D ia tt a
a n d N f a l l y B a d ia n e . A l s o o f g r e a t h e l p in S e n e g a ! w e r e A b d o u l i Ba, R e a l B asso,
C h a r l e s B e c k e r, K o l a d o C is s e , I b n o u D 'tag n e, P a t h e D i a g n e , S o u l e y m a n e B a c h i r
D i a g n e , M o u s s e D io p , W a ly C o l y F ay e, M a x , M a r i e - L o u i s e M o r e a u , M a r g o t
N d ia y e , V ic to r Sagna, O u s in a n S e n , F a to u Sow, Yoro Sylla, S a k ir T h a i m , a n d R ie n e
Acknowledgments
T o je . From th e W e s t A fric a n R e se a rc h C e n te r I received th e e x p e rt a d v ic e of
A m e r i c a n p r o f e s s o rs E i l e e n J u l i e n a n d J a n i s M a y s . T h a n k s a l s o t o S h a m i r a
J o h n s o n , P a u l a n d B e ts e y H a r n e y , J a n e H a l e , 'L i s a M c N e e , a n d Liz M e r m i n .
I a m a ls o g r a t e f u l t o Is siak a Is a a c D r a b o a n d t h e b r i l l i a n t C a n a d i a n p h o ­
t o g r a p h y t e a m , M i c h e l e t D i d i , in B u r k i n a F a s o . T h a n k s a l s o t o A m a d o u
C o u li b a ly , K alifa K o n £ a n d A b d o u l a y e S y lla in M a li. In C a m e r o o n I r e c e i v e d t h e
g e n e r o s i t y o f I r e k e B e ssik e , N g w a E m m a n u e l , N o i f e M e b o u b o , t h e la t e E n g e l ­
b e r t M v e n g , a n d E d w a r d N j o c k . M y w o rk in B e n i n w o u ld n o t h a v e b e e n p ossib le
w i t h o u t t h e a s s is t a n c e o f T o n y H u t c h i n s o n ; t h a n k s a ls o t o K n k e A l f r e d , N a t h e l i
R o b e r t s , a n d M a r t i n e d e S o u s a for t h e i r e x p e r t i s e in v o d u n . I n G h a n a M i c h a e l
O r la n s k y graciously in t r o d u c e d m e t o t h e m a n y c u l t u r a l res o u rces av aila b le . M a n y
o f t h e lo c a l folks 1 s p o k e t o in w e st a n d c e n t r a l A fric a , w h i l e e x t e n d i n g g r e a t g e n ­
e ro s ity a n d e n t h u s i a s m , a s k e d t h a t t h e i r n a m e s r e m a i n u n r e c o r d e d , a n d I t h a n k
t h e m as w ell.
O n my r e tu r n to th e U n ite d S ta te s 1 receiv ed a fello w sh ip fro m th e C e n ­
te r fo r t h e H u m a n i t i e s a t O r e g o n S t a t e U n i v e r s ity , w h i c h a ls o o ff e re d t h e o p p o r ­
t u n i t y t o w o rk w i t h a n t h r o p o l o g i s t s J o a n G r o s s , D a v i d G r o s s , a n d C o r t S m i t h ,
as well as K a m a u S a d ik i from t h e P o r t l a n d B lack E d u c a t i o n a l C e n t e r . T h a n k s also
to M i c h a e l R o b e r s o n fo r b is g e o m e t r y a d v ic e , a n d D a v i d a n d B a r b a r a T h o m a s
( n o w m a t h t e a c h e r s a t H e n d e r s o n v i l l e H i g h , N o r t h C a r o l i n a ) for i n v e s t i g a t i n g
o w a ri p a tc e r n s . A t w o m o n t h f e l l o w s h i p a t t h e U n i v e r s i t y o f O r e g o n g o t m e
th ro u g h th e su m m er, and in to my c u rr e n t p o sitio n at T h e O h io S ta te U n iv e r ­
sity. H e r e I h a v e b e e n t h a n k f u l fo r h e l p fro m P a t t i B r o s n a n , W a y n e C a r l s o n ,
Ja c q u e lin e C h a n d a , C y n th i a D illa rd , D avid H o r n , L in d sa y Jo n e s , O k e c h u k w u
O d i t a , E g o n d u R o s e m a r y O n y e j e k w e , R o b e r t R a n s o m , D a n R eff, R o s e K a p i a n ,
C a r o l y n S i m p s o n , D a a ’i y a h I S a l e e m , J e n n i f e r T e r r y , C y n t h i a T y s o n , a n d
M a n ju la W ald ro n .
T h e r e are also m a n y co lleg u es, g e o g ra p h ic a lly s c a t te r e d , w h o s e fe e d b a c k h as
b e e n i n v a l u a b l e . In p a r t i c u l a r 1 w o u ld li k e t o t h a n k M a d e l e i n e A k r i c h , Ja c k
A l e x a n d e r , M a r y J o A r n o l d i , G e o r g e A r t h u r , M a r c i a A s c h e r , J i m B a rta , S il v io
B e d i n i , T Q B e rg , J e a n - P a u l B o u r d i e r , G e o f B o w k e r , M i c h a e l T . B r o w n , P a t
C a p l i n , B ria n C a se y , J e n n i f e r C r o i s s a n t , D o n C r o w e , J i m C r u t c h f i e l d , U b i r a t a n
D 'A m b r o s i o , R o n a l d Bell, O s e i D a r k w a , M a r i a n n e d e L a e t, G a r y L e e D o w n e y ,
M u n r o e E agles, A r t u r o E sc o b a r, F l o r e n c e F a s a n e l li , J a m e s F e r n a n d e z , M a r i l y n
F ra n k e n s te in , R a y v o n F o u c h e , P au lu s G e rd e s, C h o n a t G e tz , G lo ria G ilm e r,
D a v id H a k k e n , T u r tl e H e a r t , D e b o r a h H e a t h , D a v id H ess, S t e f a n H e l m r e i c h , D arian H e n d ric k s , D a v id H u g h e s , S a n d y Jones, E sm aeli K n te h , R o g e r P. K o v a c h , G e ls a
K n ijn i k , B r u n o L ato u r, M u r r a y Leaf, Bea L u m p k i n , R o b i n M a c k a y , C a r o ! M alloy,
B en o it M a n d e lb ro t, M ik e M a rin a c c i, J o a n n a M asin g iia, Lynn M c G e e , Jam es
Ac/cnowledgnien is
M o rro w , D a v id M o s im e g e , B ria n M M u r p h y , D i a n a B aird N ’D iay e, N a n c y N o o te r ,
K a r e n N o r w o o d , S p u r g e o n E k u n d a y o P a r k e r , C li ffo rd P ic k o v e r , P a t r i c i a P o o le ,
A r t h u r P o w e ll, D e a n P r e b l e , D a n R e g a n , J i'm R a uff, S a l R e s tiv o , P ierre R o n d e a u ,
J o h n R osew all, R u d y R u c k e r, N o r a S a b e lli, J n ro n S a m p s o n , D o u g S c h u le r , P a tr ic k
( R i c k ) S c o t t , R o b S h a w , E n i d S c h i l d k r o u t , D a v i d W i l l i a m s o n S h a ffe r, L a rry
S h ir le y , D e n n i s S m i t h , G e o r g e S p ie s , S u s a n L e ig h S ta r , P a u l S to l le r , P e t e r T a y ­
lor, A g n e s T u s k a , G a r y V a n W y k , D o n n e l l W a l t o n , D M W a r r e n , D o r o t h y W a s h ­
b u r n , H e l e n W a t s o n - V e r r a n , M a r k W. W essels, P a tr ic i a S. W i l s o n , a n d C l a u d i a
Z aslavsky. L a s t a n d n o t least, t h a n k s to m y e d it o r s a t R u tg e r s , D o r e e n V a l e n t i n e
a n d M a r t h a H e lle r .
PART
-Introductio n -----------------—
CHAPTER
In troduction------------------------------------------to--------------------------- :------------------- —
fractal-----------------------------------------------—geometry-----------------:----------------------------
F r a c t a l g e o m e t r y h a s e m e r g e d as o n e o f t h e m o s t e x c i t i n g f r o n t i e r s in t h e
f u s io n b e t w e e n m a t h e m a t i c s a n d i n f o r m a t i o n t e c h n o l o g y . F r a c t a ls c a n b e s e e n
in m a n y o f t h e s w ir lin g p a t t e r n s p r o d u c e d by c o m p u t e r g r a p h i c s , a n d t h e y h a v e
b e c o m e a n i m p o r t a n t n e w t o o l fo r m o d e l i n g i n bio lo g y , geology, a n d o t h e r n a t ­
u ral s c i e n c e s . W h i l e f r a c t a l g e o m e t r y c a n i n d e e d t a k e u r i n t o t h e far r e a c h e s .
. o f h i g h - r e c h s c i e n c e , its p a t t e r n s a re su r p ris in g ly c o m m o n in t r a d i t i o n a l A f r i c a n
d e s i g n s , a n d s o m e o f its b a s i c c o n c e p t s a r e f u n d a m e n t a l t o A f r i c a n k n o w l e d g e
s y s te m s . T h i s b o o k w ill p r o v i d e a n easy i n t r o d u c t i o n t o f r a c t a l g e o m e t r y for
p e o p l e w i t h o u t a n y m a t h e m a t i c s b a c k g r o u n d , a n d it w ill s h o w h o w t h e s e s a m e
c a t e g o r i e s o f g e o m e t r i c p a t t e r n , c a l c u l a t i o n , a n d t h e o r y a r e e x p r e s s e d in
A f r ic a n cu ltu res.
M a t h e m a t i c s a n d culture
For m a n y y ears a n t h r o p o l o g i s t s h a v e o b s e r v e d t h a t t h e p a t t e r n s p r o d u c e d in d i f ­
fe re n t c u ltu re s c a n b e ch a ra c te riz e d by specific design therr.es. In E u ro p e a n d A m e r ­
ic a, fo r e x a m p l e , we o f t e n see c it ie s la id o u t in a grid p a t t e r n o f s t r a i g h t s t r e e t s
a n d r i g h t - a n g l e c o r n e r s . A n o t h e r g rid , t h e C a r t e s i a n c o o r d i n a t e s y s te m , h a s
lo ng b e e n a f o u n d a t i o n for t h e m a t h e m a t i c s used in th e s e societies. In m a n y w orks
Introductton
o f C h i n e s e a r t w e find h e x a g o n s u s e d w i t h e x t r a o r d i n a r y g e o m e t r i c p r e c i s i o n —
a c h o i c e t h a t m i g h t s e e m a r b i t r a r y w e r e it n o t fo r t h e i m p o r t a n c e o f t h e n u m ­
b e r six in t h e h e x a g r a m s o f th e i r f o r tu n e te llin g system ( t h e I Ching), in th e a n a to m y
c h a r t s for a c u p u n c t u r e (liu-qi o r “six s p irits ” ), a n d e v e n in C h i n e s e a r c h i t e c t u r e . 1
S h a p e a n d n u m b e r a r e n o t o n l y t h e u n i v e r s a l ru le s o f m e a s u r e m e n t a n d logic;
t h e y a r e a ls o c u l t u r a l to o l s t h a t c a n b e u se d for e x p r e s s i n g p a r t i c u l a r so c ia l id eas
a n d l i n k i n g d i f f e r e n t a r e a s o f life. T h e y a re , as C l a u d e L e v i - S t r a u s s w o u ld p u t it,
“g o o d to t h i n k w i t h . ”
D e s i g n t h e m e s a r e lik e t h r e a d s r u n n i n g t h r o u g h t h e s o c ia l fa b r ic ; t h e y are
less a c o m m a n d i n g f o r c e t h a n s o m e t h i n g w e c o m m a n d , w e a v i n g th e s e s t r a n d s
i n t o m a n y d i f f e r e n t p a t t e r n s o f m e a n i n g . T h e a n c i e n t C h i n e s e e m p i r e s , for
e x a m p l e , u s e d a b a s e - i o c o u n t i n g s y s te m , a n d t h e y e v e n b e g a n t h e first u n i v e r ­
sal m e t r i c s y s te m .^ S o t h e f r e q u e n t u se o f t h e n u m b e r 6 0 i n C h i n e s e k n o w l e d g e
s y s te m s c a n b e l i n k e d t o t h e c o m b i n a t i o n o f t h i s o fficia l b a se 10 n o t a t i o n w i t h
t h e i r sacred n u m b e r six. I n s o m e A m e r i c a n cities we find t h a t t h e s tre e ts a re n u m ­
b e r e d lik e C a r t e s i a n c o o r d i n a t e s , b u t in o t h e r s t h e y a r e n a m e d a f t e r h is to r i c a l
figures, a n d still o t h e r s c o m b i n e t h e tw o . T h e s e c i t y d i f f e r e n c e s ty p i c a ll y c o r r e ­
s p o n d t o d i f f e r e n t s o c i a t m e a n i n g s — a n e m p h a s i s o n h is t o r y v e rs u s efficiency, for
exam ple.
S u p p o s e t h a t v i s i t o r s fr o m a n o t h e r w o r ld w e r e to v ie w t h e g rid o f a n
A m e r i c a n city. For a c ity w i t h n u m b e r e d stre ets , t h e v is ito rs (a s s u m in g th e y c o u l d
r e a d o u r n u m b e r s ) c o u l d safely c o n c l u d e t h a t A m e r i c a n s m a d e u s e o f a c o o r d i ■n a t e s t r u c t u r e . B u t d o t h e s e A m e r i c a n s a c t u a l l y u n d e r s t a n d c o o r d i n a t e m a t h e ­
m a tic s ? C a n th e y use a c o o r d i n a t e grid t o g r a p h e q u a ti o n s ? j u s t h o w s o p h i s ti c a te d
is t h e i r m a t h e m a t i c a l u n d e r s t a n d i n g ? I n t h e f o l l o w i n g c h a p t e r , w e w ill find o u r ­
selves in a sim ilar p o sitio n , for A fric a n s e t t l e m e n t a r c h i te c t u r e is filled w ith r e m a r k ­
a b l e e x a m p l e s o f f r a c t a l s t r u c t u r e . D id p r e c o l o n i a l A f r i c a n s a c t u a l l y u n d e r s t a n d
a n d a p p l y f r a c ta l g e o m e t r y ?
A s I w ill e x p l a i n in t h i s c h a p t e r , f r a c t a l s a r e c h a r a c t e r i z e d b y t h e r e p e t i ­
t i o n o f s i m i l a r p a t t e r n s a t e v e r - d i m i n i s h i n g sc a le s. T r a d i t i o n a l A f r i c a n s e t t l e ­
m e n t s t y p i c a l l y s h o w t h i s “s e l f - s i m i l a r ” c h a r a c t e r i s t i c : c i r c l e s o f c i r c l e s o f
c i r c u l a r d w e l l i n g s , r e c t a n g u l a r w a lls e n c l o s i n g e v e r - s m a l l e r r e c t a n g l e s , a n d
s t r e e t s in w h i c h b r o a d a v e n u e s b r a n c h d o w n t o t i n y f o o t p a t h s w i t h s t r i k i n g g e o ­
m e t r i c r e p e t i t i o n . T h e f r a c ta l s t r u c t u r e w ill b e e a s ily i d e n t i f i e d w h e n w e c o m ­
p a r e a e r i a l v ie w s o f t h e s e A f r i c a n v illa g e s a n d c i t i e s w i t h c o r r e s p o n d i n g f r a c ta l
graphics sim u latio n s.
.
.
W h a t a r e w e t o m a k e o f t h i s c o m p a r i s o n ? L e t ’s p u t o u r s e l v e s b a c k in th e
s h o e s o f t h e v is ito rs fro m a n o t h e r p l a n e t . H a v i n g b e a m e d d o w n t o a n A m e r i c a n
s e t t l e m e n t n a m e d “C o rv a llis , O r e g o n , " th e y d is c o v e r t h a t t h e s t re e ts a r e n o t n u m -
F ractal geom etry
b e r e d , b u t r a t h e r t i t l e d w i t h w h a t a p p e a r t o b e a r b i t r a r y n a m e s : W a s h i n g t o n , Jef­
ferson, A d a m s , a n d so o n . A t first th e y m i g h t c o n c l u d e t h a t th e r e is n o t h i n g m a t h e ­
m a t i c a l a b o u t it. By u n d e r s t a n d i n g a b i t m o r e a b o u t t h e c u l t u r a l m e a n i n g ,
h o w e v e r* a m a t h e m a t i c a l p a t t e r n d o e s e m e r g e : t h e s e are n a m e s in h is to r i c a l s u c ­
c e s s io n . I t m i g h t b e o n l y o r d e r i n g in t e r m s o f p o s i t i o n in a series ( a n “o r d i n a l ”
n u m b e r ) , b u t t h e r e is s o m e k i n d o f c o o r d i n a t e s y s te m a t w o r k a f t e r all. A f r i c a n
d esigns h a v e to b e a p p ro a c h e d in th e s a m e way. W e c a n n o t ju st assume t h a t A fric a n
f r a c ta ls s h o w a n u n d e r s t a n d i n g o f f r a c ta l g e o m e t r y , n o r c a n w e d ism iss t h a t p o s ­
sibility. W e n e e d to lis te n t o w h a t t h e d e s i g n e r s a n d users o f th e s e s tru c tu r e s h a v e
to say a b o u t it. W h a t a p p e a r s to b e a n u n c o n s c i o u s o r a c c i d e n t a l p a t t e r n m i g h t
actu a lly h a v e a n in te n tio n a l m a t h e m a tic a l c o m p o n e n t.
O v e r a l l , t h e p r e s e n c e o f m a t h e m a t i c s in c u l t u r e c a n b e t h o u g h t o f in
t e r m s o f a s p e c t r u m fr o m u n i n t e n t i o n a l t o s e l f - c o n s c io u s . A t o n e e x t r e m e is th e
e m e r g e n c e o f c o m p l e t e l y u n c o n s c i o u s s t r u c t u r e s . T e r m i t e m o u n d s , for e x a m p l e ,
are e x c e l l e n t fra c ta ls ( t h e y h a v e c h a m b e r s w i t h i n c h a m b e r s w i t h i n c h a m b e r s )
b u t n o o n e w o u ld c l a i m t h a t te r m i t e s u n d e r s t a n d m a t h e m a t i c s . I n t h e s a m e way,
p a t t e r n s a p p e a r in t h e g ro u p d y n a m i c s o f large h u m a n p o p u l a t i o n s , b u t th e s e are
g e n e r a l ly n o t p a t t e r n s o f w h i c h a n y i n d i v i d u a l is a w a r e . U n c o n s c i o u s s t r u c t u r e s
d o n o t c o u n t as m a t h e m a t i c a l k n o w l e d g e , e v e n t h o u g h w e c a n u se m a t h e m a t i c s
t o d e s c r ib e t h e m .
M o v i n g a l o n g th i s s p e c t r u m t o w a r d t h e m o r e i n t e n t i o n a l , w e n e x t find
e x a m p le s o f d e c o ra tiv e designs w h ic h , a lt h o u g h consciously created, h a v e n o
e x p l i c i t k n o w l e d g e a t t a c h e d t o t h e m . I t is p o s s ib le , for e x a m p l e , t h a t a n a r t i s t
w h o d o e s n o t k n o w w h a t t h e w o r d “h e x a g o n ” m e a n s c o u l d s till d ra w o n e w i t h
g r e a t p r e c i s i o n . T h i s w o u ld b e a c o n s c i o u s d e s i g n , b u t t h e k n o w l e d g e is s t ric tly
im p li c it .'1 I n t h e n e x t s t e p a l o n g o u r s p e c t r u m , p e o p l e m a k e rh e s e c o m p o n e n t s
e x p l i c i t — t h e y h a v e n a m e s for t h e p a t t e r n s t h e y o b s e r v e in s h a p e s a n d n u m b e r s .
T a k i n g t h e i n t e n t i o n s p e c t r u m o n e m o r e s t e p , w e h a v e ru les fo r h o w th e s e p a t ­
t e r n s c a n b e c o m b i n e d . H e r e we c a n f i n d “a p p l i e d m a t h e m a t i c s . ” O f c o u rs e
t h e r e is a w o rld o f d i f f e r e n c e b e t w e e n t h e a p p li e d m a t h o f a m o d e r n e n g i n e e r a n d
t h e a p p li e d m a t h o f a s h o p k e e p e r — w h e t h e r o r n o t s o m e t h i n g is i n t e n t i o n a l tells
u s n o t h i n g a b o u t its c o m p l e x it y .
F in ally we m o v e to “ p u re m a th e m a tic s ," as fo u n d in th e a b s tra c t th e o rie s
o f m o d e r n a c a d e m i c m a t h e m a t i c i a n s . P u r e m a t h c a n a l s o b e v e ry s i m p l e — for
e x a m p l e , t h e d i s t i n c t i o n b e t w e e n o r d i n a l n u m b e r s (first, s e c o n d , t h i r d ) a n d c a r ­
d i n a l n u m b e r s ( o n e , tw o , t h r e e ) is a n e x a m p l e o f p u r e m a t h . B u t it w o u l d n o t
b e e n o u g h f o r p e o p l e in a s o c i e t y s i m p l y t o u se e x a m p l e s o f b o t h ty p es; th e y
w o u l d h a v e t o h a v e w o rd s for t h e s e tw o c a t e g o r i e s a n d e x p l i c i t l y r e f l e c t o n a
c o m p a r i s o n o f t h e i r p r o p e r t i e s b e f o r e w e w o u l d say t h a t t h e y h a v e a t h e o r y o f
In tro d u c tio n
6
th e d is tin c tio n b e tw e e n o rd in a l a nd c a rd in a l n u m b e rs. W h il e app lied m a t h e ­
m a t i c s m a k e s u s e o f ru le s, p u r e m a t h t e l ls us w h y t h e y w o r k — a n d h o w t o f i n d
.new o n es.
'
'
'
T h i s b o o k b e g i n s by m o v i n g a l o n g t h e s p e c t r u m ju s t d e s c r i b e d . W e w ill s t a r t b y
s h o w i n g t h a t A f r i c a n fra c ta ls a re n o t s i m p l y d u e t o u n c o n s c i o u s a c tiv ity . W e will
t h e n lo o k a t e x a m p l e s w h e r e t h e y a re c o n s c i o u s b u t i m p l i c i t d e s i g n s , f o llo w e d
b y e x a m p l e s in w h i c h A f r i c a n s h a v e d e v i s e d e x p l i c i t ru les fo r g e n e r a t i n g t h e s e
p a t t e r n s , a n d fin ally t o e x a m p l e s o f a b s t r a c t t h e o r y in t h e s e i n d i g e n o u s k n o w l ­
ed g e systems. T h e re a s o n for ta k in g s u c h a c a u ti o u s r o u t e c a n b e e xp ressed in te r m s
o f w h a t p h i l o s o p h e r K arl P o p p e r c a l l e d “falsifiability." P o p p e r p o i n t e d o u t t h a t
e v e r y o n e h a s t h e u r g e t o c o n f i r m t h e i r f a v o r i t e t h e o r i e s ; a n d so w e h a v e t o t a k e
p r e c a u t i o n s n o t to li m i t o u r a t t e n t i o n t o s u c c e s s — a t h e o r y is o n l y g o o d if you
tr y t o t e s t it fo r failu re . If w e o n ly u s e e x a m p l e s w h e r e A f r i c a n k n o w l e d g e sy s­
t e m s su c c e s sfu lly m a t c h e d f r a c ta l g e o m e t r y , w e w o u ld n o t k n o w its l i m i t a t i o n s .
T h e r e are i n d e e d gaps w h e r e t h e fam ily o f t h e o r i e s a n d p r a c t i c e s c e n t e r e d a r o u n d
fractal g eo m etry in h i g h - t e c h m a t h e m a t i c s h a s n o c o u n t e r p a r t in tr a d it io n a l Africa.
A l t h o u g h s u c h g a p s a re s i g n if i c a n t, t h e y d o n o t i n v a l i d a t e t h e c o m p a r i s o n , b u t
r a t h e r p r o v i d e t h e n e c e s s a ry q u a l i f i c a t i o n s t o a c c u r a t e l y c h a r a c t e r i z e t h e i n d i g e ­
n o u s f r a c ta l g e o m e t r y o f A fric a .
O ve rv ie w o f th e text
F o l l o w i n g t h e i n t r o d u c t i o n t o f r a c ta l g e o m e t r y i n t h e n e x t s e c t i o n , i n c h a p t e r
2 w e w ill e x p l o r e f r a c ta ls in A f r i c a n s e t t l e m e n t s . I t w ill b e c o m e c l e a r t h a t t h e
e x p l a n a t i o n o f u n c o n s c i o u s g r o u p a c t i v i t y d o e s n o t fit t h i s c a se . W h e n w e t a l k
to th e in d ig e n o u s a rc h ite c ts , th e y are q u ite e x p lic it a b o u t th o s e sa m e fractal
f e a t u r e s we o b s e r v e , a n c l u s e s e v e r a l o f t h e b a s i c c o n c e p t s o f f r a c ta l g e o m e t r y in
d is c u ssin g th e i r m a te ria l d esigns a n d a ss o c ia te d k n o w le d g e system s. T e rm ite s
m ay m a k e fractal a rc h ite c tu re s , b u t th e y d o n o t p a in t a b s tr a c t m o d e ls o f th e
s t r u c t u r e o n its- w a lls o r c r e a t e s y m b o ls fo r its g e o m e t r i c p r o p e r t i e s . W h i l e t h e s e
i n t r o d u c t o r y e x a m p l e s w o n ’t s e t t l e a ll t h e q u e s t i o n s , w e w ill a t le a s t h a v e e s t a b ­
l i s h e d t h a t t h e s e a r c h i t e c t u r a l d e s i g n s s h o u l d b e e x p l a i n e d by s o m e t h i n g m o r e
t h a n u n i n t e n t i o n a l so c ia l d y n a m i c s .
I n c h a p t e r 3 w e will e x a m i n e a n o t h e r e x p l a n a tio n : p e r h a p s fractal s e t t l e m e n t '
p a t t e r n s a re n o t u n i q u e to A fric a , a n d w e h a v e sim p ly o b s e r v e d a c o m m o n c h a r a c ­
t e r i s t i c o f all n o n - W e s t e r n a r c h i t e c t u r e s . H e r e t h e c o n c e p t o f d e s i g n t h e m e s
b e c o m e im p o rta n t. A n th ro p o lo g is ts h a v e fo u n d th a t th e design th e m e s fo u n d
in e a c h c u l t u r e a r e fairly d i s t i n c t — t h a t is, d e s p i t e t h e a r t i s t i c d i v e r s i t y w i t h i n
F ractal geometr}
e a c h s o c ie ty , m o s t o f t h e c u l t u r e ’s p a t t e r n s c a n b e c h a r a c t e r i z e d b y s p e c ific g e o ­
m e t r i c p r a c t i c e s . W e w ill see t h a t a l t h o u g h f r a c t a l d e s i g n s d o o c c u r o u t s i d e o f
A f r i c a ( C e l t i c k n o t s , U k r a i n i a n eggs, a n d M a o r i^ r a f te r s h a v e s o m e e x c e l l e n t
e x a m p l e s ) , t h e y a r e n o t e v e r y w h e r e . T h e i- r s t r o n g p r e v a l e n c e in A f r i c a ( a n d in
A f r i c a n - i n f l u e n c e d s o u t h e r n I n d i a ) is q u i t e s p e c i fic .
C h a p t e r 4 r e t u r n s to th i s e x p l o r a t i o n w i t h f r a c ta l s in A f r i c a n e s t h e t i c
d e s i g n . T h e s e e x a m p l e s are i m p o r t a n t for t w o r e a s o n s . F irst, t h e y r e m i n d us t h a t
we c a n n o t a s s u m e e x p l i c i t , f o r m a l k n o w l e d g e s i m p l y o n t h e basis o f a p a t t e r n .
I n c o n t r a s t to t h e fra c ta l p a t t e r n s o f A f r i c a n s e t t l e m e n t a r c h i t e c t u r e , t h e s e a e s ­
t h e t i c f r a c ta ls, a c c o r d i n g to t h e a r t i s a n s , w e r e m a d e “j u s t b e c a u s e it lo o k s p r e t t y
t h a t w ay.” T h e y a re n e i t h e r fo r m a l s y s te m s ( n o ru les to t h e g a m e ) n o r d o t h e a r t i ­
s a n s ’ r e p o r t e x p l i c i t t h i n k i n g (“ I d o n ’t k n o w h o w o r why, it j u s t c a m e to m e ” ).
S e c o n d , th e y p ro v i d e o n e p ossib le r o u t e by w h i c h a p a r t i c u l a r s e t o f m a t h e m a t i c a l
c o n c e p t s c a m e to b e s p r e a d o v e r a n e n o r m o u s c o n t i n e n t . T r a d e n e t w o r k s c o u l d
h a v e d iffu se d t h e fra c ta l a e s t h e t i c a c ro s s A f r i c a , r e i n f o r c i n g a d e s i g n t h e m e t h a t
m a y h a v e b e e n s c a t te r e d a b o u t in o t h e r a r e a s o f life. O f co u rs e, s u c h o r i g in sto ries
a r e n e v e r c e r t a i n , a n d all t o o easy t o i n v e n t .
P a r t 11 o f th is b oo k , s ta rtin g w i t h c h a p t e r 5, p re s e n ts th e e x p lic it d esign m e t h ­
o d s a n d s y m b o li c sy stem s t h a t d e m o n s t r a t e f r a c ta l g e o m e t r y as a n A f r i c a n k n o w l ­
e d g e s y s te m . A s in t h e i n t r o d u c t i o n t o f r a c t a l s in t h e first c h a p t e r , I. w ill a s s u m e
th e re a d e r has n o m a th e m a tic s b ac k g ro u n d a n d p rovide a n in tro d u c tio n to any
n e w c o n c e p t s a lo n g w i t h t h e A f r i c a n v e rs io n s . W e will see t h a t n o t o n ly in a r c h i ­
t e c t u r e , b u t i n t r a d i t i o n a l h a t r s t y l i n g , t e x t i l e s , a n d s c u l p t u r e , in p a i n t i n g , c a r v ­
in g , a n d i n e r a l w o r k , in r e l i g i o n , g a m e s , a n d p r a c t i c a l c ra f t, i n q u a n t i t a t i v e
t e c h n i q u e s a n d s y m b o li c sy s te m s , A f r i c a n s h a v e used t h e p a t t e r n s a n d a b s t r a c t
c o n c e p t s o f f r a c ta l geo m e rry .
C h a p t e r 10, t h e la st in p a r t n , is t h e r e s u l t o f m y c o l l a b o r a t i o n w i t h a n
A f r i c a n p h y s i c is t , P ro f e sso r C h r i s t i a n S i n a D i a t t a . A s p o n s o r for t h e F u l b r i g h t
f e l l o w s h i p t h a t e n a b l e d m y f i e ld w o rk in w e s t a n d c e n t r a l A fric a , D r. D i a t t a t o o k
t h e i d e a o f i n d i g e n o u s f r a c ta l s a n d r a n w i t h it, m o v i n g us in d i r e c t i o n s t h a t 1
w o u l d n e v e r h a v e c o n s i d e r e d o n m y o w n , a n d s till h a v e y e t t o e x p l o r e fully.
In t h e t h i r d a n d final p a r t o f t h i s b o o k w e w ill e x a m i n e t h e c o n s e q u e n c e s
o f A f r i c a n f r a c ta l g e o m e t r y : g i v e n t h a t i t d o e s e x is t, w h a t a re its s o c ia l i m p l i c a ­
tio n s ? C h a p t e r 1 1 w ill briefly r e v i e w p r e v i o u s s t u d i e s o f A f r i c a n k n o w l e d g e sy s­
te m s . W e w ill see t h a t a l t h o u g h s e v e r a l r e s e a r c h e r s h a v e p r o p o s e d ideas r e l a t e d
t o t h e f r a c ta l c o n c e p t — H e n r y L o u is G a t e s ’s “ r e p e t i t i o n w i t h r e v i s i o n , ” L e o p o l d
S e n g h o r ' s ‘“ d y n a m i c s y m m e try ," a n d W i l l i a m Fag g’s " e x p o n e n t i a l m o r p h o l o g y ” are
all g oo d ex am ples— th e r e h a v e b e e n specific o bstacles t h a t p rev en te d anthrop olo gists
a n d o t h e r s fr o m t a k i n g u p th e s e c o n c e p t s in t e r m s o f A f r i c a n m a t h e m a t i c s .
Introduction
8
C h a p t e r i 2 c o v e r s t h e p o l i t i c a l c o n s e q u e n c e s o f A f r i c a n f r a c ta l s . O n t h e
o n e h a n d , we w ill find t h e r e is n o e v i d e n c e t h a t g e o m e t r i c f o r m h a s a n y in h e r­
e n t s o c ia l m e a n i n g . I n s e t t l e m e n t d e s i g n , for e x a m p l e , p e o p l e r e p o r t b o t h o p p r e s ­
siv e a n d H b e ra to r y s o c ia l e x p e r i e n c e s w i t h f r a c t a l a r c h i t e c t u r e s . F r a c t a b v e r s u s
n o n f r a c t a l ( “E u c l i d e a n " ) g e o m e t r y d o e s n o t im p ly g o o d v e r s u s b a d . O n t h e
o t h e r h a n d , p e o p l e d o i n v e s t a b s t r a c t fo r m s w i t h p a r t i c u l a r lo c a l m e a n i n g s . T o
tak e a c o n tro v e rsia l ex a m p le , re c e n t U .S . su p rem e c o u rt d e c is io n s d e c la re d th a t
v o t i n g d is tr i c ts c a n n o t h a v e “b iz a r r e ” o r “h i g h l y i r r e g u la r ” s h a p e s , a n d s e v e r a l o f
th e s e fractal c o n to u r s h a v e b e e n re p la c e d by th e s tr a ig h t lines o f E u c lid e a n
fo r m . If f r a c ta l s e t t l e m e n t p a t t e r n s a r e t r a d i t i o n a l for p e o p l e o f A f r i c a n d e s c e n t ,
a n d E u c l i d e a n s e t t l e m e n t p a t t e r n s fo r E u r o p e a n s , is i t e t h n o c e n t r i c t o in s ist o n
o n l y E u c l i d e a n v o t i n g d i s t r i c t lines?
C h a p t e r 13 w ill e x a m i n e t h e c u l t u r a l h i s t o r y o f f r a c t a l g e o m e t r y a n d its
m a t h e m a t i c a l p re c u r s o rs in E u r o p e . W e w ill s e e t h a t t h e g a p s a r e n o t o n e - s i d e d :
j u s t as A f r i c a n s w e r e m i s s i n g c e r t a i n m a t h e m a t i c a l i d e a s i n t h e i r v e r s i o n o f
f r a c t a l g e o m e t r y , E u r o p e a n s w e r e e q u a l l y a f f e c t e d by t h e i r o w n c u l t u r a l v ie w s
a n d h a v e b e e n s lo w to a d o p t s o m e o f t h e m a t h e m a t i c a l c o n c e p t s t h a t w e r e l o n g
c h a m p i o n e d b y A f r i c a n s . I n d e e d , t h e r e is s t r i k i n g e v i d e n c e t h a t s o m e o f t h e
s o u r c e s o f m a t h e m a t i c a l i n s p i r a t i o n for E u r o p e a n f r a c t a l s w e r e o f A f r i c a n
o r i g i n . T h e f i n a l c h a p t e r w ill m o v e f o r w a r d in t i m e , h i g h l i g h t i n g t h e c o n ­
te m p o ra r y v e rs io n s o f fr a c ta l d e s ig n t h a t h a v e b e e n p ro p o s e d by A f r i c a n
a r c h i t e c t s in S e n e g a l , M a l i , a n d Z a m b i a , a n d o t h e r i l l u s t r a t i o n s o f p o s s ib le f r a c ­
ta l futures.
B u t to u n d e r s t a n d a ll th i s , w e m u s t first v is it t h e f r a c t a l p ast.
A h i s t o r i c a l i n tr o d u c t io n to f r a c t a l g e o m e tr y
T h e w o r k o f G e o r g C a n t o r ( 1 8 4 5 - 1 9 1 8 ) , w h i c h p r o d u c e d t h e first f r a c ta l , t h e
C a n t o r set (fig. 1.1), p r o v e d to b e t h e b e g i n n i n g o f a n e w o u t l o o k o n infinity. In f in ­
ity h a d l o n g b e e n c o n s i d e r e d s u s p e c t by m a t h e m a t i c i a n s . H o w c a n w e c l a i m to
b e u s i n g o n l y e x a c t , e x p l i c i t ru le s if w e h a v e a s y m b o l t h a t v a g u e l y m e a n s “ t h e
n u m b e r yo u w o u ld g e t if y o u c o u n t e d f o r e v e r" ? S o m a n y m a t h e m a t i c i a n s , s t a r t ­
in g w i t h A r i s t o t l e , h a d j u s t b a n n e d it o u t r i g h t . C a n t o r s h o w e d t h a t i t w as p o s ­
s i b le t o k e e p t r a c k o f f h e n u m b e r o f e l e m e n t s in a n i n f i n i t e s e t , a n d d i d s o in a
d e c e p t i v e l y s i m p l e fa s h io n . S t a r t i n g w i t h a s in g le s t r a i g h t li n e , C a n t o r e r a s e d t h e
m i d d l e t h i r d , le a v i n g t w o lin es. H e t h e n c a r r i e d o u t t h e s a m e o p e r a t i o n o n
t h o s e t w o lin e s , e r a s i n g t h e i r m i d d l e s a n d l e a v i n g fo u r l i n e s . I n o t h e r w o r d s , h e
u s e d a s o r t o f f e e d b a c k lo o p , w i t h t h e e n d re s u lt o f i o n e s t a g e b r o u g h t b a c k as t h e
s t a r t i n g p o i n t for t h e n e x t . T h i s t e c h n i q u e is c a l l e d “ r e c u r s i o n . " C a n t o r s h o w e d
Fractal geom etry
t h a t if t h i s re c u r s iv e c o n s t r u c t i o n w a s c o n t i n u e d fo r e v e r, it w o u l d c r e a t e a n
i n f i n i t e n u m b e r o f lin e s , a n d y e t w o u l d h a v e ze ro l e n g t h .
N o t o n ly d i d C a n t o r r e i n t r o d u c e i n f i n i t y - a s a p r o p e r o b j e c t o f m a t h e ­
m a t i c a l stu d y , b u t h is r e c u r s i v e c o n s t r u c t i o n c o u l d b e u s e d as a m o d e l for o t h e r
" p a t h o l o g i c a l c u rv e s ," s u c h as t h a t c r e a t e d b y H e l g e v o n K o c h in 1 9 0 4 (figs. 1.2,
7 .3). T h e m a t h e m a t i c a l p r o p e r t i e s o f t h e s e fig u re s w e r e e q u a l l y p e r p l e x i n g .
S m a l l p o rtio n s lo o k e d ju s t like t h e w h o le , a n d th e s e reflections w ere re p e a te d d o w n
t o in f i n i t e s i m a l sc a le s. H o w c o u l d w e m e a s u r e t h e l e n g t h o f t h e K o c h c u r v e ? If
FIGURE
1.1
T h e C a n t o r se t
In 1877 Georg Cantor came up with the idea of repeatedly subdividing a line to illustrate
the concept of an infinite set. This looping technique is called recursion. By specifying that the
recursion continues forever, C antor was able to define an infinite set.
9
F I G U R E 1. 2
T h e K o c h c u rv e
Helge v o n Koch used the same k ind of recursive loop as C antor, b u t he added lines instead of
erasing them . He began w ith a triangular shape made of four line^, th e “seed." He t h e n replaced
each o f the lines w ith a reduced version o f the original seed shape.
b
a
F I G U R E 1 .3
K o c h curve v a r ia tio n s
B^
iflt
There is nothing special about the particular shape Koch first used. For example, we can make
similar shapes that are more flat or more spiked losing variations on the seed shape (a). Nor is there
anything special about tridngles— any shape can'undergo this recursive replacement process.
Mathematician Giuseppe Peano, for example, experimented with rectangular seed shapes such as
those in (b).
Introduction
w e h o l d u p a s i x - i n c h r u l e r t o t h e c u r v e (fig. 1.4) w e g e t six i n c h e s , b u t o f
c o u r s e t h a t m isse s all t h e n o o k s a n d c r a n n i e s . If w e u s e a s m a l l e r ru ler, w e g e t
g r e a t e r l e n g t h , a n d w i t h a s m a l l e r o n e e v e n g r e a t e r l e n g t h , a n d so oix,to i n f i n ­
ity. O b v i o u s l y t h i s is n o t a v e ry usefu l w ay t o c h a r a c t e r i z e o n e o f t h e s e c u r v e s .
A n e w w ay o f t h i n k i n g a b o u t m e a s u r e m e n t w as n e e d e d . T h e a n s w e r w as to p l o t
th e s e d if fe re n t m ea s u res o f ru le r size versus l e n g t h , a n d see h o w fast w e g a in l e n g t h
as w e s h r i n k t h e r u l e r (fig. 1 .5 ). T h i s r a t e ( t h e s l o p e ) t e l ls us j u s t h o w c r i n k l e d
Ik *
o r t o r t u o u s t h e c u r v e is. F o r e x t r e m e l y c r i n k l e d c u r v e s , t h e p l o t w ill s h o w t h a t
w e r a p i d ly g a i n l e n g t h as w e s h r i n k t h e ru ler, s o it w ill h a v e a s t e e p s l o p e . For
r e l a ti v e ly s m o o t h c u r v e s , yo u d o n ’t g a in m u c h l e n g t h as y o u s h r i n k t h e ru l e r size,
so t h e p l o t h a s a s h a l l o w slop e.
T o m a t h e m a t i c i a n s t h i s s l o p e w a s m o r e t h a n j u s t a p r a c t i c a l w ay t o c h a r ­
acte riz e crin k le s. R e c a l l t h a t w h e n we first tr ie d to m e a s u r e t h e l e n g t h o f t h e K o c h
c u r v e , w e f o u n d t h a t its l e n g t h w as p o t e n t i a l l y in f in i te . Y et t h i s i n f in i te l e n g t h
fits i n t o a b o u n d e d sp ace . M a t h e m a t i c i a n F elix H a u s d o r f f ( 1 8 6 8 - 1 9 4 2 ) fo u n d t h a t
t h i s p a r a d o x c o u l d b e r e s o l v e d if w e t h o u g h t o f t h e p a t h o l o g i c a l c u r v e s as s o m e ­
h o w t a k i n g u p m o r e t h a n o n e d i m e n s i o n , as all n o r m a l lin e s d o , b u t less t h a n tw o
d im e n s i o n s , as flat sh a p e s iike squares a n d c ircle s do. I n H a u s d o r f f ’s view, t h e K o c h
c u r v e h a s a f r a c t i o n a l d i m e n s i o n , a p p r o x i m a t e l y 1.3, w h i c h is t h e s l o p e o f o u r
r u l e r - v e r s u s - l e n g t h p l p t . B e in g p u r e m a t h e m a t i c i a n s , t h e y w e r e f a s c i n a t e d w i t h
t h i s id e a o f a f r a c t i o n a l d i m e n s i o n a n d n e v e r t h o u g h t a b o u t p u t t i n g 'it--to p r a c ­
t i c a l use.
T h e c o n c e p t u a l l e a p to p ra c t ic a l a p p l i c a t i o n w as c r e a t e d by B e n o i t M a n d e l ­
b r o t (b. 1924), w h o h a p p e n e d u p o n a stu dy o f lo n g - te rm river flu c tu a tio n s by British
c iv il s e r v a n t H . E. H u r s t . H u r s t . h a d f o u n d t h a t t h e y e a rly flo o d s o f riv e rs d id n o t
h a v e a n y o n e a v e r a g e , b u t r a t h e r v a r i e d o v e r m a n y d i f f e r e n t s c a l e s — t h e r e w e re
flood y ears, flood d e c a d e s , e v e n flood c e n t u r i e s . H e c o n c l u d e d t h a t t h e o n l y way
t o c h a r a c t e r i z e t h i s t e m p o r a l w ig g lin e ss w as t o p l o t t h e a m o u n t o f f l u c t u a t i o n a t
e a c h s c a l e a n d use t h e s l o p e o f t h i s li n e . M a n d e l b r o t re a l iz e d t h a t t h i s w as
e q u i v a l e n t t o t h ^ k i n d o f s c a l in g m e a s u r e t h a t h a d b e e n , u se d for C a n t o r ’s p a t h o ­
lo g ic a l c u r v e s . A s h e b e g a n to a p p l y c o m p u t e r g r a p h i c s (figs. 1.6, 1 .7 ), h e f o u n d
t h a t t h e s e s h a p e s w e r e n o t p a t h o l o g i c a l a t all, b u t r a t h e r v e ry c o m m o n t h r o u g h ­
o u t t h e n a tu r a l w o rld . M o u ntain* ran g es h a d p e a k s w ith in peak s, trees h a d
b r a n c h e s m a d e o f b r a n c h e s , c lo u d s we,re puffs w i t h i n p u f f s ^ - e V e n h is o w n b o d y
w a s full o f re c u r s iv e s c a l i n g s t r u c t u r e s .
T h e f r a c t a l s i m u l a t i o n s fo r n a t u r a l o b j e c t s i n fig u r e 1.7 w e r e c r e a t e d j u s t
lik e t h e C a n t o r set, K o c h c u r v e , a n d o t h e r e x a m p l e s w e h a v e a lr e a d y se e n , w i t h
a s e e d s h a p e t h a t u n d e r g o e s re c u r s iv e r e p l a c e m e n t . T h e o n l y d i f f e r e n c e is t h a t
som e o f th ese s im u la tio n s require th a t c e rta in lines in th e seed sh a p e do n o t get
6 inches
6 inches
FIGURE I . 4
M e a s u r in g th e le n g th o f fr a c ta l c u r v e s
m
for
T he new curves of Rancor, Koch, an d others represented a problem in m easurem ent theory.
T he length of the curve d epen ds o n th e size of the ruler. As we shrink the ruler dow n, the length
approaches infinity.
t
OJ
>
U
O
•£
So
c
“O
4u>
D
«
e
sm aller ruler s i z e
>•
FIGURE 1 . 5
A b e t t e r w a y to m e a s u r e f r a c t a l c u r v e s
Our experiment in shrinking rulers wasn’t a total waste. In fact, it turns out that if you keep track
of how the measured length changes with ruler size, you get a very good way of characterizing the
curve. A relatively smooth fractal won’t increase length very quickly with shrinking ruler size, but
very crinkled fractals will, (a) This smooth Koch curve doesn’t add much length with shrinking
ruler size, so the plot shows only a small rise, (b) Since a small ruler can get into all the nooks and
crannies, this more crinkled Koch curve shows a steeper rise in measured length with a shrinking
ruler, (c) An extremely tortuous Koch curve has a very sreep slope for its plot.
N o te f o r m ath sticklers: T h e s e figures are p lo tted o n 0 logarithm ic graph.
F ractal geom etry
15
r e p l a c e d . T h i s is i l l u s t r a t e d fo r t h e l u n g m o d e l a t t h e b o t t o m o f fig u re 1.7. T h e
l i n e s t h a t g e t r e p l a c e d in e a c h i t e r a t i o n a r e c a l l e d “a c t i v e l i n e s . ” T h o s e t h a t d o
n o t get replaced are called “passive lines.” W e w ill bemusing t h e d is ti n c ti o n b e tw e e n
a c t i v e a n d p a s s iv e l i n e s in s i m u l a t i o n s fo p - A f r ic a n d e s i g n s as well.
M a n d e l b r o t c o i n e d t h e t e r m " f r a c t a l ” fo r t h i s n e w g e o m e t r y , a n d it is n o w
u se d in e v e ry s c i e n ti fic d i s c i p l i n e fr o m a s t r o p h y s i c s t o zoology. I t is o n e o f th e
m o s t p o w erfu l to o ls for t h e c r e a t i o n o f n e w t e c h n o l o g i e s as well as a re v o l u ti o n a ry
a p p r o a c h t o t h e a n a ly s is o f t h e n a t u r a l w o r l d . I n m e d i c i n e , fo r e x a m p l e , fra c ta l
South Africa
Fractal dimension *» 1 .0 0
Smooth Koch curve
Fractal dimension = 1.1
Great Britain
Fractal dimension =
1 .2 5
Rough Koch curve
Fractal dimension = 1.3
Norway
Fractal dimension =
1.5 2
T o r t u o u s Koch curve
Fractal dimension « 1.5
FIGURE
1.6
M e a s u r i n g n a t u r e u d t h f r a c t a l g e o m e tr y
Although the curves of Cantor and others were introduced as abstractions without physical
meaning, Benoit Mandelbrot realized chat their scaling measure, which he called “fractal
dimension,” could be put to practical use in characterizing irregular shapes in nature. The classic
example is the measurement, of coastlines. Even though it is a very crude model, we can see how
the variations of the roughness in the Koch curve are similar to the variations in these coasts.
Note that tire fractal dimension is our plot slope from figure a.5; the coastlines were measured in
the same way.
acacia tree
clouds
shell
fern
This vertical
line is passive.
These two
horizontal lines
(gray) are the
active lines that
will be replaced
by a reduced
version of this
seed shape-
we see that only the active
lines were replaced;
the passive line remains
the same. Now there
are three passive lines
(center) and four active
lines (the ends).
By the eighth iteration we can sec
the similarity to the scaling structure
of the human lungs. .
F I G U R E 1-7
S i m u l a t i n g n a t u r e w i t h f r a c t a l g e o m e tr y
In his experiments with computer graphics, Mandelbrot found that fractal shapes abound in
nature, where continual processes such as biological growth, geological change, and atmospheric
turbulence result in a wide variety of recursive scaling structures (a). T he recursive construction of
these natural shapes is basically the same as that of the other fractal shapes we have seen so far. In
some examples, like the lung model (b), certain lines of the original seed shape do not participate
in the replacement step; they are called "passive lines.” T he ones which do go through
replacement are called "active lines." Each step is referred to as an “iteration."
Fractal geom etry
d i m e n s i o n c a n be used as a d iag n o stic tool. A h e a l t h y lu n g h a s a h ig h fractal d i m e n ­
s i o n , b u t w h e n b l a c k l u n g d is e a s e b e g i n s it lo ses s o m e o f t h e fine b r a n c h i n g — a
c o n d i t i o n t h a t c a n be d e t e c t e d by m e a s u r i n g t h e f r a c ta l d i m e n s i o n o f t h e X ray.
F o r t h i s r e a s o n , B e n o i t M a n d e l b r o t w as r e c e n t l y n a m e d a n h o n o r a r y m e m b e r o f
th e F re n c h C o al M in ers U n io n .
O f c o u rs e , n o r e v o l u t i o n is w i t h o u t its c o u n t e r r e v o l u t i o n a r i e s . I t w as n o t
l o n g b e f o r e s o m e s c i e n t i s t s s t a r t e d o b j e c t i n g t h a t M a n d e l b r o t w as ig n o r i n g th e
p r e s e n c e o f t h e n a t u r a l o b j e c t s t h a t c o u l d b e d e s c r i b e d by E u c l i d e a n g e o m e try ,
s u c h as c ry s ta ls o r eggs. I t ’s tr u e t h a t n o t a ll o f n a t u r e is f r a c ta l— a n d th i s w ill b e
a n i m p o r t a n t p o i n t f o r us t o k e e p in m i n d . S o m e w r i t e r s h a v e m i s t a k e n l y
a t t e m p t e d t o p o r t r a y A f r i c a n s as “m o r e n a t u r a l ”— a d a n g e r o u s a n d m i s l e a d i n g
c l a i m , e v e n w h e n m a d e by w e l l - m e a n i n g r o m a n t i c s . S i n c e fra c ta ls are a s s o c ia te d
w i t h n a t u r e , a b o o k a b o u t “ A f r i c a n f r a c t a l s ” c o u l d b e m i s i n t e r p r e t e d as s u p p o r t
fo r s u c h r o m a n t i c o rg a n i c is t s . P o i n t i n g o u t t h a t s o m e E u c l i d e a n s h a p e s e x i s t in
t h e r e a l m o f n a t u r e m a k e s i t e a s i e r t o u n d e r s t a n d t h a t A f r i c a n f r a c ta ls a re fr o m
t h e artific ia l r e a l m o f c u l t u r e . B e f o r e m o v i n g o n t o t h e s e A f r i c a n d e s ig n s , l e t ’s
re v i e w t h e b a sic c h a r a c t e r i s t i c s o f f r a c ta l g e o m e t r y .
F i v e e s s e n t ia l c o m p o n e n t s o f f r a c ta l g e o m e t r y
RECURSION
W e h a v e s e e n t h a t fra c ta ls are g e n e r a t e d b y a c i r c u l a r p ro c e s s , a lo o p in w h i c h
t h e o u t p u t a t o n e sta g e b e c o m e s t h e i n p u t fo r t h e n e x t . R e s u lt s a re r e p e a t e d l y
r e t u r n e d , so t h a t t h e sam e o p e r a t i o n c a n b e c arrie d o u t again . T h i s is o ft e n referred
t o as “ re c u r s io n ,” a very p o w e rfu l c o n c e p t . L a t e r w e will d is tin g u is h b e t w e e n th r e e
•
u i f f c . e n i ty p e s o f r e c u r s io n , b u t f o r n o w j u s t t h i n k o f i t in te r m s o f t h i s ic e r a tiv e
f e e d b a c k lo o p . W e ’ve a lr e a d y s e e n h o w i t e r a t i o n w o r k s t o c r e a t e t h e C a n t o r s et
a n d t h e K o c h cu rv e . A l t h o u g h w e c a n c r e a t e a m a t h e m a t i c a l a b s t r a c t io n in w h i c h
t h e r e c u r s io n c o n t i n u e s forever, t h e r e a re also cases w h e r e t h e r e c u r s io n w ill “b o t ­
t o m o u t . ” In o u r g e n e r a t i o n o f t h e K o c h c u r v e , for e x a m p l e , we q u i t o n c e th e lines
g e t t o o s m a ll to p r i n t . I n fa c t, a n y p h y s i c a l l y e x i s t i n g o b j e c t w ill o n l y b e fractalw i t h i n a p a r t i c u l a r r a n g e o f scales.
SCALING
If yo u lo o k a t t h e c o a s t l i n e o f a c o n t i n e n t — ta k e th e P acific side o f N o r t h A m e r ­
ica fo r i n s t a n c e — y ou will see a ja g g e d s h a p e , a n d if y o u lo o k a t a s m a ll p i e c e o f
t h a t c o a s t l i n e — say, C a l i f o r n i a — w e c o n t i n u e t o see s i m i l a r ja g g e d n e s s . I n fa ct,
a s i m i l a r ja g g e d c u r v e c a n b e s e e n s t a n d i n g o n a cliff o v e r l o o k i n g a ro ck y C a l i ­
fo r n ia sh o r e , o r e v e n s t a n d i n g o n t h a t s h o r e lo o k i n g a t o n e rock. O f co urse, t h a t ’s
Intro d u ctio n
i8
o n ly ro u g h ly sim ilar, a n d i t ’s o n l y g o o d for a c e r t a i n r a n g e o f scales, b u t it is a s t o n ­
is h in g t o rea lize h o w w e ll t h i s w o r k s for m a n y n a t u r a l f e a t u r e s . It is t h i s " s c a l ­
in g ” p ro p e r ty o f n a tu r e t h a t a llo w s fra c ta l g e o m e try to b e so e ffe c tiv e for
m o d e l in g . T o h a v e a " s c a li n g s h a p e ” m e a n s t h a t t h e r e a r e s i m i l a r p a t t e r n s a t d if ­
f e r e n t s c a le s w i t h i n t h e r a n g e u n d e r c o n s i d e r a t i o n . E n l a r g i n g a t i n y s e c t i o n w ill
p r o d u c e a p a t t e r n t h a t lo o k s s i m i l a r t o t h e w h o l e p i c t u r e , a n d s h r i n k i n g d o w n
t h e w h o l e w ill g iv e us s o m e t h i n g t h a t lo o k s li k e a t i n y p a r t .
SELF-SIMILARITY
J u s t h o w s i m i l a r d o t h e s e p a t t e r n s h a v e t o b e t o q u a li fy as a fr a c ta l ? M a t h e ­
m a t ic i a n s d is tin g u is h b e t w e e n s t a tis tic a l self-sim ilarity, as in t h e c a s e o f t h e c o a s t ­
l i n e , a n d e x a c t s e l f - s i m i l a r i t y , as i n t h e c a s e o f t h e K o c h c u r v e . I n e x a c t
s e l f - s im i la r it y w e n e e d t o b e a b l e t o s h o w a p r e c i s e r e p l i c a o f t h e w h o l e in a t
l e a s t s o m e o f its p a r t s . I n t h e K o c h c u r v e a p r e c i s e r e p l i c a o f t h e w h o l e c o u l d
b e f o u n d w i t h i n a n y s e c t i o n o f t h e f r a c t a l ( " s t r i c t l y s e l f - s i m i l a r ” ), b u t t h i s i s n ’t
t r u e fo r all f r a c ta l s . T h e b r a n c h i n g f r a c t a l s u s e d t o m o d e l t h e lu n g s a n d a c a c i a
tr e e (fig. 1 .7), fo r e x a m p l e , h a v e p a r t s (e .g ., t h e s t e m ) t h a t d o n o t c o n t a i n a tin y
im a g e o f t h e w h o l e . U n l i k e t h e K o c h c u r v e , t h e y w e r e n o t g e n e r a t e d by r e p l a c ­
in g e v e r y l i n e in t h e s e e d s h a p e w i t h a m i n i a t u r e v e r s i o n o f t h e s e e d ; i n s t e a d ,
w e u s e d s o m e p a s s iv e li n e s t h a t w e r e j u s t c a r r i e d t h o u g h t h e i t e r a t i o n s w i t h o u t
c h a n g e , in a d d i t i o n t o a c t i v e l i n e s t h a t c r e a t e d a g r o w i n g t i p b y t h e u s u a l
recu rsiv e re p la c e m e n t.
INFINITY
S i n c e f r a c ta ls c a n b e l i m i t e d t o a fi n it e r a n g e o f sca les, i t m a y s e e m li k e in f in ity
is ju st a h is to rical artifact, a t best a H o ly G r a i l w h o se q u e s t allo w e d m a t h e m a t i c i a n s
s e r e n d ip it o u s l y t o s t u m b l e acro ss fractals. It is t h i s k i n d o f o m i s s i o n t h a t h a s m a d e
m a n y p u r e m a t h e m a t i c i a n s r a t h e r n o n p l u s s e d a b o u t t h e w h o l e f r a c ta l affair,
a n d in s o m e c ases d o w n r i g h t h o s t i l e (cf. K r a n t z 1 9 8 9 ) . T h e r e is n o w ay to c o n ­
n e c t fractals t o t h e idea o f d i m e n s i o n w i t h o u t u s i n g in fin ity , a n d for m a n y m a t h ­
e m a t i c i a n s t h a t is t h e i r c r u c i a l ro le.
FRACTIONAL
DIMENSION
H o w c a n it be t h a t t h e K o c h c u r v e , o r a n y m e m b e r o f its f r a c t a l fam ily, h a s in f i­
n i t e l e n g t h i n a fi n it e b o u n d a r y ? W e a r e u s e d to t h i n k i n g o f d i m e n s i o n as o n l y
w h o l e n u m b e r s — t h e o n e - d i m e n s i o n a l li n e , t h e t w o - d i m e n s i o n a l p l a n e — b u t
t h e t h e o r y o f m e a s u r e m e n t t h a t g o v e r n s fracta ls allow s d i m e n s i o n s t o be frac tion s.
C o n s id e r , for e x a m p l e , t h e i n c r e a s i n g d i m e n s i o n o f t h e K o c h c u rv e s in figure 1.6.
A b o v e t h e t o p , we c o u ld g o as c lo s e as we like t o a o n e - d i m e n s i o n a l lin e. B elo w
F ractal geometry
t h e b o t t o m , we c o u ld m a k e t h e c u r v e s o j a g g e d t h a t it s t a r t s to fill i n tw o d im e n sio n a l areas of th e p la n e . In b e tw e e n , w e n e e d a n in -b e tw e e n d im e n sio n .
L o o k in g fo r f r a c ta ls in A f r ic a n c u l t u re
A s w e e x a m i n e A f r i c a n d e s i g n s a n d k n o w l e d g e sy s tem s , t h e s e five e s s e n t i a l
c o m p o n e n t s will b e a u se fu l w ay t o k e e p t r a c k o f w h a t d o e s o r d o e s n o t m a t c h
f r a c ta l g e o m e try . S i n c e s c a l i n g a n d s e lf- s im i la r it y a r e d e s c r i p t i v e c h a r a c t e r i s t i c s ,
o u r first s te p will b e to lo ok for th e s e p r o p e r tie s in A f r i c a n designs. O n c e we e s t a b ­
lish t h a t t h e m e , we c a n ask w h e t h e r o r n o t th e s e c o n c e p t s h a v e b e e n in t e n ti o n a ll y
a p p li e d , a n d s t a r t to lo o k for t h e o t h e r t h r e e e s s e n t i a l c o m p o n e n t s . W e w ill n o w
t u r n t o A f r i c a n a r c h i t e c t u r e , w h e r e w e find s o m e o f t h e c l e a r e s t i l l u s t r a t i o n s o f
i n d i g e n o u s se l f - s im i la r d esig n s.
19
CHAPTER
-* * » ■ ,
:
2 _
in--------------------------------------------------—A frican-----------------------------------------------settlem ent--------------------------------------------architecture— ^------------------------- --------------
A rc h ite c tu r e o fte n p ro v id es e x c e lle n t e x a m p le s o f c u ltu ra l d e sig n th em es,
b e c a u s e a n y t h i n g t h a t is g o i n g t o b e so m u c h a p a r t o f o u r l i v e s — a s t r u c t u r e
t h a t m a k e s u p o u r b u i l t e n v i r o n m e n t , o n e in w h i c h w e w ill liv e , w o r k o r p la y —
is likely t o h a v e its d e s i g n i n f o r m e d by o u r s o c i a l c o n c e p t s . T a k e re l ig i o u s a r c h i ­
t e c t u r e for e x a m p l e . S e v e i a l c h u r c h e s h a v e b e e n u u i l t u s i n g a t r i a n g u l a r f l o o r
p la n to sym bolize th e C h r i s t i a n tr in ity ; o th e r s h a v e used a cross s h a p e . T h e
R o m a n P a n t h e o n was d iv id e d in to th r e e v e rtic a l lev els: th e b o t t o m w ith
s e v e n n i c h e s r e p r e s e n t i n g t h e h e a v e n l y b o d i e s , t h e m i d d l e w i t h t h e i 2 z o d ia c
signs, a n d o n to p a h e m i s p h e r e s y m b o liz in g th e o r d e r o f t h e c o s m o s as a
w h o l e . 1 B u t w e d o n ’t n e e d to lo o k t o g r a n d i o s e m o n u m e n t s ; e v e n t h e m o s t m u n ­
d a n e s h a c k w ill i n v o l v e g e o m e t r i c d e c i s i o n s — s h o u l d it b e s q u a r e o r o b l o n g ?
p i t c h e d r o o f o r f la t? f a c e n o r t h o r w e s t ? — a n d so c u l t u r e w ill p l a y a r o l e h e r e
a s w ell.
A t first g l a n c e A f r i c a n a r c h i t e c t u r e m i g h t s e e m so v a r i e d t h a t o n e w o u ld
c o n c l u d e its s t r u c t u r e s h a v e n o t h i n g in c o m m o n . A l t h o u g h t h e r e is g r e a t d i v e r ­
sity a m o n g t h e m a n y c u l t u r e s o f A f r i c a , e x a m p l e s o f f r a c ta l a r c h i t e c t u r e c a n be
f o u n d in e v e r y c o r n e r o f t h e A f r i c a n c o n t i n e n t . N o t all a r c h i t e c t u r e in A f r i c a
is f r a c ta l— f r a c ta l g e o m e t r y is n o t t h e o n l y m a t h e m a t i c s u s e d in A f r i c a —-hu t its
r e p e a t e d p r e s e n c e a m o n g s u c h a w id e v a r i e t y o f s h a p e s is q u i t e s t r ik i n g .
Fractals in A fr ic a n se ttle m en t architecture
In e a c h c a s e p r e s e n t e d h e r e w e w ill c o m p a r e t h e a e r i a l p h o t o o r a r c h i t e c ­
t u r a l d i a g r a m o f a s e t t l e m e n t t o a c o m p u t e r - g e n e r a t e d f r a c ta l m o d e l . T h e f r a c ­
ta l s i m u l a t i o n w ill m a k e t h e s e l f - s im i la r a s p e c t s o f t h e p h y s i c a l s t r u c t u r e m o r e
e v i d e n t , a n d in s o m e cases it will e v e n h e l p us u n d e r s t a n d t h e lo cal c u ltu ra l m e a n ­
in g o f t h e a r c h i t e c t u r e . S i n c e t h e A f r i c a n d e s i g n e r s u s e d t e c h n i q u e s lik e i t e r a ­
t i o n in b u i l d in g th e s e s t ru c tu r e s , o u r v ir tu a l c o n s t r u c t i o n t h r o u g h frac tal g rap h ics
w ill g iv e us a c h a n c e to s e e h o w t h e p a t t e r n s e m e r g e t h r o u g h t h i s p ro cess.
R e c t a n g u l a r f r a c t a l s in s e t t l e m e n t a r c h i t e c t u r e
I f y ou fly o v e r t h e n o r t h e r n p a r t o f C a m e r o o n , h e a d i n g t o w a r d L a k e C h a d a lo n g
t h e L o g o n e R iv e r , y ou will s e e s o m e t h i n g lik e figure 2 .1 a. T h i s a erial p h o t o s h o w s
t h e c ity o f L o g o n e - B ir n i in C a m e r o o n . T h e K o t o k o p e o p le , w h o f o u n d e d th is city
c e n t u r i e s a g o , u se t h e lo c a l c la y t o c r e a t e h u g e r e c t a n g u l a r b u i l d i n g c o m p l e x e s .
T h e la r g e st o f t h e s e b u il d in g s , in t h e u p p e r c e n t e r o f t h e p h o t o , is t h e p a l a c e o f
t h e ch ief, o r “M i a r r e ” (fig. 2 .1 b ). E a c h c o m p l e x is c r e a t e d by a pro cess o f t e n c alle d
“a r c h i te c t u r e by a c c r e t io n , ” in th is case a d d in g r e c t a n g u la r en closu res t o p reex istin g
r e c t a n g l e s . S i n c e n e w e n c l o s u r e s o f t e n i n c o r p o r a t e t h e w a lls o f t w o Qt m o r e . o f
t h e o ld o n e s , e n c l o s u r e s t e n d to g e t l a r g e r a n d la r g e r as yo u g o o u t w a r d f r o m t h e
c e n te r . T h e e n d re s u lt is t h e c o m p l e x o f r e c t a n g l e s w i t h i n r e c t a n g l e s w i t h i n r e c ­
t a n g l e s t h a t we see in t h e p h o t o .
S i n c e t h i s a r c h i t e c t u r e c a n be d e s c r ib e d in te r m s o f s elf-s im ilar s c a l in g — it
m a k e s use o f t h e sa m e p a t t e r n a t sev eral differe nt scales— it is easy to sim u la te using
a c o m p u te r- g e n e ra te d fractal, as we see in figures 2 . i c - e . T h e seed shape o f th e m o d el
is a r e c t a n g le , b u t e a c h sid e is m a d e u p o f b o t h a c t i v e lin e s (gray) a n d passive lines
(b la c k ). A f t e r . t h e first it e r a t io n we see h o w a sm all v e rs io n o f th e original re c ta n g le
is re p r o d u c e d by e a c h o f t h e a c ti v e lines. O n e m o r e it e r a t io n gives a ra n g e o f scales
t h a t is a b o u t t h e sa m e as t h a t o f t h e p a la c e ; t h i s is e n la r g e d in figure 2 . i e .
D u r i n g m y v isit t o L o g o n e - B i r n i in t h e s u m m e r o f 1 9 93 , t h e M i a r r e k in d l y
a llo w e d m e t o c l i m b o n t o t h e p a la c e r o o f a n d ta k e th e p h o t o s h o w n in figure 2 . if.
1 asked sev eral o f th e K o to k o m e n a b o u t th e v a r i a ti o n in scale o f t h e i r a rc h ite c tu re .
T h e y e x p l a i n e d it in t e r m s o f a c o m b i n a t i o n o f p a t r i l o c a l h o u s e h o l d e x p a n s i o n ,
a n d t h e h i s t o r i c n e e d for d e f e n s e . " A m a n w o u l d lik e h i s s o n s t o live n e x t to
h i m , " t h e y s a id , “a n d so w e b u i l d by a d d i n g w alls t o t h e f a t h e r ’s h o u s e . ” I n t h e
p ast, i n v a s i o n s b y n o r t h e r n m a r a u d e r s w e re c o m m o n , a n d so a la rg e r d e f e n s iv e
w a ll w a s a ls o n e e d e d . S o m e t i m e s t h e a s s e m b ly o f f a m i l i e s w o u ld o u t g r o w th i s
d e f e n s iv e e n c l o s u r e , a n d so th e y w o u ld t u r n t h a t w all i n t o h o u s i n g , a n d b u ild a n
e v e n la rg e r e n c l o s u r e a r o u n d it. T h e s e s c a l i n g a d d i t i o n s c r e a t e d t h e t r a d i t i o n o f
s e l f - s im i la r s h a p e s we s till see to d a y , a l t h o u g h t h e p o p u l a t i o n is far b e lo w t h e
a. A n aerial view of the city of Logone-Birni in Cameroon.
The largest building complex, in the center, is the palace
of the chief.
b. A closer view of the palace.
T he smallest rectangles, in the
center, are the royal chambers.
Pho to co urtesy M usee de I ' H o m m e , Paris.
c. Seed shape for the fractal
simulation'of the palace.
T he active lines, in gray,
will be replaced by a scaleddown replica of the entire
seed.
e. Enlargment of the third
iteration.
d. First three iterations of the fractal simulation.
FIGURE
2.1
L o g o n e -B irn i
(figure c o n t i n u e s )
f. Photo by the author taken from the roof of the palace.
g. The guti, the
royal insignia,
painted on the
palace walls.
13y permission
Le chemin de la lumiere
h. T he spiral path taken by visitors to the throne.
B y permission o f L e b e u f 1 969.
of Lebeuf 1969.
F I G U R E 2 . 1 (continued.)
In s id e L o g o n e-B irn i
In tro d u c tio n
o r i g i n a l 1 8 0 ,0 0 0 e s t i m a t e d Tor L o g o n e - B i r n i ’s p e a k in t h e n i n e t e e n t h c e n tu r y . A t
t h a t t i m e t h e r e w a s a g i g a n t i c w a ll, a b o u t 10 f e e t t h i c k , t h a t e n c l o s e d t h e
p erim eter o f th e e n tire settlem en t.
T h e w o m e n I s p o k e w i t h w e r e m u c h less i n t e r e s t e d i n e i t h e r p a t r i l i n e a g e
o r m ilita r y h is to ry ; t h e i r re s p o n s e s c o n c e r n i n g a r c h i t e c t u r a l s c a l in g w e r e prim arily
a b o u t t h e c o n t r a s t b e t w e e n t h e ra w e x t e r i o r w alls a n d t h e s t u n n i n g w a t e r p r o o f
fi n is h t h e y c r e a t e d f o r c o u r t y a r d s a n d i n t e r i o r r o o m s . T h i s b e g a n b y s m o o t h i n g
w e t w a lls f l a t . w i t h s p e c i a l s t o n e s , a p p l y i n g a r e s in c r e a t e d f r o m a p l a n t e x t r a c t ,
a n d t h e n a d d i n g . b e a u t i f u l l y a u s t e r e d e c o r a t i v e lin e s .
T h e m o s t i m p o r t a n t o f t h e s e d e c o r a t i v e d r a w in g s is t h e guri, a royal insig nia
(fig. 2.1 g). T h e c e n t r a l m o t i f o f t h e g u ti s h o w s a re c t a n g le in s id e a r e c t a n g l e inside
a r e c t a n g l e ; it is a k i n d o f a b s t r a c t m o d e l t h a t t h e K o t o k o t h e m s e l v e s h a v e c r e ­
a t e d . T h e r e a s o n f o r c h o o s i n g s c a l i n g r e c t a n g l e s as a s y m b o l o f r o y a l t y b e c o m e s
c le a r w h e n w e lo o k a t t h e passage t h a t o n e m u s t t a k e to v is it t h e M i a r r e (fig. 2 . i h ) .
T h e p assag e as a w h o l e is a r e c t a n g u l a r spiral. E a c h t i m e y o u e n t e r a s m a ll e r scale,
y o u a re r e q u i r e d t o b e h a v e m o r e p o lite ly . By t h e t i m e y o u a r r i v e a t t h e t h r o n e
y o u a r e s h o e l e s s a n d s p e a k w i t h a v e r y c u l t u r e d f o r m a l i t y . 2 T h u s t h e f r a c ta l
s c a l i n g o f t h e a r c h i t e c t u r e is n o t s i m p l y t h e re s u lt o f u n c o n s c i o u s s o c i a l d y n a m ­
ics; it is a su b je c t o f a b s t r a c t r e p r e s e n t a t i o n , anti e v e n a p r a c t ic a l t e c h n i q u e a p p lied
to social ran k in g .
T o th e w est n ear th e N ig e ria n b order th e landscape o f C a m e ro o n becom es
m u c h g r e e n e r ; t h i s is t h e f e r tile h i g h g ra s sla n d s r e g i o n o f t h e B a m i le k e . T h e y to o
h a v e a f r a c t a l s e t t l e m e n t a r c h i t e c t u r e b a s e d o n r e c t a n g l e s (fig. 2 . 2 a ) , b u t i t h a s
n o c u l t u r a l r e l a t i o n to t h a t o f t h e K o t o k o . R a t h e r t h a n t h e t h i c k c la y o f L o g o n e B ir n i , t h e s e h o u s e s a n d t h e a t t a c h e d e n c l o s u r e s a r e b u i l t f r o m b a m b o o , w h i c h ,
is v e r y s t r o n g a n d w id e ly a v a i l a b l e . A n d t h e r e w a s n o m e n t i o n o f k i n s h i p ,
d e f e n s e , o r p o l i t i c s w h e n I a s k e d a b o u t t h e a r c h i t e c t u r e ; h e r e 1 w a s t o l d it is p a t ­
t e r n s o f a g r i c u l t u r a l p r o d u c t i o n t h a t u n d e r l i e t h e s c a l in g . T h e g r a s s l a n d so il a n d
c l i m a t e a re e x c e l l e n t for f a r m i n g , a n d t h e g a r d e n s n e a r t h e B a m i l e k e h o u s e s t y p ­
ic ally g ro w a d o z e n d if fe re n t p l a n t s all in a s in g le s p a c e , w i t h e a c h - t a k i n g its c h a r ­
a c t e r i s t i c v e r t i c a l p l a c e . B u t t h i s is l a b o r i n t e n s i v e , a n d s o m o r e d i s p e r s e d
p l a n t i n g s — ro w s o f c o r n a n d g r o u n d - n u t — a r e u s e d in t h e w i d e r s p a c e s f a r t h e r
fr o m t h e h o u s e . S i n c e t h e s a m e b a m b o o m e s h c o n s t r u c t i o n is u s e d fo r h o u s e s ,
h o u s e e n c l o s u r e s , a n d e n c l o s u r e s o f e n c l o s u r e s , t h e r e s u l t is a s e l f - s im i la r a r c h i ­
tectu re. U n lik e th e d efen siv e la b y rin th o f K o to k o a rc h ite c tu re , w h e re th ere
w e re o n l y a few w e l l - p r o t e c t e d e n tr y w a y s , t h e f a r m i n g a c t i v i t i e s r e q u i r e a lo t o f
m o v e m e n t b e t w e e n e n c l o s u r e s , so a t all s c a l e s we s e e g o o d - s iz e d o p e n i n g s . T h e
f r a c t a l s i m u l a t i o n in figures 2 . 2 b , c s h o w s h o w t h i s s c a l i n g s t r u c t u r e c a n b e m o d ­
e l e d u s i n g a n o p e n s q u a r e as t h e s e e d s h a p e .
FIGURE 2.2
B a m ile k e se ttlem en t
(a) Plan of Bamileke settlement from about i960, (b) Fractal simulation of Bamileke architecture.
In the first iteration (“seed shape”), the two active lines are shown in gray, (c) Enlarged view of
fourth iteration.
( a , Begum 1952; reprinted with permission f r o m o r s t o m ) .
26
introduction
C irc u la r fra cta ls in s e ttle m e n t a rc h ite c tu r e
M u c h o f s o u t h e r n A f r i c a is m a d e u p o f a r i d p l a i n s w h e r e h e r d s o f c a t t l e - a n d o t h e r
l i v e s t o c k a re ra is e d . R i n g - s h a p e d l i v e s t o c k p e n s , o n e fo r e a c h e x t e n d e d fam ily,^
c a n b e s e e n in t h e a erial p h o t o in figure 2.3a, a B a-ila s e t t l e m e n t in s o u t h e r n Z a m ­
bia. A d ia g r a m o f a n o t h e r B a -ila s e t t l e m e n t (fig. 2 .3 d ) m a k e s th e s e liv e sto c k e n c l o ­
sures ( “k ra a ls ” ) m o r e clear. T o w a r d t h e b a c k o f e a c h p e n w e find t h e fam ily liv in g
q u a r t e r s , a n d t o w a r d t h e f r o n t is t h e g a t e d e n t r a n c e fo r l e t t i n g l i v e s t o c k in a n d
o u t. For th is r e a s o n t h e f r o n t e n t r a n c e is a s s o c ia te d w i t h lo w s t a tu s ( u n c l e a n , a n i ­
m a l s ) , a n d t h e b a c k e n d w i t h h i g h s t a t u s ( c l e a n , p eop le).'* T h i s g r a d i e n t o f s t a ­
t u s is re f le c te d by t h e size g r a d i e n t in t h e a r c h i t e c t u r e : t h e f r o n t is o n l y f e n c i n g ,
as w e g o t o w a r d t h e b a c k s m a l l e r b u i l d i n g s (f o r s t o r a g e ) a p p e a r , a n d t o w a r d t h e
v e ry b a c k e n d a r e t h e la r g e r h o u s e s . T h e t w o g e o m e t r i c e l e m e n t s o f t h i s s t r u c ­
t u r e — a r i n g s h a p e o v e r a l l , a n d a s t a t u s g r a d i e n t i n c r e a s i n g w i t h size fr o m f r o n t
t o b a c k — e c h o e s t h r o u g h o u t e v e r y s c a l e o f t h e B a -il a s e t t l e m e n t .
T h e s e t t l e m e n t as a w h o l e h a s t h e s a m e s h a p e : it is a r i n g o f rin gs. T h e s e t ­
t l e m e n t , like t h e li v e s t o c k p e n , h a s a f r o n t / b a c k s o c i a l d i s t i n c t i o n : t h e e n t r a n c e
is lo w s t a t u s , a n d t h e b a c k e n d is h i g h s t a tu s . A t t h e s e t t l e m e n t e n t r a n c e t h e r e
a re n o fam ily e n c l o s u r e s a t a ll fo r t h e first 2 0 y a r d s o r s o , b u t t h e f a r t h e r b ack, w e
go, t h e la r g e r t h e fa m ily e n c l o s u r e s b e c o m e .
A t t h e b a c k e n d o f t h e i n t e r i o r o f t h e s e t t l e m e n t , w e see a s m a ll e r d e t a c h e d
r i n g o f h o u s e s , l i k e a s e t t l e m e n t w i t h i n t h e s e t t l e m e n t . T h i s is t h e c h i e f ’s
e x t e n d e d family. A t t h e b a c k o f t h e i n t e r i o r o f tine c h i e f ’s e x t e n d e d fa m il y rin g ,
t h e c h i e f h a s h i s o w n h o u s e . A n d if w e w e r e t o v ie w a s i n g l e h o u s e f r o m a b o v e ,
w e w o u ld see t h a t it is a r i n g w i t h a s p e c i a l p l a c e a t t h e b a r k o f t h e i n t e r i o r : t h e
h o u s e h o l d altar.
S i n c e w e h a v e a s i m i l a r s t r u c t u r e a t all sc a le s, t h i s a r c h i t e c t u r e is e a sy to
m o d e l w i t h fractals. F ig u re 2 . 3 b s h o w s t h e first t h r e e i t e r a t i o n s . W e b e g i n w i t h
a s e e d s h a p e t h a t c o u l d be t h e o v e r h e a d v i e w o f a s i n g l e h o u s e . T h i s is c r e a t e d
by a c t i v e li n e s t h a t m a k e u p t h e r i n g - s h a p e d w a lls, as w e ll as a n a c t i v e l i n e a t
t h e p o s i t i o n o f t h e a l t a r a t t h e b a c k o f t h e i n t e r i o r . T h e o n l y p a s s iv e li n e s a r e
those a d ja c e n t to th e e n tr a n c e . In th e n e x t ite ra tio n , we h a v e a sh a p e th a t c o u ld
be t h e o v e r h e a d v ie w o f a fa m ily e n c l o s u r e . A t t h e e n t r a n c e t o t h e f a m ily e n c l o ­
su re w e h a v e o n l y f e n c i n g , b u t as w e go t o w a r d t h e b a c k w e h a v e b u i l d i n g s o f
i n c r e a s i n g size. S i n c e t h e s e e d s h a p e u se d o n l y p a s s iv e l i n e s n e a r t h e e n t r a n c e
a n d in c r e a s i n g ly la rg e r lin e s t o w a r d t h e b a c k , t h i s i t e r a t i o n o f o u r s i m u l a t i o n h a s
t h e s a m e size g r a d i e n t t h a t t h e real fam ily e n c l o s u r e sh o w s. Fin ally , t h e t h i r d i t e r ­
a tio n p ro v id es a stru ctu re t h a t c o u ld b e th e o v e rh e a d v iew o f th e w h o le s e ttle ­
m e n t . A t th e e n t r a n c e to th e s e t t l e m e n t w e h a v e o n ly fe n c in g , b u t as we go to w a rd
FIGURE 2 .3
B a -ila
( a) A e r i a l p h o t o o f B a - i l a s e t t l e m e n t b e f o r e 1 9 4 4 - ( b ) F r a c t a l g e n e r a t i o n o f B a - i l a s i m u l a t i o n .
N o t e t h a t t h e s e e d s h a p e h a s o n l y a c t i v e l i n e s ( g r a y ) e x c e p t fo r t h o s e n e a r t h e o p e n i n g ( b l a c k ) ,
(a, A m e r i c a n Geograph ic Institute.)
In tro d u c tio n
t h e b a c k w e h a v e e n c l o s u r e s o f i n c r e a s i n g size. A g a i n , b y h a v i n g t h e s e e d s h a p e
use o n l y p a s s iv e li n e s n e a r t h e e n t r a n c e a n d i n c r e a s i n g l y l a r g e r li n e s t o w a r d th e
b a c k , th is i t e r a t i o n o f o u r s i m u l a t i o n h a s t h e sam e size g r a d i e n t t h a t the'T eal s ettle m e n t sh ow s.
I n e v e r v i s i t e d t h e B a - il a m y self; m o s t o f m y i n f o r m a t i o n c o m e s fr o m t h e
c la s s ic e t h n o g r a p h y b y E d w i n S m i t h a n d A n d r e w D a l e , p u b l i s h e d in 1 92 0.
W h i l e t h e i r c o l o n i a l a n d m i s s i o n a r y m o t i v a t i o n s d o n o t i n s p i r e m u c h tr u s t,
t h e y o f t e n s h o w e d a s t r o n g c o m m i t m e n t t o w a r d u n d e r s t a n d i n g t h e B a - il a p o i n t
o f v ie w for s o c i a l s t r u c t u r e . T h e i r a n a l y s i s o f B a - il a s e t t l e m e n t a r c h i t e c t u r e
p o i n t s o u t f r a c t a l a t t r i b u t e s . T h e y t o o n o t e d t h e s c a l i n g o f h o u s e size, fr o m
t h o s e less t h a n 12 f e e t w i d e n e a r t h e e n t r a n c e , t o h o u s e s m o r e t h a n 4 0 f e e t w id e
a t t h e b a c k , a n d e x p l a i n e d it as a s o c ia l s t a t u s g r a d i e n t ; “ t h e r e b e i n g a w o r l d o f
d if fe re n c e b e t w e e n t h e s m a ll h o v e l o f a careless n o b o d y a n d t h e s p a c i o u s d w e lli n g
o f a c h i e f ’ ( S m i t h a n d D a l e 1 9 6 8 , 1 1 4 ).
It is in S m i t h ’s d i s c u s s i o n o f re lig io u s b eliefs, h o w e v e r , t h a t t h e m o s t s t r i k ­
in g f e a t u r e o f t h e B a - i l a ’s f r a c ta l a r c h i t e c t u r e is i l l u m i n a t e d . U n l i k e m o s t m i s ­
s i o n a r ie s o f h i s ti m e , S m i t h w as a s t r o n g p r o p o n e n t o f r e s p e c t fo r lo c a l re lig io n s.
H e w as n o r e l a t i v i s t — u n d e r s t a n d i n g a n d r e s p e c t w ere' s t r a t e g i e s f o r c o n v e r ­
s i o n — b u t h i s d e l i g h t in t h e in d i g e n o u s s p i rit u a l s t r e n g t h c o m e s ac ro s s c le a r ly in
h i s w r it in g s a n d p r o v i d e d h i m w i t h i n s i g h t i n t o t h e s u b t l e r e l a t i o n o f t h e so c ia l,
s a c r e d , a n d p h y s i c a l s t r u c t u r e o f t h e B a - il a a r c h i t e c t u r a l p l a n .
In this village th e r e are a b o u t 25 0 h u ts , b uilt m ostly o n th e edge o f a circle four
h u n d r e d yards in d ia m e te r. Inside th is circle th e r e is a sub sid iary o n e o c c u p ie d
by th e c h ie f, his family, a n d c a tt le . It is a village in itself, a n d th e form o f it in
th e p la n is th e form o f th e g reater n u m b e r o f Ba-ila villages, w h ic h d o n o t a tta in
to th e d im e n sio n s o f S h a l o b a ’s cap ital. T h e o p e n space in t h e c e n te r of t h e v il­
lage is also broken by a second subsidiary village, in w h ic h reside im p o rta n t m e m ­
bers o f th e c h i e f ’s family, a n d also by th r e e o r four m i n i a t u r e h u t s su r ro u n d e d
by a fence: th e s e are th e mantlu a mizhivno ( “th e m a n e s ’ h u t s ”) w h e re offerings
are m a d e to th e ancestral spirits. T h u s early d o we see traces o f th e nll-pervnding
religious c o n sc io u sn e ss o f th e Ba-ila.
( S m i t h a n d D ale 1968, 1 13)
I n t h e first i t e r a t i o n o f t h e c o m p u t e r - g e n e r a t e d m o d e l t h e r e is a d e t a c h e d
a c t i v e li n e in s id e t h e r i n g , a t t h e e n d o p p o s i t e t h e e n t r a n c e . T h i s w a s m o t i v a t e d
by t h e r i n g c o m p r i s i n g t h e c h i e f ’s fam ily, b u t it also d e s c r i b e s t h e l o c a t i o n o f t h e
s a c r e d a l t a r w i t h i n e a c h h o u s e . A s a l o g i c ia n w o u ld p u t it, t h e c h i e f ’s f a m ily rin g
is t o t h e w h o l e s e t t l e m e n t as t h e a l t a r is t o t h e h o u s e . It is n o t a s t a t u s g r a d i e n t ,
as we saw w i t h t h e f r o n t - b a c k axis, b u t r a t h e r a r e c u r r i n g f u n c t i o n a l ro le b e t w e e n
d i f f e r e n t sca les: “T h e w o r d a p p l i e d t o t h e c h i e f 's r e l a t i o n t o h is p e o p l e is latlela:
in t h e e x t r a c t s g iv e n a b o v e w e t r a n s l a t e it ‘to r u l e , 1 b u t it h a s t h i s o n l y as a sec-
Fractals in A fr ic a n se ttle m en t architecture
o n d a r y m e a n i n g . K u le la is p r i m a r i ly t o n u r s e , t o c h e r i s h ; it is t h e w o r d a p p li e d
t o a w o m a n c a r i n g for h e r c h i l d . T h e c h i e f is t h e f a t h e r o f t h e c o m m u n i t y ; t h e y
a re h is c h i l d r e n , a n d w h a t h e d o e s is lela t h e m ” ( S m i t h a n d D a le 1 9 6 8 , 3 0 7 ) .
T h i s r e l a t i o n s h i p is e c h o e d t h r o u g h o u t ' f a m i l y a n d s p i r i t u a l tie s a t all
scales', a n d is s t r u c t u r a l l y m a p p e d t h r o u g h t h e se l f - s im i la r a r c h i t e c t u r e . T h e
n e s t i n g o f c i r c u l a r s h a p e s — a n c e s t r a l m i n i a t u r e s t o c h i e f ’s h o u s e r i n g t o c h i e f ’s
e x t e n d e d fam ily r i n g to che g re a t o u t e r ring— was n o t a s t a tu s g r a d i e n t , as we saw
f o r t h e e n c l o s u r e v a r i a t i o n fr o m f r o n t t o b a c k , b u t s u c c e s s iv e i t e r a t i o n s o f lela.
A very d if fe re n t c ir c u la r fracta l a r c h i t e c t u r e c a n b e s e e n in t h e fam ou s s t o n e
b u i l d i n g s in t h e M a n d a r a M o u n t a i n s o f C a m e r o o n . T h e v a r i o u s e t h n i c g ro u p s
o f t h i s a r e a h a v e t h e i r o w n s e p a r a t e n a m e s , b u t c o l l e c t i v e l y a re o f t e n re f e rre d t o
as K ird i, t h e F u l a n i w o rd for “ p a g a n , ” b e c a u s e o f t h e i r s t r o n g r e s i s t a n c e a g a i n s t
c o n v e r s i o n t o Is la m . T h e i r b u i l d i n g s a r e c r e a t e d fr o m t h e s t o n e r u b b l e t h a t
c o m m o n ly covers th e M a n d a r a m o u n t a in terrain. M u c h o f th e s to n e h as n a tu ra l
f r a c t u r e l i n e s t h a t t e n d t o s p l i t i n t o t h i c k f l a t s h e e t s , so t h e s e r e a d y - m a d e
b ric k s— a l o n g w i t h d e f e n s i v e n e e d s — h e l p e d t o in s p ire t h e c o n s t r u c t i o n o f t h e i r
h u g e castlelike c o m p lex es. But r a th e r th a n b e in g th e E u c lid e a n sh apes o f E u ro ­
p e a n c a s t le s , t h i s A f r i c a n a r c h i t e c t u r e is fr a c ta l.
’ F ig u re 2 .4 a s h o w s t h e b u i l d i n g c o m p l e x o f t h e c h i e f o f M o k o u l e k , o n e o f
t h e M o f o u s e t t l e m e n t s . A n a r c h i t e c t u r a l d i a g r a m o f M o k o u l e k , d r a w n by F re n c h ,
researchers from th e o r s t o m scie n c e in s titu te , show s its o v erall stru c tu re (fig. 2.4b).
It is prim arily c o m p o s e d o f th r e e s t o n e e n c lo s u re s ( t h e large circles), e a c h o f w h i c h
s u r r o u n d s ti g h t l y s p i r a le d g ra n a r ie s ( s m a ll c i r c le s ) . T h e s e e d s h a p e for t h e s i m ­
u l a t i o n r e q u i r e s a c ir c le , m a d e o f p a s s iv e li n e s , a n d t w o d i f f e r e n t sets o f a c t i v e
lin e s (fig. 2.4 c ) . I n s id e t h e c ir c le is a s c a l i n g s e q u e n c e o f s m a ll a c t i v e lines; th e s e
w ill b e c o m e t h e g r a n a r ie s . O u t s i d e t h e c ir c le t h e r e is a la r g e a c t i v e lin e ; t h i s w ill
r e p l i c a t e t h e e n c l o s u r e filled w i t h g ra n a r ie s . By t h e f o u r t h i t e r a t i o n w e h a v e c r e ­
a t e d t h r e e e n c lo s u r e s filled w i t h sp iral c lu s te rs o f g ra n a r ie s , plus o n e unfilled. T h e
re a l d i a g r a m o f M o k o u l e k s h o w s s e v e r a l u n f i l l e d c i r c l e s — e v i d e n c e t h a t n o t
e v e r y t h i n g in t h e a r c h i t e c t u r a l s t r u c t u r e c a n b e a c c o u n t e d fo r by frac tals. N e v ­
e r t h e le s s , a n i m p o r t a n t f e a t u r e is s u g g e s te d by t h e s i m u l a t i o n .
I n t h e first i t e r a t i o n we see t h a t t h e large e x t e r n a l a c t i v e li n e is t o t h e left
o f t h e c ir c le . B u t s i n c e it is a t a n a n g l e , t h e n e x t i t e r a t i o n finds t h i s a c t i v e lin e
a b o v e a n d t o t h e r i g h t. If we fo llo w t h e i t e r a t i o n s , we c a n see t h a t t h e dynam ic
constrwction o f t h e c o m p l e x lias a s p ira l p a t t e r n ; t h e r e p l i c a t i o n s w h o r l a b o u t a
c e n t r a l l o c a t io n . T h i s sp iral d y n a m i c c a n be m issed w i t h j u s t a s t a t i c view — I c e r ­
t a i n ly d i d n ’t see it b e fo r e I tr ie d t h e s i m u l a t i o n — b u t o u r p a r t i c i p a t i o n . i n t h e v i r ­
tu a l c o n s t r u c t i o n m a k e s t h e spiral q u i t e e v i d e n t . 3 T h e sim ilarity b e t w e e n t h e sm all
s p irals o f g ra n a r ie s in s id e t h e e n c l o s u r e s a n d t h i s l a r g e - s c a l e s p ira l s h a p e o f t h e
c
FIGURE 2 . 4
M o k o u le k
( a ) M o k o u l e k , C a m e r o o n . T h e s m a l l b u i l d i n g s in s i d e t h e s t o n e w a l l a r e g r a n a r i e s . T h e r e c t a n g u l a r
b u i l d i n g ( t o p r i g h t ) h o l d s t h e s a c r e d a l t a r , ( b ) A r c h i t e c t u r a l d i a g r a m o f M o k o u l e k . ( c ) F ir s t t h r e e
i t e r a t i o n s o f t h e M o k o u l e k s i m u l a t i o n . T h e s e e d s h a p e is c o m p o s e d o f a c i r c l e d r a w n w i t h p a s s i v e
lin es ( b la c k ) a n d w it h gray a c t i v e lin es b o t h inside a n d o u tsid e th e c irc le , (d ) F o u rth ite ra tio n o f
the M o k o u le k sim u lation ,
f a a n d b, b y permission f r o m S e i g n o b o s 1 9 8 2 .)
Fractals in A fr ic a n settlement architecture
c o m p l e x as a w h o l e i n d i c a t e s t h a t t h e f r a c t a l a p p e a r a n c e o f t h e a r c h i t e c t u r e is
n o t m e r e ly d u e t o a r a n d o m a c c u m u l a t i o n o f v a rio u s - s iz e d c i r c u l a r fo rm s. T h e
i d e a o f c i r c l e s o f i n c r e a s i n g size, s p i r a l i n g f r o m a c e n t r a l p o i n t , h a s b e e n a p p li e d
a t tw o d i f f e r e n t sc ales, a n d t h i s s t r u c t u r a l c o h e f e n c e is c o n f i r m e d b y t h e a r c h i ­
tects’ o w n concepts.
I n o u r s im u la tio n th e activ e lin e b e c a m e lo c a te d to w a rd th e c e n te r o f th e
s p i ra l. T h e M o f o u a ls o t h i n k o f t h e i r a r c h i t e c t u r e as s p i r a l i n g f r o m t h i s c e n t r a l
l o c a t i o n , w h i c h h o l d s t h e i r s a c r e d altar. T h e a l t a r is a k i n d o f c o n c e p t u a l “a c t i v e
l i n e ” in t h e i r s c h e m a ; it is r e s p o n s ib l e f o r t h e i t e r a t i o n s o f life. S e i g n o b o s ( 1 9 8 2 )
n o te s t h a t th is a re a o f t h e c o m p l e x is t h e site o f b o t h religio us a n d p o litic a l a u t h o r ­
ity; it is t h e l o c a t i o n for r i t u a l s t h a t g e n e r a t e c y c le s o f a g r i c u l t u r a l f e r til it y a n d
a n c e s t r a l s u c c e s s io n . T h i s g e o m e t r i c m a p p i n g b e t w e e n t h e s c a l in g c ir c le s o f t h e .
a r c h i t e c t u r e a n d t h e s p i r i t u a l c y cle s o f life is r e p r e s e n t e d in t h e i r d i v i n a t i o n
( “f o r t u n e t e l l i n g ” ) r i tu a l , in w h i c h t h e p r i e s t c r e a t e s c o n c e n t r i c c ir c le s o f sto n e s '
a n d p la c e s h i m s e l f a t t h e c e n te r. A s in t h e guti p a i n t i n g in L o g o n e - B ir n i , in w h i c h
t h e K o t o k o h a d m o d e l e d t h e i r s c a l i n g r e c t a n g l e s , t h e M o f o u h a v e a ls o c r e a t e d
th e ir o w n scalin g sim u latio n .
By t h e t i m e 1 a r r iv e d a t M o k o u l e k in 1 9 9 4 t h e c h i e f h a d d ie d , a n d t h e o w n ­
e r s h i p o f t h i s c o m p l e x h a d b e e n p ass ed o n to h i s w id o w s. T h e n e w c h i e f to l d me
thac t h e d esig n o f th is a r c h i t e c t u r e , in c l u d i n g t h a t o f h is n e w c o m p l e x , b e g a n w ith
a p re c i s e k n o w l e d g e o f t h e a g ri c u lt u ra l y ield . T h i s v o l u m e m e a s u r e w as t h e n c o n ­
v e r t e d t o a n u m b e r o f g r a n a r ie s , a n d t h e s e w e r e a r r a n g e d in spirals. T h e d e sig n
is t h u s n o t s i m p l y a m a t t e r o f a d d i n g o n g r a n a r i e s as t h e y a r e n e e d e d ; in fact, it
h a s a m u c h m o r e q u a n t i t a t i v e basis t h a n m y c o m p u t e r m o d e l , w h i c h I sim p ly did
by e y e b a ll.
N o t a ll c i r c u l a r a r c h i t e c t u r e s , in A f r i c a h a v e t h e k i n d o f c e n t r a l i z e d
l o c a t i o n t h a t w e sa w in M o k o u l e k . T h e S o n g h a i v i l l a g e o f L a b b e z a n g a in M a l i
(fig. 2 . 5 a ) , for e x a m p l e , s h o w s c i r c u l a r sw irls o f c i r c u l a r h o u s e s w i t h o u t a n y
s in g le fo c u s . B u t c o m p a r i n g t h i s t o t h e f r a c t a l im a g e o f figure 2 .5 b , w e see t h a t
a l a c k o f c e n t r a l fo cu s d o e s n o t m e a n a l a c k o f s e lf-s im ila r ity . It is i m p o r t a n t to
r e m e m b e r c h a t w h ile “s y m m e t r y ” in E u c l i d e a n g e o m e t r y m e a n s s i m ila rity w i t h i n
o n e s c a l e (e.g ., s i m i l a r i t y b e t w e e n o p p o s i t e s id e s in b i l a t e r a l s y m m e t r y ) , f r a c ta l
g e o m e t r y is b a s e d o n s y m m e t r y b e t w e e n d i f f e r e n t sc a le s. E v e n t h e s e d e c e n t r a l ­
ized sw ir ls o f c i r c u l a r b u i l d i n g s s h o w a s c a l i n g s y m m e t r y . .
P a u l S to l le r , a n a c c o m p l i s h e d e t h n o g r a p h e r o f t h e S o n g h a i , te lls m e t h a t
t h e r e c t a n g u l a r b u i l d i n g s t h a t c a n b e s e e n in figure 2 . 5 a a r e d u e to Is la m ic in flu ­
e n c e , a n d t h a t t h e o r i g i n a l s t r u c t u r e w o u l d h a v e b e e n c o m p l e t e l y c ir c u la r .
T h a n k s t o P e t e r B ro a d w e ll, a c o m p u t e r p r o g r a m m e r f r o m S i l i c o n G r a p h i c s In c.,
' w e w e r e a b le t o r u n a q u a n t i t a t i v e te s t o f t h e p h o t o t h a t c o n f i r m e d w h a t o u r eyes
In tro d u c tio n
32
FIGURE 2 .5
Labbezanga
(a) Aerial view of the village of Lahhetanga in Mali, (b) Fractal graphic.
( a , photo by G e o rg G c r s te r ; b , by p erm issio n o f B enoit M a n d e lb ro t.}
w e r e t e l l i n g us: t h e S o n g h a i a r c h i t e c t u r e c a n b e c h a r a c t e r i z e d by a f r a c ta l d i m e n ­
s i o n s i m i l a r to t h a t o f t h e c o m p u t e r - g e n e r a t e d f r a c ta l o f figure 2.5b.**
T h i s k i n d o f d e n s e c i r c u l a r a r r a n g e m e n t o f c ir c le s , w h i l e o c c u r r i n g in all
so r ts o f v a r i a t i o n s , is c o m m o n t h r o u g h o u t i n l a n d w e s t A f r i c a . B o u r d i e r a n d
T r i n h ( 1 9 8 5 ) , for e x a m p l e , d e s c r ib e a sim ila r c ircu lar a r c h i t e c t u r e in B u r k i n a Faso.
T h e s c a l i n g o f i n d i v i d u a l b u i l d i n g s is b e a u t i f u l l y d i a g r a m m e d in t h e i r c o v e r
i l l u s t r a t i o n (fig. 2 . 6 a ) , a p o r t i o n o f o n e o f t h e large b u i l d i n g c o m p l e x e s c r e a t e d
by t h e N a n k a n i society. A s for t h e S o n g h a i , fo r e ig n c u l t u r a l i n f lu e n c e s h a v e n o w
i n t r o d u c e d r e c t a n g u l a r b u il d in g s as w ell. I n t h e N a n k a n i c o m p l e x t h e o u t e r m o s t
e n c l o s u r e ( t h e p e r i m e t e r o f t h e c o m p l e x ) is s o c ia lly c o d e d a s m a l e . A s w e m o v e
in , t h e s u c c e s s iv e e n c l o s u r e s b e c o m e m o r e f e m a l e a s s o c i a t e d , d o w n t o t h e c i r ­
c u l a r w o m a n ’s dego (fig. 2 . 6 b ) , t h e c i r c u l a r f i r e p l a c e , a n d f i n a l l y t h e s c a l i n g
s t a c k s o f p o t s (fig. 2 .6 c ).
U s i n g a t e c h n i q u e q u i t e c lo s e t o t h a t o f t h e K o t o k o w o m e n , t h e w o m e n
o f N a n k a n i a ls o w a t e r p r o o f a n d d e c o r a t e t h e s e w alls. T h e r e c u r r e n t i m a g e o f a
Fractals m A fr ic a n se ttle m e n t architecture
t r i a n g l e in th e s e d e c o r a t i o n s (s e e w alls o f d e g o ) r e p r e s e n t s t h e zalanga, a n e s t e d
s t a c k o f c a l a b a s h e s ( c i r c u l a r b o w ls c a r v e d f r o m g o u r d s ) t h a t e a c h w o m a n k e e p s
in h e r k i t c h e n (fig. 2 .6 d ). S i n c e th e s e c a l a b a s h e s a r e s t a c k e d fr o m large to sm all,
th ey (a n d th e rope th a t h o ld s th e m ) form a tr ia n g le — th u s th e trian g u lar
•v
d e c o r a t i o n s also r e p r e s e n t s c a l in g c irc le s , j u s t in a m o r e a b s t r a c t way. T h e s m a l l ­
e s t c o n t a i n e r in a w o m a n ’s z a la n g a is t h e /cumpio, w h i c h is a s h r i n e for h e r soul.
W h e n s h e dies, th e zalanga, alo n g w i t h h e r p o ts , is sm a sh e d , a n d h e r soul is released
t o e t e r n i t y . T h e e t e r n i t y c o n c e p t , ' a s s o c i a t e d w i t h w e l l - b e i n g , is s y m b o li c a ll y
FIGURE 2 . 6
N a n k a n i home
(a) D r a w i n g o f a N a n k a n i h o m e , ( b ) T h e w o m a n ’s m a i n r o o m (dego)
in s i d e t h e N a n k a n i h o m e , ( c ) A s c a l i n g s t a c k o f p o t s in d i e f i r e p l a c e ,
(d) T h e ?a!an ga .
(a, Bonrd ier a n d T rinh 1 5 8 5 ; co urte sy o f the authors; b - d , p/totos fro m
B ourdier a n d T rin h 19 8 5 , by permission o f the a u th o rs .)
33
Introduction
34
r e p r e s e n t e d by t h e c o il s o f a s e r p e n t o f i n f i n i t e l e n g t h , s c u l p t e d i n t o t h e w alls
o f th e s e hom es.
F r o m t h e 2 0 - m e t e r d i a m e t e r o f t h e b u i l d i n g c o m p l e x t o t h e 0 . 2 - r f te t e r
k u m p i o — a n d n o t s im p ly a t o n e o r t w o le v e ls in b e t w e e n , b u t w i t h d o z e n s o f selfs i m i l a r s c a l in g s — t h e N a n k a n i f r a c t a l s p a n s t h r e e o r d e r s o f m a g n i t u d e , w h i c h is
c o m p a ra b le to th e re s o lu tio n o f m o s t c o m p u te r screens. M o re o v e r, th ese scalin g
circles a re far from u n c o n s c i o u s a c c i d e n t : as in sev eral o t h e r a r c h i t e c t u r e s w e h a v e
e x a m i n e d , t h e y h a v e m a d e c o n s c i o u s u s e o f t h e s c a l in g in t h e i r s o c i a l s y m b o l ­
ism. In th is case, t h e m o s t p r o m i n e n t s y m b o li s m is t h a t o f b i r t h i n g . W h e n a c h il d
is b o r n , fo r e x a m p l e , i t m u s t r e m a i n i n t h e i n n e r m o s t e n c l o s u r e o f t h e w o m e n ’s
r d e g o u n t i l it c a n c ra w l o u t by itself. E a c h s u c c e s siv e e n t r a n c e is— s p a t ia l ly as w e ll
as so c ia lly — a r i t e o f p assag e, s t a r t i n g w i t h t h e b io l o g i c a l e n t r a n c e o f t h e c h i l d
f r o m t h e w o m b . I t le a v e s e a c h o f t h e s e n e s t e d c h a m b e r s as t h e n e x t i t e r a t i o n in
life ’s sta g e s is b o r n . T h e z a la n g a m o d e l s t h e e n t i r e s t r u c t u r e i n m i n i a t u r e , a n d its
d e s t r u c t i o n in t h e e v e n t o f d e a t h m a p s t h e j o u r n e y i n re v e r s e : f r o m t h e c ir c le s
o f t h e la r g e s t c a l a b a s h to t h e t i n y k u m p i o h o l d i n g t h e s o u l — f r o m m a t u r e a d u l t
t o t h e e t e r n a l r e a l m o f a n c e s t o r s w h o d w e ll i n " t h e e a r t h ’s w o m b . ” T h e r e is a
c o n s c i o u s s c h e m e to t h e s c a l i n g c i r c l e s o f t h e N a n k a n i : it is a r e c u r s i o n w h i c h
b o t t o m s - o u t a t in f in ity .
B r a n c h in g fr a c ta ls
W h i l e A f r i c a n c i r c u l a r b u i l d i n g s a r e ty p i c a ll y a r r a n g e d in c i r c u l a r c l u s t e r s , t h e
p a t h s t h a t le a d t h r o u g h t h e s e s e t t l e m e n t s a r e ty p i c a ll y n o t c i r c u la r . L i k e t h e
b r o n c h i a l passag es t h a t o x y g e n a t e t h e f o u n d a l v e o l i of t h e lu n g s , t h e r o u t e s t h a t
n o u r i s h c i r c u l a r s e t t l e m e n t s o f t e n t a k e a b r a n c h i n g f o r m (e .g ., figure 2 . 7 ) . B u t
d esp ite m y u n a v o id a b ly o rg a n ic ist m e ta p h o r, th e s e c a n n o t be sim p ly re d u c e d to
u n c o n s c i o u s tr a c e s o f m i n i m u m e ff o rt. F o r o n e t h i n g , c o n s c i o u s d e s i g n c r i t e r i a
a r e e v i d e n t in c o m m u n i t i e s in w h i c h t h e r e is a n a r c h i t e c t u r a l t r a n s i t i o n f r o m c i r ­
c u l a r t o r e c t a n g u l a r b u i l d in g s , s i n c e t h e y c a n c h o o s e t o e i t h e r m a i n t a i n o r e ra s e
t h e b r a n c h i n g fo rm s.
D iscussion c o n c e r n i n g s u c h d ecisio ns are a p p a r e n t in t h e s e t t l e m e n t o f B anyo,
C a m e ro o n , w h e re th e tra n s itio n h a s a lo n g h isto ry (H u ra u lt 1975)- 1 fo u n d th a t
few c i r c u l a r b u i l d i n g s w e r e le f t, b u t t h o s e t h a t w e r e s t il l i n t a c t s e r v e d as a n
e m b o d i m e n t o f c u l t u r a l m e m o r y . T h i s ro le w a s h o n o r e d i n t h e c a s e o f t h e c h i e f ’s
c o m p l e x a n d e x p l o i t e d in t h e c a s e o f a b l a c k s m i t h ’s s h o p , w h i c h w a s t h e s i t e o f
o c c a s i o n a l t o u r i s t v is its. A f t e r p a s s i n g a p p r o v a l by t h e g o v e r n m e n t o f f ic ia ls
a n d t h e s u l t a n , 1 w as g r e e t e d b y t h e o f f i c i a l c i t y s u r v e y o r , w h o — c o n s i d e r i n g
t h e fa c t t h a t h is r a is o n d ’e t r e w as E u c l i d e a n i z i n g t h e s t r e e t s — s h o w e d s u r p r is in g
Fractals in A fr ic a n se ttle m e n t architecture
35
F I G U R E 2.7
B r a n c h in g p a th s in a S e n e g a le s e s e t t l e m e n t
{a) Aerial p h o to o f a traditional s e ttle m e n t in n o rth e a st Senegal. T h e space betw een enclosure
walls, serving as roads an d footpaths, creates a b ra n c h in g pattern , (b) A branch in g fractal c a n be
created by the background of a scaling set of circular shapes.
(a, courtesy Institut Geographique du Senegal.)
a p p r e c i a t i o n fo r m y p r o j e c t a n d h e l p e d m e l o c a t e t h e m o s t f r a c ta l a r e a o f t h e
c it y (fig. 2 .8 a ) . A t t h e u p p e r left o f t h e p h o t o w e se e a p o r t i o n o f t h e E u c l i d e a n
grid t h a t c o v e r s t h e re s t o f t h e city, b u t m o s t o f t h i s a re a is s till fr a c ta l. T h e l o c a ­
t i o n o f t h i s c a r e f u ll y m a i n t a i n e d b r a n c h i n g — f a n n i n g o u t fr o m a la r g e p laza
t h a t is b o r d e r e d by t h e p a l a c e o f t h e s u l t a n a n d t h e g r a n d m o s q u e — is n o
c o i n c i d e n c e . By m a r k i n g m y p o s i t i o n o n t h e a e r i a l p h o t o as 1 t r a v e l e d t h r o u g h
(fig. 2 . 8 b ) , I w as l a t e r a b le t o c r e a t e a m a p by d ig i ta l ly a l t e r i n g t h e p h o t o im ag e
(fig. 2 .8 c ) . T h i s p r o v i d e s a s t a r k o u t l i n e — l o o k i n g m u c h lik e t h e v e i n s i n a
le a f— o f t h e f r a c ta l s t r u c t u r e o f th i s t r a n s p o r t a t i o n n e t w o r k . I m ay h a v e p lu n g e d
t h r o u g h a w all o r t w o in c r e a t i n g th i s m a p , b u t it c e r t a i n l y u n d e r e s t i m a t e s t h e
fine b r a n c h i n g o f t h e f o o t p a t h s , as 1 d i d n o t a t t e m p t t o i n c l u d e t h e i r e x t e n s i o n s
i n t o p r i v a t e h o u s i n g e n c lo s u r e s .
H o w d o e s t h e c r e a t i o n o f th e s e s c a l in g b r a n c h e s i n t e r a c t w i t h t h e k i n d s o f
i t e r a t i v e c o n s t r u c t i o n a n d s o c ia l m e a n i n g we h a v e s e e n a s s o c ia te d w ich o t h e r
e x a m p l e s o f f r a c t a l a r c h i t e c t u r e ? A g o o d i l l u s t r a t i o n c a n b e f o u n d in t h e
Position 1— outside palace
FIGURE
Position 2 — road below mosque
2.8
B r a n c h i n g p a t h s in B a n y o
(a) Aerial phoco of the city of Banyo,
Cameroon, (b) Successive views of the
branching paths, as marked on the photo above.
The clay walls require their own roof, which
comes in both thatched and metal versions
along the walkway in the last photo, (c) Aerial
photo of Banyo with only public paths showing,
(a, c o u r t e s y N a t i o n a l I n s t i t u t e o f C a r t o g r a p h y ,
Cameroon.)
Position 3 —narrow walkway
FIGURE 2 . 9
S t r e e t s o f Cairo
(a) M a p o f s t r e e t s o f C a i r o , 1 8 9 8 . ( b ) F r a c t a l s i m u l a t i o n fo r C a i r o s t r e e t s , ( c ) E n l a r g e d v i e w
o f fourth iteration.
Introduction
38
b r a n c h i n g stre e ts o f N o r t h A f r i c a n c ities. F ig ure 2 .9 a s h o w s a m a p o f C a i r o , E gypt,
i n 1 8 9 8 . T h e m a p w as c r e a t e d b y a n i n s u r a n c e c o m p a n y , a n d I h a v e c o l o r e d t h e
s t r e e t s b l a c k t o m a k e t h e s c a l i n g b r a n c h e s m o r e a p p a r e n t . F ig u r e 2 . 9 b s h o w s its
c o m p u te r sim u la tio n . D ela v a l ( 1 9 7 4 ) h as d escrib ed th e m o r p h o g e n e s is o f S a h a ­
r a n c i t i e s in t e r m s o f s u c c e s s iv e a d d i t i o n s s i m i l a r t o t h e l i n e r e p l a c e m e n t in t h e
f r a c ta l a l g o r i t h m s w e h a v e u se d h e r e . T h e first “s e e d s h a p e " c o n s i s t s o f a m o s q u e
c o n n e c t e d by a w id e a v e n u e t o t h e m a r k e t p l a c e , a n d su c c e s s iv e i t e r a t i o n s o f c o n ­
s t r u c t i o n a d d su c c e s s iv e c o n t r a c t i o n s o f t h i s form .
S i n c e t h e s e f r a c ta l S a h a r a n s e t t l e m e n t a r c h i t e c t u r e s p r e d a t e Is la m (s e e
D e v is s e 1 9 8 3 ) , it w o u ld b e m i s l e a d i n g to s e e t h e m as a n e n t i r e l y M u s l i m i n v e n ­
t i o n ; b u t g i v e n t h e p r e v i o u s o b s e r v a t i o n s a b o u t t h e i n t r o d u c t i o n o f Is la m i c
a r c h i te c t u r e as a n i n t e rru p t io n o f c ir c u la r fractals in s u b - S a h a r a n A fric a , it is im p o r­
t a n t t o n o t e t h a t Is la m i c c u l t u r a l in f l u e n c e s h a v e m a d e s t r o n g c o n t r i b u t i o n s to
A f r i c a n f r a c ta ls as w ell. H e a v e r ( 1 9 8 7 ) d e s c r ib e s t h e " a r a b e s q u e " a r t i s t i c fo r m in
N o r t h A f r i c a n a r c h i t e c t u r e a n d d e s i g n in t e r m s t h a t r e c a l l s e v e r a l f r a c ta l c o n ­
c e p t s ( e .g ., “c y c l i c a l r h y t h m s ” p r o d u c i n g a n “ i n d e f i n i t e l y e x p a n d a b l e ” s t r u c ­
t u r e ) . H e d is c u s s e d t h e s e p a t t e r n s as v is u a l a n a l o g u e s t o c e r t a i n I s l a m i c s o c ia l
c o n c e p t s , a n d w e will e x a m i n e his idea s in g re a te r d e ta il in c h a p t e r 12 o f th is b oo k.
C o n c lu s io n
T h r o u g h o u t t h i s c h a p t e r , w e h a v e s e e n t h a t a w id e v a r i e t y o f A f r i c a n s e t t l e m e n t
a r c h i t e c t u r e s c a n be c h a r a c t e r i z e d as fra ctals. T h e i r p h y s i c a l c o n s t r u c t i o n m a k e s
u se o f s c a l in g a n d i t e r a t i o n , a n d t h e i r s e lf- s im i la r it y is c le a r ly e v i d e n t fr o m c o m ­
p a r i s o n to fractal-grap hic s im u la tio n s. C h a p t e r 3 wilt s h o w t h a t fracta l a r c h i t e c t u r e
is n o t sim p ly a ty pic al c h a r a c t e r i s t i c o f n o n - W e s t e r n s e t t l e m e n t s . T h i s a l o n e d o e s
n o t a ll o w us t o c o n c l u d e a n i n d i g e n o u s A f r i c a n k n o w l e d g e o f f r a c t a l g e o m e t r y ;
in f a c t , I w ill a rg u e in c h a p t e r 4 t h a t c e r t a i n f r a c ta l p a t t e r n s in A f r i c a n d e c o r a ­
ti v e a r t s are m e re ly t h e res u lt o f a n i n t u i t i v e e s t h e ti c . B u t as w e h a v e a lr e a d y s e e n ,
t h e fractals in A fr i c a n a rc h i te c t u r e a re m u c h m o r e t h a n t h a t . T h e i r d e sig n is lin k ed
t o c o n s c i o u s k n o w l e d g e s y s te m s t h a t s u g g e s t s o m e o f t h e b a s i c c o n c e p t s o f f r a c ­
ta l g e o m e t r y , a n d in la t e r c h a p t e r s w e w ill fin d m o r e e x p l i c i t e x p r e s s i o n s o f t h i s
i n d i g e n o u s m a t h e m a t i c s in a s t o n i s h i n g v a r i e t y a n d fo r m .
CHAPTER
3
-Fractals-------------------------------------------ire
-crosS'Cultural—c o m p a r is o n -
T h e f r a c t a l s e t t l e m e n t p a t t e r n s o f ' A f r i c a s t a n d in s h a r p c o n t r a s t t o t h e C a r te - ,
s i a n g rid s o f E u r o - A m e r i c a n s e t t l e m e n t s . W h y t h e d if f e r e n c e ? O n e e x p l a n a t i o n
c o u ld b e t h e d if f e r e n c e in so c ia l s t r u c t u r e . E u r o - A m e r i c a n c u lt u r e s are o rg a n iz e d
by w h a t a n t h r o p o l o g i s t s w o u ld c a ll a “s t a t e s o c i e t y . ” T h i s i n c l u d e s n o t j u s t t h e
m o d e r n n a t i o n - s t a t e , b u t re f e rs m o r e g e n e r a l l y t o a n y s o c i e t y w i t h a la r g e
p o l i t i c a l h i e r a r c h y , l a b o r s p e c i a l i z a t i o n , a n d c o h e s i v e , f o r m a l c o n t r o l s — w h a t is
s o m e t i m e s c a ll e d “ t o p - d o w n ” o r g a n i z a t i o n . P r e c o l o n i a l A f r i c a n c u lt u r e s in c l u d e d
m a n y s t a t e s o c i e t i e s , as w e ll as a n e n o r m o u s n u m b e r o f s m a ll e r , d e c e n t r a l i z e d
so c ia l g ro u p s , w i t h l i t t l e p o l i t i c a l h i e r a r c h y — t h a t is, s o c i e ti e s t h a t a re o rg a n iz e d
“b o t t o m - u p ” r a t h e r t h a n “ t o p - d o w n .” 1 B u t if f r a c t a l a r c h i t e c t u r e is s i m p l y t h e
a u t o m a t i c re su lt o f a n o n s t a t e social o r g a n iz a tio n , t h e n we sh o u ld see fractal s e t t l e ­
m e n t p a t t e r n s in t h e i n d i g e n o u s s o c ie tie s o f m a n y p a rts o f t h e w orld. I n this c h a p ­
t e r w e w ill e x a m i n e t h e s e t t l e m e n t p a t t e r n s f o u n d in t h e i n d i g e n o u s s o c i e t i e s
o f t h e A m e r i c a s a n d t h e S o u t h P a c if ic , b u t o u r s e a r c h w ill t u r n u p v e ry few f r a c ­
tals. R a t h e r t h a n d i v i d i n g t h e w o r l d b e t w e e n a E u c l i d e a n W e s t a n d f r a c t a l
n o n - W e s t , w e w ill f i n d t h a t e a c h s o c i e t y m a k e s use o f its p a r t i c u l a r d e s i g n
t h e m e s in o rg a n i z in g its b u il t e n v i r o n m e n t . A f r i c a n a r c h i t e c t u r e t e n d s t o b e frac­
tal b e c a u s e t h a t is a p r o m i n e n t d e s i g n t h e m e in A f r i c a n c u l t u r e . I n fact, th is c u l ­
tu r a l s p e c i f i c i t y o f d e s i g n t h e m e s is t r u e n o t o n l y fo r a r c h i t e c t u r e , b u t fo r m a n y
introduction
40
o t h e r ty p e s o f m a t e r i a l d e s i g n a n d c u l t u r a l p r a c t i c e s as w e ll. W e w ill b e g i n o u r
su rv e y w i t h a b r i e f lo o k a t t h e d e s i g n t h e m e s in N a t i v e A m e r i c a n so c ie tie s, w h i c h
i n c l u d e d b o t h h i e r a r c h i c a l s t a t e e m p i r e s as w e l l as s m a l l e r , d e c e n t r a l i z e d tr ib a l
cu ltu res.
'N ative A m e r i c a n d e sig n
T h e A n c e s t r a l P u e b l o s o c i e t y d w e l l e d in w h a t is n o w t h e s o u t h w e s t e r n U n i t e d
S tates aro u n d 1 1 0 0
c .e
.
A e r i a l p h o t o s o f t h e s e sites (fig. 3 .1 ) a re s o m e o f t h e m o s t
f a m o u s e x a m p l e s o f N a t i v e A m e r i c a n s e t t l e m e n t s . B u t as we c a n se e fr o m th i s
v a n t a g e p o i n t , t h e a r c h i t e c t u r e is p r i m a r i ly c h a r a c t e r i z e d by a n e n o r m o u s circu la r
fo r m c r e a t e d fr o m s m a lle r re c ta n g u la r c o m p o n e n t s — c e r t a i n l y n o t t h e s a m e shapea t t w o d i f f e r e n t scales. T h i s j u x t a p o s i t i o n o f t h e c ir c le a n d t h e q u a d r i l a t e r a l ( r e c ­
tan gle or cross-shaped) form is n o t a c o in c i d e n c e ; th e tw o forms are t h e m o s t im p o r­
t a n t d e s i g n t h e m e s in t h e m a t e r i a l c u l t u r e o f m a n y N a t i v e A m e r i c a n s o c i e ti e s ,
in c lu d in g b o th N o r th a n d S o u th c o n tin e n ts A s far as a r c h i t e c t u r e is c o n c e r n e d , t h e r e are n o e x a m p l e s o f t h e n o n ! in e a r
s c a l i n g w e saw in A f r i c a . T h e o n l y N a t i v e A m e r i c a n a r c h i t e c t u r e s t h a t c o m e
c lo s e a r e a few i n s t a n c e s o f l i n e a r c o n c e n t r i c fig ures (fig. 3 . 2 a ) . S h a p e s a p p r o x ­
i m a t i n g c o n c e n t r i c c i r c l e s c a n b e s e e n i n t;he P o v e r t y P o i n t c o m p l e x i n n o r t h -
h
a
FIGURE 3 .I
E u c l i d e a n g e o m e t r y in N a t i v e A m e r i c a n a r c h i t e c t u r e
(a) Aerial photo of Bandelter, one of the Ancestral Pueblo settlements (starting around 1100 c .e .)
in norluwestern New Mexico, (b) Aerial photo of Pueblo Bonito, another Ancestral Pueblo .
settlement (starting around 950 C. E. ) . N o te t h a t they are mostly rectangular at the smallest scale
and circular at the largest scale.
( a , p h o io by T o m B aiter; b , photo by G e o rg G e r s te r .)
F ra c ta l m cross-cultural comparison
41
e r n L o u i s i a n a , f o r e x a m p l e , a n d t h e r e w e r e c o n c e n t r i c c i r c l e s o f t e p e e s in t h e
C h e y e n n e c a m p s . T h e s t e p - p y r a m i d s o f M e s o a m e r i c a l o o k li k e c o n c e n t r i c
s q u a r e s w h e n v i e w e d fr o m a b o v e . B u t l i n e a r c o n c e n t r i c fig ures a r e n o t f r a c ta ls.
F irst, t h e s e a re l i n e a r layers: t h e d i s t a n c e b e t w e e n li n e s is alw a y s t h e s a m e , a n d
t h u s t h e n u m b e r o f c o n c e n t r i c c ircles w i t h i n t h e largest circle is fin ite. T h e n o n ­
l i n e a r s c a l i n g o f f r a c t a l s r e q u i r e s a n e v e r - c h a n g i n g d i s t a n c e b e t w e e n lin e s ,
figure
3.2
L i n e a r c o n c e n tr ic f o r m s in N a tiv e A m e r i c a n a r c h i t e c t u r e
(a) N a t i v e A m e r i c a n a r c h i t e c t u r e is t y p i c a l l y b a s e d o n q u a d r i l a t e r a l grid s o r a c o m b i n a t i o n o f
circ ula r a n d g ri d fo r m s . T h e o n l y e x a m p l e s o f s c a l i n g s h a p e s a re t h e s e li n e a r c o n c e n t r i c fo rm s. In
the P o v e r t y P o i n t c o m p l e x , fo r e x a m p l e , c o n c e n t r i c c i r c l e s w e r e u se d, a n d c o n c e n t r i c sq u a r e s c a n be
seen if w e l o o k a t t h e M e x i c a n s t e p p y r a m i d s f r o m a b o v e . T h e s e fo r m s a re b e t t e r c h a r a c t e r i z e d as
E u c li d e a n t h a n f r a c t a l fo r t w o re a so n s: ( b ) First , t h e y a re lin e a r . H e r e is a n e x a m p l e o f a n o n l i n e a r
c o n c e n t r i c c i r c l e . W h i l e t h e l i n e a r v e r s i o n m u s t h a v e a f in it e n u m b e r o f c i r c l e s , t h is figure c o u l d
h a v e a n i n f i n i t e n u m b e r a n d st il l fit in t h e s a m e b o u n d a r y , ( c ) S e c o n d , t h e y o n l y s c a l e w i t h r e s p e c t
to o n e p o i n t ( t h e c e n t e r ) . H e r e is a n e x a m p l e o f c i r c l e s w i t h m o r e g l o b a l s c a l i n g s y m m e t r y .
/rUroduction
w h i c h m e a n s t h e r e c a n b e a n i n f i n i t e n u m b e r in a f i n i t e s p a c e (fig. 3 . 2 b ) . S e c ­
o n d , e v e n n o n lin e a r c o n c e n t r ic c ircles are o n ly self-sim ilar w ith re sp e c t to a
sin g le locus ( t h e c e n t e r p o i n t ) , r a t h e r t h a n h a v in g t h e g lo b a l se lf-sitn ilarity o f
f r a c t a l s (fig. 3 . 2 c ) .
T h e i m p o r t a n c e o f t h e c i r c l e is d e t a i l e d i n a f a m o u s p a s s a g e b y B l a c k E l k
( 1 9 6 1 ) , in w h i c h h e e x p l a i n s t h a t “e v e r y t h i n g a n I n d i a n d o e s is in a c i r c l e , a n d
t h a t is b e c a u s e t h e P o w e r o f t h e W o r l d alw a y s w o rk s in c ir c le s , a n d e v e r y t h i n g
trie s t o be r o u n d . ” B u t h e g o e s o n to n o t e t h a t h is p e o p l e t h o u g h t o f t h e i r w o r ld
as “ t h e circle o f t h e four q u a r t e r s ." A s im i la r c o m b i n a t i o n o f t h e c ir c le a n d q u a d r i ­
la t e r a l fo r m c a n b e s e e n m a n y N a t i v e A m e r i c a n m y t h s a n d a rtifa c ts; it is n o t t h e i r
o n l y d e s i g n t h e m e , b u t i t c a n b e f o u n d i n a s u r p r is in g n u m b e r o f d i f f e r e n t s o c i ­
eties. B u rla n d (1 9 6 5 ) , for e x a m p le , s h ow s a c e re m o n ia l r a ttle c o n s is tin g o f a w o o d e n
h o o p w i t h a cross i n s i d e fr o m s o u t h e r n A l a s k a , a N a v a j o s a n d p a i n t i n g s h o w i n g
f o u r e q u i d i s t a n t sta lk s o f c o r n g r o w in g fro m a c i r c u la r lak e, a n d a P a w n e e bu ffaloh i d e d r u m w i t h f o u r a r r o w s e m a n a t i n g f r o m its c i r c u l a r c e n t e r . N a b o k o v a n d
E a s t o n ( 1 9 8 9 ) d e s c r i b e t h e c u l t u r a l s y m b o l i s m o f t h e t e p e e in t e r m s o f its c o m ­
b i n a t i o n o f c i r c u l a r h i d e e x t e r i o r a n d t h e fo u r m a i n s t r u t s o f t h e i n t e r i o r w o o d
s u p p o r t s . W a t e r s ( 1 9 6 3 ) p r o v i d e s a n e x t e n s i v e i l l u s t r a t i o n o f t h e c u l t u r a l s ig ­
n if ic a n c e o f c o m b i n i n g t h e c i r c u la r a n d cross fo rm in his c o m m e n t a r y o n t h e H o p i
c r e a tio n m yth.
T h e fo u rfo ld s y m m e t r y o f t h e q u a d r i l a t e r a l fo rm h a s le a d t o s o m e s o p h i s ­
t i c a t e d c o n c e p t u a l s t r u c t u r e s i n N a t i v e A m e r i c a n k n o w l e d g e s y s te m s . I n N a v a j o
s a n d p a i n t i n g , for e x a m p l e , t h e c r u c i f o r m s h a p e r e p r e s e n t s t h e “ fo u r d i r e c t i o n s ”
c o n c e p t , s i m i l a r to t h e C a r t e s i a n c o o r d i n a t e s y s te m . W h i l e o r d e r l y a n d c o n s i s ­
t e n t , it is by n o m e a n s s i m p l e (s e e W i t h e r s p o o n a n d P e t e r s o n 1 9 9 5 ) . T h e fo u r
N a V a j o d i r e c t i o n s a r e a ls o a s s o c i a t e d w i t h c o r r e s p o n d i n g s u n p o s i t i o n s ( d a w n ,
day , e v e n i n g , n i g h t ) , y e a r l y s e a s o n s ( s p r i n g , s u m m e r , fa ll, w i n t e r ) , p r i n c i p a l
c o lo r s ( w h i t e , b l u e , y e llo w , b l a c k ) , a n d o t h e r q u a d r i l a t e r a l d i v i s i o n s ( b o t a n i c a l
c a t e g o r i e s , p a r t i t i o n s o f t h e life c y c le , e t c . ) . T h e s e a r e f u r t h e r b r o k e n i n t o i n t e r ­
s e c t in g b ip o la rities (e.g., t h e e a s t /w e s t s u n p a t h is b r o k e n by t h e n o r t h / s o u t h d i r e c ­
t i o n s ) . C o m b i n e d w i t h c i r c u l a r c u r v e s ( u s u a lly r e p r e s e n t i n g o r g a n i c s h a p e s a n d
p ro c e s s e s ), t h e r e s u lti n g s c h e m a a r e Tich c u l t u r a l re s o u rc e s f o r i n d i g e n o u s m a t h e ­
m a t ic s (see M o o r e 1994)- B ut, e x c e p t for m i n o r re p e t it io n s ( l ik e t h e s m a ll c ir c u la r
k i v a s in t h e C h a c o c a n y o n s i t e o f fig. 3 . 1 ) t h e r e is n o t h i n g p a r t i c u l a r l y f r a c ta l
a b o u t th e s e q u a d rila te ra l designs.
M a n y M e s o a m e r i c a n . c i t i e s , s u c h a s t h e M a y a n s ’ T e o t i h u a c a n , t h e A z te c 's
T e n o c h t i t l a n , a n d t h e T o l t e c ’s T u l a , e m b e d d e d a w e a l t h o f a s t r o n o m i c a l k n o w l ­
e d g e in t h e i r r e c t a n g u l a r l a y o u t s , a l i g n i n g t h e i r s t r e e t s a n d b u i l d i n g s w i t h h e a v ­
e n l y o b j e c t s a n d e v e n t s ( A v e n i 1 9 8 0 ) . J. T h o m p s o n ( 1 9 7 0 ) a n d K l e i n ( 1 9 8 2 )
Fractals in cross-cultural com parison
d e s c r ib e t h e q u a d r i l a t e r a l figure as a n u n d e r l y i n g t h e m e in M e s o a m e r i c a n g e o ­
m e t r i c t h i n k i n g , fr o m s m a l l - s c a l e m a t e r i a ! c o n s t r u c t i o n t e c h n i q u e s s u c h as
w e a v i n g , t o t h e h e a v e n l y c o s m o l o g y o f t h e f o u r . s e r p e n t s . R o g e l i o D iaz, o f th e
.M a th e m a tic s M u seu m a t,th e U n iv e rs ity -o f Q u e re ta r o , p o in ts o u t th a t th e skin
p a t t e r n s o f t h e d i a m o n d b a c k r a t t l e s n a k e w e r e u se d by t h e M a y a n s t o s y m b o liz e
th i s c o n c e p t (fig. 3 . 3 a ) .
C o m p a r i n g t h e M a y a n s n a k e p a t t e r n w i t h a n A f r i c a n w e a v in g b a s e d o n th e
c o b r a s k i n p a t t e r n (fig. 3 . 3 b ) , w e c a n see h o w g e o m e t r i c m o d e l i n g o f s i m i l a r n a t ­
u ra l p h e n o m e n a i n t h e s e tw o c u l t u r e s re s u lts in v e ry d i f f e r e n t r e p r e s e n t a t i o n s .
T h e N a t i v e A m e r i c a n e x a m p l e e m p h a s i z e s t h e E u c l i d e a n s y m m e t r y w ithin one
size fr a m e ( “size f r a m e ” b e c a u s e t h e t e r m “s c a l e ” is c o n f u s i n g in t h e c o n t e x t o f
s n a k e s k i n ) . T h i s M a y a n p a t t e r n is c o m p o s e d o f four s h a p e s o f t h e s a m e size, a
fo u r fo ld sy m m e try . B u t t h e A f r i c a n e x a m p l e e m p h a s iz e s f r a c ta l s y m m e t ry , w h i c h
is n o t a b o u t s i m i l a r i t y b e t w e e n r i g h t / l e f t o r u p / d o w n , b u t r a t h e r s i m i l a r i t y
betw een d iffe r e n t size fr a m e s . T h e A f r i c a n s n a k e p a t t e r n s h o w s d i a m o n d s w i t h i n
d i a m o n d s w i t h i n d i a m o n d s . N e i t h e r d e s i g n is n e c e s s a r il y m o r e a c c u r a t e : c o b r a
s k i n d o e s i n d e e d e x h i b i t a f r a c ta l p a t t e r n — t h e s n a k e ’s " h o o d , ” its t w i n w h i t e
p a t c h e s , a n d t h e i n d i v i d u a l sc a le s t h e m s e l v e s a re all d i a m o n d s h a p e d — a n d yet
s n a k e s k i n p a t t e r n s ( t h a n k s t o t h e a r r a n g e m e n t o f t h e s c a le s ) a re also c h a r a c ­
te r i s t i c a l l y in d i a g o n a l ro w s, so t h e y a re a c c u r a t e l y m o d e l e d as E u c l i d e a n s t r u c ­
t u r e s as w e ll. E a c h c u l t u r e c h o o s e s t o e m p h a s i z e t h e c h a r a c t e r i s t i c s t h a t b e s t fit
its d e s i g n t h e m e .
T h e r e a re a few c a s e s in w h i c h N a t i v e A m e r i c a n s h a v e u s e d s c a l i n g g e o ­
m e t r i e s in a r t i s t i c d e s ig n s . S e v e r a l o f t h e s e w e r e n o t , h o w e v e r , p a r t o f t h e t r a ­
d i t i o n a l r e p e r t o i r e . ^ N a v a j o b l a n k e t s , f o r e x a m p l e , w e r e o r i g i n a l l y q u i t e lin e a r;
-it w a s -o n l y c n - e x a m i n i n g P e r s i a n - r u g a ::t-hat N a v a j o w e a v e r s b e g a n t o u se m o r e
s c a l i n g s t y le s o f d e s i g n ( a n d e v e n t h e n t h e d e s i g n s w e r e m u c h m o r e E u c l i d e a n
t h a n t h e P e r s i a n o r i g in a l s ; see K e n t 1 9 8 5 ) . T h e P u e b l o " s t o r y t e l l e r ” figures h a v e
s o m e s c a l i n g p r o p e r t i e s , b u t th e y a r e o f r e c e n t ( 1 9 6 0 s ) o r i g i n . P o t t e r y a n d c a l a ­
b a s h ( c a r v e d g o u r d ) a r t i s a n s in A f r i c a o f t e n c r e a t e s c a l i n g by a l l o w i n g t h e
d e s i g n a d a p t i v e l y t o c h a n g e p r o p o r t i o n a c c o r d i n g to t h e t h r e e - d i m e n s i o n a l fo rm
o n w h i c h it is i n s c r i b e d (s e e “a d a p t i v e s c a l i n g ” in c h a p t e r 6 ) , b u t th i s w a s q u i t e
ra r e i n N a t i v e A m e r i c a n p o t t e r y u n t i l t h e 19 60s.
F in a lly , t h e r e a r e t h r e e N a t i v e A m e r i c a n d e s i g n s t h a t a re b o t h i n d i g e n o u s •
a n d f r a c ta l . T h e b e s t c a s e is t h e a b s t r a c t fig u ra tiv e a r t o f t h e H a i d a , K w a k i u t l ,
T l i n g u t , a n d o t h e r s in t h e P acific N o r t h w e s t ( H o l m 1 9 6 5 ). T h e figures, p rim a rily
c a r v i n g s , h a v e t h e k i n d o f g lo b a l , n o n l i n e a r s e lf- s im i la r it y n e c e s s a r y t o q ualify
as f r a c ta ls a n d c le a r ly e x h i b i t r e c u r s i v e s c a l i n g o f u p t o t h r e e o r fo u r it e r a t i o n s .
T h e y a ls o m a k e u se o f a d a p t i v e s c a l i n g , as il l u s t r a t e d by t h e s h r i n k i n g se rie s o f
FIGURE
3.3
S n a k e s h i n m o d e ls in N a t i v e A m e r i c a n a n d A f r i c a n c u l t u r e s
U ) R o g e lio D iaz o f t h e M a t h e m a t ic s M u se u m a t th e U n iv e r s ity o f Q u e r e t a r o sh o w s h o w the sk in
p a tterns o f the d ia m o n d b a c k ra ttlesn ak e w ere used by the M a y a n s to sy m bo lize a c o s m o lo g y based
on quad rilateral structure, (b) T h e M a n d ia c k w e avers o f G u in e a - B is s a u h a v e also cre a ted an
a b stra c t d esign based on a s n a k e s k in p a tte r n , h u t c h o s e to e m p h a siz e t h e fractal c h a ra c te ristic s .
Fractals m cross-cwltural comparison
figures o n t h e d i m i n i s h i n g h a n d l e s o f s o u p ladles. R e s e a r c h e r s s i n c e A d a m s
(1 9 3 6 ) h a v e p o i n t e d to t h e s im ila rity w ith e arly C h i n e s e art, w h i c h also h a s s o m e
b e a u ti f u l e x a m p l e s o f s c a l i n g fo r m , a n d its s t y le o f c u r v a t u r e a n d b i l a t e r a l s y m ­
m e try c o u ld i n d e e d b e c u l t u r a l l y rie d '’to t h e s e N e w W o r l d d e s i g n s t h r o u g h a n
a n c i e n t c o m m o n o r i g in . H o w e v e r , 1 d o u b t t h a t is t h e c a s e for t h e s c a l in g c h a r - .
a c t e r is tic s . T h e P a c i f i c N o r t h w e s t a r t a p p e a r s t o h a v e d e v e l o p e d its s c a l in g
s t r u c tu r e as t h e r e s u lt o f c o m p e t i t i o n b e t w e e n a r t is a n s fo r in c r e a s i n g ly e l a b o r a t e
c a rv in g s (F a ris 1 9 8 3 ) . A l t h o u g h s o m e r e s e a r c h e r s h a v e a t t r i b u t e d t h e c o m p e t i ­
tio n pressure to E u ro p e a n trad in g influences, th e d e v e l o p m e n t o f t h e scaling designs
was c le a rly a n i n t e r n a l i n v e n t i o n .
T h e o t h e r t w o t r a d i t i o n a l N a t i v e A m e r i c a n d e s i g n s d o n o t q ua lify as fr a c ­
tals q u i t e as w e ll. O n e i n v o l v e s t h e s a w - t o o t h p a t t e r n f o u n d in s e v e r a l b a s k e t
a n d w e a v i n g d e s ig n s . W h e n t w o s a w - t o o t h row s i n t e r s e c t a t a n a n g le , t h e y c r e ­
a te a t r ia n g l e m a d e fr o m t r i a n g u l a r edges. B u t th e s e ty p ic a lly h a v e o n ly tw o i t e r ­
a t i o n s o f s c a l e , a n d t h e r e is n o i n d i c a t i o n in t h e e t h n o g r a p h i c l i t e r a t u r e t h a t
th e y are s e m a n t i c a l l y tie d to ideas o f r e c u r s io n o r s c a l in g (see T h o m a s a n d S lo c k ish 1 9 8 2 , 1 8 ). T h e o t h e r is a n a r r a n g e m e n t o f s p i r a l a r m s o f t e n f o u n d o n
c o il e d b a s k e t s . A g a i n , t h i s is s e l f - s i m i l a r o n l y w i t h r e s p e c t to t h e c e n t e r p o i n t ,
b u t t h e r e a r e s o m e n o n l i n e a r s c a l i n g v e r s i o n s ( t h a t is, d e s i g n s t h a t r a p i d l y g e t
s m a ll e r as y o u m o v e f r o m b a s k e t e d g e t o c e n t e r ) . H o w e v e r , t h e s e d e s i g n s
g e n e r a l ly a p p e a r t o b e a f u s io n b e t w e e n t h e c i r c u l a r fo r m o f t h e b a s k e t a n d t h e
cru cifo rm s h a p e o f th e arm s: a g a in m o r e a c o m b i n a ti o n o f tw o E u c lid e a n
shapes th a n a fractal.
O n e o f t h e m o s t c o m m o n e x a m p l e s o f t h i s f u s i o n b e t w e e n t h e c ir c le a n d
t h e cro ss is t h e “ b if o ld r o t a t i o n ” p a t t e r n in w h i c h t h e a r m s c u r v e in o p p o s i t e
d i r e c t i o n s , as s h o w n in fig u re 3 .4 a . F ig u r e 3 . 4 b s h o w s t h e fig u re o f 2 b a t fromM i m b r e s p o t t e r y w i t h a m o r e c o m p l e x v e r s i o n o f t h e b if o ld r o t a t i o n . E u c l i d e a n
s y m m e t r y h a s b e e n e m p h a s i z e d in t h i s figure; for e x a m p l e , t h e e a rs a n d m o u t h
o f th e b a t h a v e b e e n m a d e to lo o k s i m i la r to in c r e a s e t h e b i l a te r a l sy m m e try , a n d
th e belly is d r a w n as a r e c t a n g l e . F ig u re 3 . 4 c s h o w s t h e figure o f a b a t fr o m a n
A fric a n d esign ; it is a zigzag s h a p e t h a t e x p a n d s in w i d t h fro m t o p to b o t t o m , r e p ­
r e s e n tin g t h e w i n g o f t h e b a t. H e r e w e see n e g l e c t o f t h e b i l a t e r a l s y m m e t r y o f
t h e bat, a n d a n e m p h a s i s o n t h e s c a l in g fo lds o f a s i n g le w in g. A g a i n , t h e N a t i v e
A m e r i c a n r e p r e s e n t a t i o n m a k e s use o f its q u a d r i l a t e r a l / c i r c u l a r d esig n t h e m e , ju st
as th e A f r i c a n r e p r e s e n t a t i o n o f t h e b a t e m p h a s iz e s s c a l i n g d e s i g n .
T h e r e is p l e n t y o f c o m p l e x i t y a n d s o p h i s t i c a t i o n in t h e in d i g e n o u s g e o m ­
etry a n d n u m e r ic - s y s te m s o f t h e A m e r i c a s (s e e A s c h e r 1 9 91 , 8 7 - 9 4 ; C lo s s 1986;
Eglash i 9 9 8 b ) , b u t w i t h t h e im p re s s iv e e x c e p t i o n o f t h e Pacific N o r t h w e s t c a r v ­
ings, fra c ta ls a r e a l m o s t e n t i r e l y a b s e n t in N a t i v e A m e r i c a n d e sig n s.
A rk a n s a s p o ttery
(a)
Pim a b a sk e t
S o u th w e ste rn pottery m otif
T h e c i r c u l a r a n d q u a d r i l a t e r a l f o r m s w e r e o f t e n c o m b i n e d i n N a t i v e A m e r i c a n d e s i g n s as a
fo u rfo ld o r b ifold rota tio n .
(b)
T h i s i m a g e o f a b a t , fr o m a M i t n b r e s p o t t e r y i n S o u t h w e s t e r n
N a t i v e A m e r ic a n tra dition , sh o w s an em p ha sis o n c ircu la r and
q u a d rila te r a l fo r m . T h e ear a n d t h e m o u t h , fo r e x a m p le , are m ad e
t o l o o k s i m i l a r t o e m p h a s i z e b i l a t e r a l s y m m e t r y , a n d t h e b e l l y is
d r a w n as a r e c t a n g l e . It a l s o s h o w s t h e w i n g b o n e s a s a b i f o l d
ro ta tio n pattern.
(c)
T h i s A f r ic a n scu lp tu re o f a bat, from th e L eg a so c ie ty o f Zaire, pays
l i t t l e a t t e n t i o n t o t h e b i l a t e r a l s y m m e t r y o f t h e b a t ’s b o d y b u t g i v e s a n
em p h a sis o n th e s c a lin g sym m etry o f the w in g folds, s h o w n as an
e x p a n d i n g z ig z a g p a t t e r n .
FIGURE 3 . 4
T h e bifold r o ta ti o n in N a t i v e A m e r i c a n d e s ig n
( a : L e f t , fr o m Miles 19 6 3 . C e n t e r , fr om S o u t h w e s t I n d i a n C r a f t A r t s by C l m a Lee T a n n e r . C o p y r ig h t
1 9 6 8 b y the A r i z o n a B o a rd o f Regents. R e p r i n t e d b y permission 0/ the U n i v e r s i t y o f A r i z o n a Press.
Right, c o u r t e s y D o n C - r o w e - b , f r o m Z a s l o w 1 9 7 7 , c o u r t e s y o f the a u t h o r , c , cou rt esy o f D a n i e l D i e b u y c k . )
Fractals in cross-culcuTal comparison
D e s i g n s o f A s i a a n d t h e S o u t h P a c ific
S e v e r a l o f t h e S o u t h P a cific c u l t u r e s s h a r e a t r a d i t i o n o f d e c o r a t i v e c u r v e d a n d
s p ira l fo rm s, w h i c h in c e r t a i n M a o r i v e r s i o n s —- p a r tic u la r l y t h e i r r a f t e r a n d t a t • to o p a t t e r n s — w o u ld c e r t a i n l y c o u n t a s ’f r a c ta l (s e e H a m i l t o n 1 9 7 7 ). T h e s e are
s t r o n g l y su g g e s tiv e o f t h e c u r v a t u r e o f w a v e s a n d sw irling, w ater. C l a s s i c J a p a n ­
ese p a i n t i n g s o f w a t e r w a v e s w e r e a ls o p r e s e n t e d as fra c ta l p a t t e r n s in M a n d e l ­
b r o t 's ( 1 9 8 2 ) s e m i n a l t e x t ( p l a t e C 1 6 ) . T h e s e m a y h a v e s o m e h i s t o r i c r e l a t i o n
to sc a lin g p a t t e r n s in C h i n e s e a r t (see W a s h b u r n a n d C r o w e 1988, fig. 6 .9 ) , w h i c h
are b a s e d o n s w ir lin g fo r m s o f w a t e r a n d c lo u d s , a b s t r a c t e d as s p i r a l s c a l i n g '
s t r u c t u r e s . W h i l e b o t h t h e J a p a n e s e a n d C h i n e s e p a t t e r n s a re e x p l i c i t l y a s s o c i­
a te d w i t h a n effort t o i m i t a t e n a t u r e , t h e s e M a o r i d e sig n s are r e p o r t e d t o b e m o r e
a b o u t c u l t u r e — in p a r t ic u l a r , th e y e m p h a s iz e m i rro r -im a g e s y m m e t rie s , w h i c h a re
a s s o c i a t e d w i t h t h e i r c u l t u r a l t h e m e s o f c o m p l i m e n t a r i t y in s o c i a l r e l a t i o n s
( H a n s o n 1 9 8 3 ).
In a l m o s t all o t h e r in d i g e n o u s e x a m p l e s , h o w e v e r , t h e P a c if ic I s la n d e r p a t ­
t e r n s a r e q u i t e E u c l i d e a n . S e t t l e m e n t l a y o u t , f o r i n s t a n c e , is t y p i c a l l y in o n e
o r tw o ro w s o f r e c t a n g u l a r b u i l d i n g s n e a r t h e c o a s ts , w i t h c i r c u l a r a r r a n g e m e n t s
o f r e c t a n g l e s a ls o o c c u r r i n g i n l a n d (s e e F r a s e r 1 9 6 8 ) . T h e b u i l d i n g c o n s t r u c ­
t i o n is g e n e r a l l y b a s e d o n a c o m b i n a t i o n o f r e c t a n g u l a r g rid s w i t h t r i a n g u l a r
o r c u r v e d a r c h roofs. O c c a s i o n a l l y t h e s e t r i a n g u l a r fa c e s a re d e c o r a t e d w i t h t r i ­
angles, b u t o th e rw is e n o n s c a lin g d esig n s d o m i n a te b o th s tru c tu r a l a n d d e c o ­
r a t i v e patterns.-^
A g a i n , it is i m p o r t a n t to n o t e t h a t t h i s la c k o f fra c ta ls d o e s n o t im ply a lack
o f s o p h i s t i c a t i o n in t h e i r m a t h e m a t i c a l t h i n k i n g . F o r e x a m p l e , A s c h e r ( 1 9 9 1 )
h a s a n aly z ed s o m e o f t h e a l g o r i t h m i c p r o p e r tie s o f W a rlp ir i (P acific Is la n d e r) sa n d
d r a w in g s . S i m i l a r s t r u c t u r e s a re a ls o f o u n d ' i n " A f r i c a ; w h e r e th e y a re called"
lu so n a . Buc w h i l e t h e lu s o n a t e n d t o use s i m i l a r p a t t e r n s a t d i f f e r e n t sc a le s (as
w e w ill se e in c h a p t e r 5 ) , t h e W a r l p i r i d r a w i n g s t e n d t o use d i f f e r e n t p a t t e r n s at
d i f f e r e n t sc ales. A s c h e r c o n c l u d e s t h a t t h e W a r l p i r i m e t h o d o f c o m b i n i n g d if ­
f e r e n t g r a p h m o v e m e n t s is a n a l o g o u s to a lg e b r a ic c o m b i n a t i o n s , b u t t h e A f r i c a n
lu s o n a a re b e s t d e s c r i b e d as fractals.
C o m p l i c a t i n g m y c h a r a c t e r i z a t i o n o f t h e S o u t h P acific as d o m i n a t e d by
E u c l i d e a n p a t t e r n s is t h e e x t e n s i v e i n f l u e n c e o f I n d i a . I t is p e r h a p s n o c o i n c i ­
d e n c e c h a t th e t r ia n g l e of tr ia n g le s m e n t i o n e d a b o v e is m o s t c o m m o n in I n d o n e ­
sia. In a r c h i t e c t u r e , a f a m o u s e x c e p t i o n t o t h e g e n e r a l l y E u c l i d e a n fo r m is t h a t
o f B o rob ud ur, a te m p le o f I n d i a n re lig io u s o rig in in J a v a . A l t h o u g h n o r t h e r n I n d i a
t e n d s t o w a r d E u c l i d e a n a r c h i t e c t u r e , e x p l i c i t re c u r s iv e d e s ig n is s e e n in s e v e r a l
t e m p l e s in s o u t h e r n I n d i a — t h e K a n d a r y a M a h a d e o in K h a j u r a h o is o n e o f t h e
Introduction
48
c l e a r e s t e x a m p l e s — a n d is r e l a t e d to re c u r s iv e c o n c e p t s in re l ig i o u s c o sm o lo g y .
T h e s e s a m e a re a s in s o u t h e r n I n d i a a ls o h a v e a v e r s i o n o f t h e lu s o n a d r a w in g s ,
a n d m a n y o t h e r ex am p les o f fractal design. Interestingly, th e s e e x a m p le s from s o u t h ­
e r n I n d i a a r e t h e p r o d u c t s o f D r a v i d i a n c u l t u r e , w h i c h is s u s p e c t e d t o h a v e sig­
n i f i c a n t h i s t o r i c a l r o o t s in A f r i c a .
E u r o p e a n desig n s
M o s t t r a d i t i o n a l E u r o p e a n f r a c ta l d e sig n s, lik e t h o s e o f J a p a n a n d C h i n a , a r e d u e
t o i m i t a t i o n o f n a t u r e — a t o p i c we w ill t a k e u p i n t h e f o l lo w in g c h a p t e r . T h e r e
a r e a t le a s t tw o s t e l l a r e x c e p t i o n s , h o w e v e r , t h a t a r e w o r t h n o t i n g . O n e is t h e
s c a l in g i t e r a t i o n s o f t r ia n g l e s in t h e floor tile s o f t h e C h u r c h o f S a n t a M a r i a in
C o s t n e d i n R o m e (see p l a t e 5 .7 in W a s h b u r n a n d C r o w e 1 9 8 8 ) . I h a v e n o t b e e n
a b le to d e t e r m i n e a n y t h i n g a b o u t t h e i r c u l t u r a l o r i g in s , b u t t h e y a r e c le a rly
a r t is t ic i n v e n t i o n r a t h e r t h a n i m i t a t i o n o f s o m e n a t u r a l f o r m . T h e o t h e r c a n b e
f o u n d in c e r t a i n v a r i e ti e s o f C e l t i c i n t e r l a c e d esig n s. N o r d e n f a l k ( 1 9 7 7 ) su gg ests
t h a t th e s e a re h is to ric a lly r e l a te d to t h e sp iral d e sig n s o f p r e - C h r i s t i a n C e l t i c re li­
g i o n , w h e r e t h e y t r a c e t h e f lo w o f a v it a l life fo r c e . T h e y a r e g e o m e t r i c a l l y
classified as a n E u le r i a n p a t h , w h i c h is clo sely a s s o c ia te d w i t h m a t h e m a t i c a l k n o t
t h e o r y (cf. J o n e s 1 9 9 0 , 9 9 ) .
C o n c l u s io n
F ra c t a l s t r u c t u r e is by n o m e a n s u n i v e r s a l in t h e m a t e r i a l p a t t e r n s o f i n d i g e n o u s
so c ie tie s . I n N a t i v e A m e r i c a n d e sig n s, o n l y t h e P acific N o r t h w e s t p a t t e r n s s h o w
a s t r o n g f r a c ta l c h a r a c t e r i s t i c ; E u c l i d e a n s h a p e s o t h e r w i s e d o m i n a t e t h e a r t a n d
a r c h i t e c t u r e . E x c e p t fo r t h e M a o r i s p i r a l d e s i g n s , f r a c t a l g e o m e t r y d o e s n o t
a p p e a r to be a n i m p o r t a n t a s p e c t o f in d i g e n o u s S o u t h Pacific p a t t e r n s e ith er. T h a t
is n o t t o say t h a t f r a c ta l d e s i g n s a p p e a r n o w h e r e b u t A f r i c a — s o u t h e r n I n d i a is
full o f f r a c ta l s , a n d C h i n e s e flu id sw irl d e s i g n s a n d C e l t i c k n o t p a t t e r n s a re
a lm o s t c e r t a in l y o f i n d e p e n d e n t o rig in .”* T h e i m p o r t a n t p o i n t h e r e is t h a t t h e frac­
tal d e s i g n s o f A f r i c a s h o u l d n o t b e m i s t a k e n for a u n i v e r s a l o r p a n c u l t u r a l p h e ­
n o m e n o n ; th e y are c u ltu ra lly specific. T h e n e x t c h a p t e r w ill e x a m i n e th e
q u e s t i o n o f t h e i r m a t h e m a t i c a l sp ecificity.
CHAPTER
---------------— -------------------In te n tio n
an d-------------------------------------------------in v e n tio n --------------------------------------------in--------------------------------------------------—design------------------------------ 1----—
B efo re w e c a n d is c u ss t h e f r a c t a l s h a p e s in A f r i c a n s e t t l e m e n t a r c h i t e c t u r e s as
g e o m e tric k n o w le d g e , we n e e d t o t h i n k carefully a b o u t th e re la tio n b e tw e e n m a te rial d e sig n s a n d m a t h e m a t i c a l u n d e rs ta n d i n g. D e s ig n s a re b e s t s e e n as p o s i ti o n e d
o n a r a n g e o r s p e c t r u m o f ^ i n t e n t i o n . A t t h e b o t t o m o f t h e ra n g e a re u n i n t e n ­
t i o n a l p a t t e r n s, c r e a t e d a c c i d e n t a l l y as t h e b y - p r o d u c t o f s o m e o t h e r a c ti v it y .
In t h e m i d d l e o f rh e r a n g e a re d e s i g n s t h a t a r e i n t e n t ' o n n !
in tu itiv e ,
w ith n o rules o r g u id e l in e s t o e x p l a i n its c r e a t i o n . A t t h e u p p e r e n d o f t h e r a n g e ,
we h a v e t h e i n t e n t i o n a l a p p l i c a t i o n o f e x p l i c i t r u l e s t h a t w e a re a c c u s t o m e d to
a s s o c i a t i n g w i t h m a t h e m a t i c s . T h e f o l l o w i n g s e c t i o n s w ill e x a m i n e t h e f r a c ta l
d e s i g n s t h a t o c c u r in v a r i o u s p o s i t i o n s a l o n g t h i s i n t e n t i o n a l i t y s p e c t r u m .
F ra c ta ls fr o m u n c o n s c io its a c tiv ity
A n e x c e l le n t e x a m p le o f u n i n t e n t i o n a l fractals c a n b e fo u n d in t h e w ork o f M ic h a e l
Batty a n d Paul L ongley (1 9 8 9 ) , w h o e x a m i n e d t h e sh a p e o f large-scale u r b a n sprawl
s u r r o u n d i n g E u r o p e a n a n d A m e r i c a n c i t i e s (fig. 4 . 1 ) . W h i l e t h e b l o c k s o f t h e s e
citie s a re ty p ic a lly laid o u t in r e c t a n g u l a r g rids, a t v e ry large s cales— a r o u n d 100
sq u a re m i le s — w e c a n see t h a t t h e p r o c e s s o f p o p u l a t i o n g r o w t h h a s c r e a t e d a n
irre g u la r p a t t e r n . T h i s ty p e o f f r a c t a l , a “d if f u s i o n l i m i t e d a g g r e g a t i o n , ” also
50
In tro d u c tio n
F I G U R E 4.1
U r b a n s p r a w l in L o n d o n
L arg e-sca le urban spraw l
g e n e ra lly h as a fra ctal
stru ctu re. T h e u rb a n spraw l
fra ctals o n ly ex ist at v e ry
l a r g e s c a l e s — a b o u t 1 0 0 sq.
m ile s — a n d re s u lt fro m th e
u n co n sciou s accu m u lation
o f u rb an p o p u la tio n d yn am ics.
A t le v e ls o f co n s c io u s in te n t
( e .g ., th e grid o f c i t y b l o c k s ) ,
E u r o p e a n citie s are ty p ic a lly
E u c l i d e a n . A r e a is 1 0 x 1 0
k ilom eters.
( .R e p r in t e d w i t h p e r m i s s i o n f r o m
B a tty et al. 1 9 8 9 )
o c c u r s in c h e m i c a l s y s te m s w h e n p a r t i c l e s in a s o l u t i o n a r e a t t r a c t e d t o a n e l e c ­
t r o d e . F ra c t a l u r b a n s p r a w l is c le a r ly t h e r e s u l t o f u n c o n s c i o u s s o c ia l d y n a m i c s ,
n o t c o n s c i o u s d esig n . A t t h e s m a ll e r sc ale s in w h i c h t h e r e is c o n s c i o u s p l a n n i n g ,
E u r o p e a n a n d A m e r i c a n s e t t l e m e n t a r c h i t e c t u r e s a r e ty p i c a ll y E u c l i d e a n .
F r a c t a l s fro m n a tu r e :.m im e s is v e rs u s m o d e lin g
I t m i g h t b e t e m p t i n g t o t h i n k t h a t t h e c o r ^ a s t j ^ t w e e n . t h e E u c li .d e jn - d e s ig n s
o f E u ro p e a n d t h e fra c ta l d e sig n s o f A f r i c a c a n be e x p l a i n e d by t h e i m p o r t a n t ro le
o f t h e n a t u r a l e n v i r o n m e n t in A f r i c a n s o c i e ti e s . B u t t h i s a s s u m p t i o n t u r n s o u t
to be w rong; if a n y th in g , th e r e is a t e n d e n c y for in d ig en o u s societies to .fay o r E u clid ­
e a n s h a p e s . P h y s i c is t K h . S . , M a m e d o v o b s e r v e d s u c h a c o n t r a s t in h i s r e f le c t io n s
o n h i s y o u t h in a n o m a d i c c u l t u r e :
My p aren ts an d c o u n try m en . . .
u p to t h e s e c o n d w o r ld w a r h a d b e e n
n o m a d s . • •. O u ts i d e ou r n o m a d t e n t s we w ere liv in g in a w o n d e rfu l k in g d o m
o f variou s c u rv e d lines a n d forms. S o w h y w e re th e a e s t h e t i c signs n o t form ed
from t h e m , h a v i n g in s te a d p re s e rv e d g e o m e t r i c p a t t e r n s . . . ? [Jjn t h e cities
where th e straight-line geometry’ was p re d o m in a n t th e a esth etic signs were formed
. . . w ith n a tu re playing th e d o m i n a ti n g role. . . . [TJhe n o m a d did n o t n eed the
“p o rtra it” o f a n o ak to be carried w ith him elsewhere because h e co uld view all
sorts o f oaks every day and every h o u r . . . while for th e tow nsfolk th e ir in c lin a ­
tion to n a tu re was m ore a result of nostalgia.
( M a m e d o v 1986, 5 1 2 - 5 1 3 )
I n t e n t i o n a n d i n v e n t i o n in d e s ig n
. C o n t r a r y t o r o m a n t i c p o r t r a i t s o f t h e “n o b l e s a v a g e ” li v i n g as o n e w i t h
n a tu r e , m o s t in d i g e n o u s s o c i e t i e s s e e m q u i t e i n t e r e s t e d i n d i f f e r e n t i a t i n g t h e m ­
selves fro m t h e i r s u r r o u q d ii ig s . I t is t h e i n h a b i t a n t s o f larg e s t a t e s o c i e ti e s , s u c h
as t h o s e o f m o d e r n J i u r o p e } w h o y e a r n ,fo m i m i c t h e n a t u r a l . W h e n E u r o p e a n
de signs a r e fr a c ta l , it is u s u a l ly d u e _t_P.an .effo rt.to m i m i c n a t u r e . A f r i c a n fra c ta ls
based o n m i m i c r y o f n a t u r a l fo r m a re r e l a t i v e l y ra r e ; t h e ir i n s p i r a t i o n t e n d s to
co m e fr o m t h e r e a l m o f c u l t u r e .
H o w s h o u l d w e p l a c e s u c h n a t u r e - b a s e d d e s i g n s in o u r i n t e n t i o n a l i t y s p e c ­
tr um ? T h a t d e p e n d s o n t h e d i f f e r e n c e b e t w e e n m i m e s is.an d.m O -d elin g^> 4 im esjs^
is a r ^ a t t e m p t to m i r r o r t h e im a g e o f a p a r t i c u l a r o b j e c t , a g o al e x p l i c i t l y s t a t e d
by P l a t o a n d A r i s t o t l e as t h e e s s e n c e o f a r t , o n e t h a t w a s s u b s e q u e n t l y fo l lo w e d
in E urope for m a n y c e n tu r ie s (see A u e r b a c h 1953). A p h o to g r a p h is a goo d e x a m p le
o f m im esis. A p h o t o m i g h t c a p t u r e t h e f r a c ta l im a g e o f a t r e e , b u t it w o u ld b e
foolish to c o n c l u d e t h a t t h e p h o t o g r a p h e r k n o w s f r a c ta l g e o m e try . If a r t i s a n s a re
simply try in g t o c o p y a p a rt ic u l a r n a t u r a l o b je c t, t h e n t h e sc alin g is a n .u .n in te n d e d
b y -p r o d u c t.
T h e m o s t i m p o r t a n t a t t n b u t e s j : h a t s e p a r a t e m i m e s is f r o m Q n o d e l i n g e r e
a b s t r a c t i o n a n d g e n e r a l i z a t i o n .( A b s t r a c ti o n ^ is a n a t t e m p t t o l e a v e o u t m a n y o f
th e c o n c r e t e d e t a i l s o f t h e s u b j e c t by c r e a t i n g a s i m p l e r figure w h o s e s t r u c t u r e
is s till'ro u g h ly a n a lo g o u s t o t h e o r i g i n a l — o f t e n c a ll e d a “stylized" r e p r e s e n t a t i o n ,
in t h e a rts C jG en etaliz atio n ^ ir i e a n s s e l e c t i n g a n a n a l o g o u s s t r u c t u r e t h a t is^comm o n to all e x a m p l e s o f t h e s u b j e c t ; w h a t is o f t e n re f e r r e d t o as a n “u n d e r l y i n g ”
form o r law .1 F or e x a m p l e , M a n d e l b r o t ( 1 9 8 1 ) p o i n t s t o t h e E u r o p e a n B e a u x A r t s :
style as a n a t r e m p t n o t m e r e ly ro i m i t a t e n a t u r e , b u t t o “guess its law s.” H e n o t e s
th a t th e in te rio r o f t h e Paris o p e ra h o u s e m a k e s use o f scalin g a rc h e s -w ith in -a rc h e s ;,
. a p a t t e r n t h a t g e n e ra liz e s s o m e o f t h e s c a l in g c h a r a c t e r i s t i c s o f n a t u r e , but- is nota co py o f a n y o n e p a r t i c u l a r n a t u r a l o b j e c t .
S i n c e t h e u l t i m a t e g e n e r a l i z a t i o n is a m a t h e m a t i c a l m o d e l ,, w h y d i d n ’t
d e s i g n p r a c t i c e s s u c h as t h e B e a u x A r t s s t y l e r e s u l t i n a n e a r l y d e v e l o p m e n t o f
fra c ta l g e o m e t r y ? For E u r o p e a n s , s u c h l u s h o r n a m e n t a t i o n w as p r e s e n t e d —-an d
g e n e r a l ly a c c e p t e d — as e m b o d y i n g t h e opposite o f m a t h e m a t i c s ; i t w as ani e ff o rt
to c r e a t e d e sig n s t h a t c o u ld o n ly b e u n d e r s t o o d in i r r a ti o n a l, e m o t i o n a l , o r i n t u ­
itiv e te r m s . E v e n s o m e m o v e m e n t s a g a i n s t t h i s a t t e m p t , s u c h as t h e u s e o f d i s ­
t i n c t l y E u c l i d e a n fo r m s in t h e h i g h m o d e r n a r t s s t y le , s i m p l y r e i n f o r c e d t h e
a s s o c i a t i o n b e c a u s e it o n l y o f f e r e d a r e v e r s a l , s u g g e s t i n g t h a t “ m a t h e m a t i c a l ”
shapes (cubes, sp h eres, e tc .) c o u ld be e s th e tic a lly a p p re c ia te d . W i t h rare
e x c e p t i o n s ( e .g ., T h o m p s o n 1 9 1 7 ) , m i m e s i s o f n a t u r e in p r e - W W 11 E u r o p e a n
a rt t r a d i t i o n s m e r e ly f u r t h e r e d t h e a s s u m p t i o n t h a t E u c l i d e a n g e o m e t r y w as t h e
o n ly tr u e g e o m e t r y . 2
5*
In tro d u c tio n
T h e d if f e r e n c e b e tw e e n ^ m im e s js a n d m o d e l i n g p ro v i d e s tw o d i f f e r e n t posit i o n s a l o n g t h e j n t e n t i o n 3 l j C y _ s g e c t r u m . T h e le a s t i n t e n t i o n a l w o u ld b e m e r e ly
h o l d i n g a m i r r o r t o n a t u r e —-for e x a m p l e , if s o m e o n e w a s j u s t s h o o t i n g , a c a m ­
e ra o r p a i n t i n g a realistic p ic t u r e o u td o o r s a n d h a p p e n e d t o i n c l u d e a fra c ta l o b j e c t
( c l o u d , tr e e , e t c . ) . T h i s m i m e s is d o e s n o t c o u n t as m a t h e m a t i c a l t h i n k i n g . M o r e |
i n t e n t i o n a l is a stylized r e p r e s e n t a t i o n o f n a t u r e . If t h e a r t i s t h a s r e d u c e d t h e n a t - i
ural im ag e t o a s t ru c tu r a ll y a n a l o g o u s c o l l e c t i o n o f m o r e s i m p l e e l e m e n t s , s h e h a s \
c re a te d a n ab stract m odel. W h e t h e r o r n o t s u c h a b stra c tio n s m o v e to w a rd m o re J
m a t h e m a t i c a l m o d e l s is a m a t t e r o f lo c al p r e f e r e n c e .
T h e t w o e x a m p l e s o f A f r i c a n r e p r e s e n t a t i o n s o f n a t u r e w e o b s e r v e d in
th e previous c h a p te r c e rta in ly sh o w th a t th e a rtisa n s h a v e g o n e b e y o n d
m e r e m im e s is. T h e M a n d i a c k c o b r a p a t t e r n w e saw in fig u re 3 .2 . s h o w s a s t r ic t ly
sy ste m a tic scalin g p a tte rn . T h is te x tile d esig n c o n v e y s th e sc a lin g p ro p e rty
o f 't h e n a tu r a l c o b ra s k in p a t t e r n — d ia m o n d s a t m a n y sc a le s— in a sty lized or
a b s t r a c t way. W e c a n t a k e th is- i d e a a s t e p f u r t h e r b y e x a m i n i n g a n o t h e r
B w a m i b a t s c u l p t u r e (fig . 4 . 2 ) . T h i s s p i r a l p a t t e r n is a l s o a s t y l i z e d r e p r e ­
s e n t a t i o n o f t h e n a t u r a l s c a l i n g o f t h e b a t ’s w in g , b u t it.is a d i f f e r e n t g e o m e t r i c
d e s i g n t h a n t h e e x p a n d i n g zigzag p a t t e r n w e s a w in fi g u r e 3 .4 c . It is m o r e s ty 1-
FIGURE
4-2
Stylized, s c u l p t u r e o f a b a t
A nother Legn bat sculpture, but unlike the zigzag design
we saw in figure 3.4c, here the scaling of the wing folds is
represented by a spiral.
(B y p erm ission o f th e M u se u m o f A f r i c a n A r t, N .Y .)
/m ention an d invention in design
iz e d i n t h e s e n s e o f b e i n g f u r t h e r a b s t r a c t e d f r o m t h e o r i g i n a l n a t u r a l b a t ’s
w i n g . T h i s p r o v i d e s f u r t h e r e v i d e n c e t h a t t h e sc u l p t o r s w e r e f o c u s e d o n t h e
s c a l i n g p r o p e r t i es— t h e g e n e r a l i z e d u n d e r l y i n g f e a t u r e — a n d n o t p a r t i c u l a r c o n ­
c r e te d etails.
f
T h e g r e a t e s t d a n g e r o f t h i s b o o k is t h a t r e a d e r s m i g h t m i s i n t e r p r e t its ^
\ m e a n i n g in te rm s o f p rim itiv ism . T h e fa c t t h a t A f r i c a n j r acta ls are_rarely t h e r e s u l t 1
o f i m i t a t i n g n a t u r a l fo r m s Jh e lp s r e m i n d us t h a t t h e y a re n o t d u e t o “p r i m i t i v e s
l i v in g c lo s e t o n a t u r e . ” B u t e v e n fo r t h o s e r a r e cases i n w h i c h A f r i c a n f r a c ta ls
are re p r e s e n ta tio n s o f n a tu r e , it is c lea rly a s £ ^f'C ^scio \js,abstractiQ n , n o t a m im e tic
re f le c tio n . T h e g e o m e t r i c c h i n k i n g t h a t g o e s i n t o chesg^exam ples m a y ja g s i m p le,
b u t it is q u i t e i n t e n t i o n a l .
T h e fr a c ta l e s t h e t i c
J u s t as w e saw h o w d e s i g n s b a s e d o n n a t u r e r a n g e fr o m u n c o n s c i o u s t o i n t e n ­
t i o n a l , a r t i f i c i a l d e s i g n s a ls o v a r y a l o n g a r a n g e o f i n t e n t i o n , w i t h s o m e s i m p l y
th e result o f an in tu itiv e in s p ira tio n , a n d o th e r s a m o re self-conscious a p p lic a ­
t i o n o f ru le s o r p r i n c ip l e s . T h e e x a m p l e s o f A f r i c a n f r a c ta l s in fig u re 4 .3 d i d n o t
a p p e a r to be r e l a t e d t o a n y t h i n g o t h e r t h a n t h e a r t i s a n ’s e s t h e t i c i n t u i t i o n or
s e n s e o f b e auty. A s far as 1 c o u l d d e t e r m i n e fr o m d e s c r i p t i o n s in t h e l i t e r a t u r e
a n d m y o w n fie ld w o rk , t h e r e w e r e n o e x p l i c i t ru le s a b o u t h o w t o c o n s t r u c t th e s e
d e s i g n s , a n d n o m e a n i n g w as a t t a c h e d to t h e p a r t i c u l a r g e o m e t r i c s t r u c t u r e o f
t h e fig u re s o t h e r t h a n l o o k i n g g o o d . I n p a r t i c u l a r , I s p e n t s e v e r a l w e e k s in
D a k a r w a n d e r i n g t h e s t r e e ts a s k i n g a b o u t c e r t a i n f r a c ta l fa b ric p a t t e r n s a n d j e w ­
e l r y d e s i g n s , a n d cite i n s i s t e n c e c h a t t h e s e p a t t e r n s w e r e “j u s t fo r l o o k s ” w as so
a d a m a n t t h a t , if s o m e o n e f i n a l l y h a d o f f e r e d a n e x p l a n a t i o n , I w o u l d h a v e
v i e w e d it w i t h s u s p ic io n .
S i n c e s o m e p r o f e s s i o n a l m a t h e m a t i c i a n s r e p o r t t h a t t h e i r id e a s w e r e p u re
i n t u i t i o n — a s u d d e n flash o f i n s i g h t , o r “A h a ! ” as M a r t i n G a r d n e r p u t s it— we
s h o u l d n ’t d i s c o u n t t h e g e o m e t r i c t h i n k i n g o f a n a r t i s a n w h o r e p o r t s “1 c a n ’t tell
y o u h o w I c r e a t e d r h a t , it j u s t c a m e t o m e . " E s t h e t i c p a t t e r n s c l e a r ly q u a li fy as
i n t e n t i o n a l d e s ig n s . O n t h e j i t h e r hand,, t h e r e is n ’t m u c h we c a n s a y a b o u t t h e
m a t h e m a ti c a l ideas b e h i n d th e s e p a tte rn s ; th e y will h a v e t o r e m a in a m ystery unless
s o m e t h i n g m o r e is rev eale d a b o u t t h e i r m e a n i n g o r d i e a r t is a n ’s; c o n s t r u c t i o n te c h niqu£S.sl t is w o r t h n o t i n g , h o w e v e r , t h a t t h e y d o c o n t r i b u t e to t h e f r a c ta l d e sig n
t h e m e in A f r i c a . E s t h e t i c p a t t e r n s h e l p i n s p ir e p r a c t i c a l d esig n s, a n d v ic e .y e rsa .
S i n c e a n c i e n t tr a d e n e t w o r k s w e r e w e ll e s t a b l i s h e d , t h e d if fu sio n o f e s t h e t i c p a t ­
t e r n s is p r o b a b l y o n e p a r t o f t h e e x p l a n a t i o n for h o w fra c ta ls c a m e t o be so w i d e ­
s p r e a d ac ro s s t h e A f r i c a n c o n t i n e n t .
FIGURE 4 .3
E s t h e t i c fr a c ta l s
( a ) M e u r a n t ( q u o t e d in R e i f 1 9 9 6 ) r e p o r t s
t h a t t h e M b u t i w o m e n w h o c r e a t e d t h is
f r a c t a l d e s i g n , a b a r k - c l o t h p a i n t i n g , t o ld
h i m t h e d e s i g n w a s n o t “ t e l l i n g s t o r ie s , "
n o r w a s it “ r e p r e s e n t i n g a n y vp a r t i c u i a r
o b j e c t . ” ( b ) S c a l i n g p a t t e r n s c a n be
fo u n d in m a n y A f r i c a n d e c o r a t iv e designs
t h a t a r e r e p o r t e d t o b e “ ju st fo r b e a u t y . ”
U p p e r l e f t , S h o o w a R a f f i a c l o t h ; l o w e r l e ft,
S e n e g a l e s e t i e d y e ; rig ht, S e n e g a l e s e
pendant.
( a , co u rt e sy G e o r g e s M e n i a n l . b : U ppe r le ft ,
B r it is h M u s e u m ; l o w e r l e f t , f r o m M u s e c R oy al
d e l A / r i q u c C e n t r a l , Belgiu m ; right, photo
co urte sy I F A N , D a k a r .)
Im e n iio n a n d in v e n tio n in design
55
FIGURE 4 . 4
T h e q u in c u n x fra cta l
A c u s t o m e r in T o u b a , S e n e g a l , s e l e c t s a f r a c t a l q u i n c u n x p a t t e r n f o r h is l e a t h e r n e c k b a g . T h e
q u i n c u n x is h i s t o r i c a l l y i m p o r t a n t b e c a u s e o f its u se b y e a r l y A f r i c a n A m e r i c a n “ m a n o f s c i e n c e "
B enjam in Banneker.
O f c o u r s e , t h e r e a r e p l e n t y o f A f r i c a n d e s i g n s t h a t a re s t r i c t l y E u c l i d e a n ,
b u t e v e n t h e s e c a n o c c u r i n “ f r a c t a l i z e d ” v e r s io n s . O n e p a r t i c u l a r l y i n t e r e s t i n g
ex a m p l e is t h e quincunx (fig. 4 . 4 ) . T h e b a sic q u i n c u n x is a p a t t e r n o f five sq uares,
w i t h o n e a t t h e c e n t e r a n d o n e a t e a c h c o r n e r . T h e d e s i g n is c o m m o n i n S e n e ­
gal, w h e r e it is sa id t o r e p r e s e n t t h e “ li g h t o f A l l a h . ” T h e q u i n c u n x is h i s t o r i ­
cally i m p o r c a n t b e c a u s e t h e im a g e w as r e c o r d e d a s j a e i n g o f re lig io u s s i g n if i c a o c e
to t h e early A f r i c a n A m e r i c a n “ m a n o f s c i e n q e ” B e n j a m i n B a n n e k e r . S i n c e e v i ­
d e n c e s h o w s t h a t B a n n e k e r ’s g r a n d f a t h e r ( B a n n a k a ) c a m e f r o m S e n e g a l , t h e
q u i n c u n x is a f a s c i n a t i n g p o s s i b i l i t y fo r g e o m e t r y in t h e A f r i c a n d i a s p o r a (s e e
E glnsh 19 9 7 c fo r d e t a i l s ) . B e c a u s e o f t h e f r a c t a l e s t h e t i c , t h i s re l ig i o u s s y m b o l h
is o f t e n . a r r a n g e d in a r e c u r s i v e p a t t e r n — fiv e s q u a r e s q f five s q u a r e s — a s s h o w n / .
in figure 4 .4 in t h e d e s i g n fo r a l e a t h e r n e c k bag.
J ‘
F in a lly , t h e r e a r e a ls o e x a m p l e s o f t h e f r a c t a l e s t h e t i c i n c o m m o n h o u s e ­
h o l d f u r n i s h i n g s . E u r o - A m e r i c a n f u r n i t u r e is d i f f e r e n t i a t e d by f o r m a n d f u n c ­
ti o n — sto o ls are s tr u c tu r e d d if fe re n tly fro m c h a irs, w h ic h are s tr u c tu r e d
d i f f e r e n t l y f r o m c o u c h e s . B u t in A f r i c a n h o m e s o n e o f t e n se e s d i f f e r e n t sizes
o f t h e s a m e s h a p e (fig. 4 .5 ). A s i m i l a r d i f f e r e n c e h a s b e e n n o t e d in c r o s s - c u l tu r a l
co m p a riso n s o f h o u sin g . W h e r e a s E u r o - A m e r ic a n s w o u ld n e v e r t h i n k to h a v e
a g o v e r n e r ’s m a n s i o n s h a p e d l i k e a p e a s a n t ’s s h a c k ( o r v ic e v e r s a ) , p r e c o l o n i a l
A f r i c a n a r c h i t e c t u r e ty p i c a l l y u s e d t h e s a m e fo r m a t d i f f e r e n t sizes (as w e saw
for t h e sta tu s d i s t i n c t i o n s in t h e B a -ila s e t t l e m e n t in c h a p t e r 2 ). I t is u n f o r t u n a t e \
f chat t h i s A f r i c a n s t r u c t u r a l c h a r a c t e r i s t i c is ty p i c a l l y d e s c r i b e d in t e r m s o f a ]
!
lack— as d i e a b s e n c e o f s h a p e d i s t i n c t i o n s r a t h e r t h a n a s t h e p r e s e n c e o f a se a l- /
\ ing d e s i g n t h e m e .
Introduction
56
FI GURE 4 .5
T h e fr a c t a l e s t h e t i c i n h o u s e h o l d o b j e c t s
A fr ic 3 n s to o ls , chairs, an d ben ch es are often created in a scaling series.
(P h o to co u rtesy o f A f r i c a P l a c e , I n c .)
C o n c lu s i o n
W e n o w h a v e s o m e g u id e lin e s to h e l p d e t e r m i n e w h ic h fracta l designs s h o u ld c o u n t
as m a t h e m a t i c s , w h i c h s h o u l d n o t , a n d w h i c h a re i n b e t w e e n . F ig u r e 4.-6 s u m ­
m arizes t h i s s p e c t r u m . F r a c t a ls p r o d u c e d b y ^ u n c o n s c i o u s a c t i v i t y , o r as t h e u n i n ­
t e n t i o n a l b y - p r o d u c t fr o m so m e o t h e r p u rp o s e , c a n n o t b e a t t r i b u t e d t o i n d i g e n o u s
c o n c e p t s . B u t s o m e a r t i s t i c a c t i v i t i e s , s u c h as t h e c r e a t i o n o f sty liz e d r e p r e s e n -
Llumtentioiial
Unconscious activity
•urban sprawl
Accidental/ractals
• “mirror” portrait of nature
(moneys; e.g., photography)
I n te n tio n a l
I n te n tio n a l
but implicit
and explicit
I
1
Construction techniques
C o n s c i o u s m e o f n a t u r a l s c a lin g
•stylistic abstraction of natural scaling
Esthetic design
•intuitive fractal design theme
f ig u r e
4 .6
F r o m u n c o n s c i o u s a c c i d e n t to e x p lic it d e s i g n
Knowledge systems
/ n t e n t i o n a n d i n v e n t i o n in d e s ig n
c a tio n s o f n a t u r e o r p u r e l y e s t h e t i c d e s i g n s , d o s h o w i n t e n t i o n a l a c t i v i t y fo cu s ed
o n f r a c ta ls. S u c h e x a m p l e s m a y be r e s t r i c t e d in t e r m s o f g e o m e t r i c t h i n k i n g —
tja.e-mt-isans m a y o n l y r e p o r t t h a t t h e d e s i g n s u d d e n l y c a m e t o t h e m in a flash o f
in t u it io n - V -b u t th e s e a r e c le a r ly d i s t i n g u i s h e d fr o m th o s e w h i c h a re u n c o n s c i o u s
o T "a c c id e n ta l. T h e f o l l o w i n g c h a p t e r s will c o n s i d e r e x a m p l e s t h a t a re n o t o n ly
i n t e n t i o n a l , b u t also in c l u d e e n o u g h e x p li c it in f o r m a t i o n a b o u t design te c h n i q u e s
a n d k n o w le d g e sy stem s to be easily id en tifiab le as m a t h e m a t i c a l p ra ctice a n d ideas.
PA RT
-A frican------------------ —
“fractal-m athem atics-
CHAPTER
5
G eom etric
algorithms-
T h e w ord (J^g o rith m J) d e riv e s fro m th e n a m e o f a n A r a b m a t h e m a tic ia n ,
A l-K h w arizm i (c. 7 8 0 - 8 5 0
c
.e .),
w h o s e b o o k H is a b a l'ja b r w ’ al-m u q a b a lci ( C a l ­
c u l a t i o n by R e s t o r a t i o n a n d R e d u c t i o n ) a l s o g a v e us t h e w o r d “ a l g e b r a . ”
A l t h o u g h A l - K h w a r i z m i f o c u s e d o n n u m e r i c p r o c e d u r e s fo r s o l v i n g e q u a t i o n s ,
t h e m o d e r n t e r m ^ a l g o r i t h m / a p p li e s to a n y for m a lly s p e c ifie d p r o c e d u r e . A p e o m e t r i c a l g o r i t h m g iv es e x p l i c i t i n s t r u c t i o n s for g e n e r a t i n g a part.ifi.u la r s e t o f s p atia l p a t t e r n s . W e h a v e a l r e a d y s e e n h o w i t e r a t i o n s o f s u c h p a t t e r n - g e n e r a t i n g
p r o c e d u r e s c a n p r o d u c e f r a c t a l s o n a c o m p u t e r s c r e e n ; i n t h i s c h a p t e r we will
e x a m i n e t w o i n d i g e n o u s a l g o r i t h m s t h a t a ls o use i t e r a t i o n t o p r o d u c e s c a l i n g
designs: t h e 4 5 -degreg_-angle p o q s tr u e tjo n s o f t h e M a n g b e t u , a n d t h e lu s o n a d r a w ­
ings o f t h e C h o k w e .
g e o m e t r y in M c in g b e tu d e s i g n
T h e M a n g b e t u o c c u p y t h e D e l e R i v e r a r e a in t h e n o r t h e a s t e r n p a r t o f t h e
D e m o c r a t i c R e p u b l i c o f C o n g o (f o rm a lly Z a ir e ) . A r c h a e o l o g i c a l e v i d e n c e s h o w s
ir o n s m e l t i n g in t h e are a s i n c e 2 3 0 0
b .c
. e .,
b u t th e M a n g b e tu , c o m in g from drier
lands a ro u n d p re s e n t-d a y U g a n d a , d id n o t arriv e u n til a b o u t 1000
c
.e.
T hrough
b o th c o n flic t a n d c o o p e ra tio n , th e y e x c h a n g e d c u ltu ra l tra d itio n s w ith o th e r
61
62
A fr ic a n fra c ta l m athem atics
s o c i e ti e s o f t h e a re a : B a n t u - s p e a k i n g p e o p le s s u c h a s t h e B u d a , B u a a n d L ese, a n d
U b a n g i a n - s p e a k i n g p e o p l e s s u c h as t h e A z a n d e , B a n g b a , a n d B ara-m bo. A r o u n d
180 0 a n um b,er o f s m a ll c h ie f d o m s w e re c o n s o l id a t e d i n t o tjfie first M a n g b e t u k i n g ­
d o m . A l t h o u g h it la s te d o n l y t w o g e n e r a t i o n s , a t r a d i t i o n o f c o u r t ly p re s tig e c o n t i n u e d e v e n in sm all villages a n d spread t o m a n y o f th e M a n g b e t u ’s tr a d in g p artn ers.
T h i s c o m b i n a t i o n o f c u l t u r a l d iv e r s it y , e x c h a n g e , a n d p r e s t i g e r e s u l t e d in a
th r iv in g artistic tra d itio n .
/
A d e t a i l e d a c c o u n t o f M a n g b e t u h is t o r y a n d t r a d i t i o n s c a n b e f o u n d in
African Reflections: A n from Northeastern Zaire. S c h i l d k r o u t a n d K e im ( iggo) beg in
- t h e i r a n a ly s is b y s h o w i n g t h a t t h e m o s t f a m o u s a s p e c t o f Man_gbe.Lti-.3rt, t h e
‘‘rjat.u ra lis tic.Jb o k ,’’ w a s a c t u a l l y q u i t e rare in t h e t r a d i t i o n a l M a n g b e t u s o c i e t y
o f t h e n i n e t e e n t h c e n t u r y . D u r i n g a r e s e a r c h e x p e d i t i o n t o t h e C o n g o in *914
( th e o rig in o f th e p h o to s used h e re ) , m a m m alo g ist
H e r b e r t L a n g b e c a m e f a s c i n a t e d w i t h life lik e c a r v i n g s
o f h u m a n figures, a n d as w o r d s p r e a d t h a t h e w a s p a y ­
in g h i g h p r i c e s fo r t h e m , m o r e o f t h e s e c a r v i n g s w e r e
p ro d u c e d . O t h e r co lle c to r s c a m e to b uy th ese piec e s,.an d
e v e n t u a l l y t h e e c o n o m i c r e w a r d s for p r o d u c i n g n a t u ­
ra l is t ic M a n g b e t u a r t b e c a m e so s t r o n g t h a t it r e p l a c e d
o t h e r s t y le s .
S c h ild k ro u t a n d K eim sh o w t h a t o rig in ally th e
m o s t i m p o r t a n t e s t h e t i c w as n o t n a tu r a lis m , b u t a b stra c t
g e o m e t r i c d esig n . T h e in d i g e n o u s f a s c t n a t i o n w i t h a r t i ­
fic e a n d a b s t r a c t i o n w a s j g n o r e d - by c o l o n i z e r s , a n d
t h e i r p r e c o n c e p t i o n s o f A f r i c a n s as nature_-Ioving
“ c h i l d r e n o f t f i e f o r e s t” b e c a m e a self-fu lfillin g e x p e c ­
t a t i o n . B u t t h e a r t if a c ts a n d p h o t o g r a p h i c r e c o r d s f ro m
t h e 191 4 e x p e d i t i o n p r o v i d e us w i t h e x c e l l e n t e x a m p l e s
o f t r a d i t i o n a l M a n g b e t u p a t t e r n s , as w e ll as a n o p p o r ­
t u n i t y to in fe r s o m e o f t h e i r t e c h n i q u e s .
F ig u re 5.1 s h o w s t h e d e c o r a t i v e e n d o f a n iv o ry
h a t p i n . L ik e t h e a r c h i t e c t u r e a n d e s t h e t i c p a t t e r n s we
h a v e s e e n , t h i s is c le a r ly a s c a l i n g d e s i g n , b u t t h e p r e ­
c i s i o n o f t h e p a t t e r n s u g g e s ts t h a t t h e r e m a y b e a m o r e
FIGURE 5 . I
M a n g e b e t u iv o ry s c u l p t u r e
( T ra n sp a re n cy no . 3 9 3 5 , photograph b y L i n t o n G a rd in e r , co urte sy
A m e r i c a n M u s e u m o f R l a t v r a l H is t o ry.)
G eom etric algorithms
fo rm a l g e o m e t r i c pro cess a t w ork. S im il a r d e s ig n c a n be s e e n a t w ork in th e M a n g b e t u ’s g e o m e t r i c style o f p e rs o n a ! a d o r n m e n t . F ig ure 5 .2 a s h o w s a M a n g b e t u h a i r ­
sty le, p o p u l a r d u r i n g t h e t i m e t h a t th i s c a r v i n g w a s ,p r e a t e d ( a b o u t 1 9 1 4 ) , w h i c h
f e a t u r e d a d is k a n g l e d t o t-he v e r t i c a l a t 4 5 d e g r e e s . M e n o f t e n w o re a h a t w i t h
t h e t o p f l a t t e n e d , f o r m i n g t h e s a m e a n g l e , as s e e n in figure 5 .2 b . j u s t as a p l a n e
c u t s d i a g o n a l l y t h r o u g h t h e t o p o f t h e h e a d s in t h e iv o r y s c u l p t u r e o f figure 5.1,
real M a n g b e t u h e a d d r e s s e s also t e r m i n a t e d in a 4 5 ' d e g r e e a n g le .
T h i s w a s o n ly o n e p a r t o f a n e l a b o r a t e g e o m e t r i c e s t h e t i c b a s e d o n m u l ­
t ip le s o f t h e 4 5 - d e g r e e a n g le . F igu re 5 .2 b s h o w s a n iv o ry h a t p i n , e n d i n g in a disk
p e r p e n d i c u l a r to it, in s e r t e d p e r p e n d i c u l a r t o t h e h a t . T o its r i g h t, a s m a ll ivory
a r r o w p i n n e d t o t h e h a t p o i n t s h o riz o n ta lly , t h u s f o r m i n g a n a n g l e o f 135 d e g re e s
w ith th e h a tp i n . E a c h p a rt o f th e e n s e m b le w as a lig n e d by a m u ltip le o f th e
4 5 - d e g r e e a n g le . T h i s a d o r n m e n t sty le i n c l u d e d a rtificia l e l o n g a t i o n o f t h e h e a d ,
w h i c h is c le a r ly v is ib le in t h e p h o t o g r a p h in figure 5 .2 b . E l o n g a t i o n w as a c c o m ­
p l i s h e d b y w r a p p i n g a c l o t h b a n d a r o u n d t h e h e a d o f i n f a n t s ; t h e w o m a n in
fig u re 5 . 2 a is w e a v i n g o n e o f t h e s e b a n d s . H e a d e l o n g a t i o n r e s u l t e d in a n a n g l e
o f 135 d e g r e e s b e t w e e n t h e b a c k o f t h e h e a d a n d t h e n e c k .
FIGURE 5.2
Q e o m e tr i c d e sig n in M a n g b e tu p e r s o n a l a d o r n m e n t
(a) M a n g b e t u w o m a n w e a v i n g h e a d b a n d , (b ) M a n g b e t u c h i e f .
fa, negative n o . 1 1 1 9 1 9 , photograph by H . L a n g , co urtesy A m e r ic a n M u s e u m o f N a tu r a l H isto ry;
b , n e g a t i v e n o . 2 2 4 1 0 5 , p h o t o g r a p h by H . L a n g , c o u r t e s y A m e r i c a n M u s e u m o f N a t u r a l H i s t o r y . )
63
A fr ic a n fractal m athem atics
64
W h i l e t h e M a n g b e t u g e o m e t r i c c o n c e p t i o n o f t h e b o d y m a y h a v e i n s p ir e d
t h e 4 5 'd e g re e -a n g le d esign t h e m e , th o s e desig ns were c erta in ly n o t lim ited to sim ple
m i m i c r y o f a n a t o m y . W e c a rt c l e a r l y s e e t h i s in t h e i r m u s ic a l i n s t r u m e n t s . T h e
d r u m in figure 5 .3 a, for e x a m p l e , h a s its u p p e r s u r f a c e c u t a t a 4 5 ' d e g r e e a n g l e
t o t h e v ertical. T h e s t r i n g e d i n s t r u m e n t s h o w n in figure 5 .3 b h a s a r e s o n a t o r t h a t
m e e t s th e v e r t ic a l t u n i n g s t e m a t a 13 5 ' d e g r e e a n g le . E v e n in t h e c a s e o f a n t h r o ­
p o m o r p h i c d esig ns, t h e a r t is a n s e l a b o r a t e d o n t h e h u m a n fo r m in w ays t h a t s h o w
b
F I G U R E 5 .3
Q e o m e tric d esig n in M a n g b e tit m u s ic a l in s tr u m e n ts
(a) Drum, (b) Harp.
(a , negative no. t 1 1 8 9 6 , J)/Kitogm|?h b y H - L ang, c o u rtesy A m e ric a n M u s e u m o f N a tu r a l H isto ry ;
b , c o u rtesy R i c t b c r g M u se u m Z u ric h , p h o tofpaph by W euscctn a n d K a u f.)
G e o m e tric algorithms
c re a tiv e — a n d n o t m erely im ita tiv e — a p p lic a tio n s of g e o m e tric a l th in k in g ..
F or e x a m p l e , t h e r e is a n a n t h r o p o m o r p h i c d e c o r a t i v e m o t i f a t t h e e n d o f t h e
t u n i n g s t e m s h o w n in fig u re 5 . 3 b , b u t t h e s e h u m a n h e a d s a r e n o r s i m p l y m i m ­
i c k i n g h u m a n fo r m . I n fig u re 5 . 2 b w e sa'W t h a t t h e M a n g b e t u h a d a 13 5 - d e g r e e
a n g l e b e t w e e n t h e b a c k o f t h e h e a d a n d t h e n e c k . T h e c a r v e d h e a d s in fig u re
5 .3b h a v e a 9 0 -d e g re e a n g le b e tw e e n t h e b ac k o f th e h e a d a n d th e n e c k . S u c h
d i s t o r t i o n s i n d i c a t e a c t i v e g e o m e t r i c t h i n k i n g r a t h e r t h a n p a s s iv e r e f l e c t i o n o f
n a t u r a l a n a t o m i c a l a n g le s ( w h i c h , r e c a l l i n g t h e a r t ifi c ia l h e a d e l o n g a t i o n , w e re
n o t so n a t u r a l to b e g i n w i t h ) .
T h e r e are also purely a b stra c t d esig ns t h a t m a k e use o f m u ltip les o f 45 degrees,
,a s w e see in figure 5 .4 . M o d e r n M a n g b e t u r e p o r t t h a t t h e c r e a t i o n o f a d e s i g n
re f le c te d t h e a r t i s a n ’s d e s ire t o “m a k e it b e a u t i f u l a n d s h o w t h e i n t e l l i g e n c e o f
th e c re a to r” (S c h ild k ro u t a n d K eim 1990,
1 0 0 ). T h i s sug gests a n o t h e r r e a s o n f o r a r t i ­
sa n s to a d h e r e to a n g le s t h a t a r e m t i l r i p l e s o f
45 d e g r e e s: if t h e r e w e r e n o ru le s t o fo llow ,
t h e n it w o u ld h a v e b e e n d if fic u lt t o c o m p a r e
d e sig n s a n d d e m o n s t r a t e o n e ’s i n g e n u i ty . By
r e s t r i c t i n g t h e p e r m is s ib l e a n g l e s t o a s m a ll
se t, th e y w e r e b e t t e r a b l e t o d i s p l a y t h e i r
g e o m e t r i c a c c o m p lis h m e n t s . C o m b in in g th is 45-d eg v ee-an g le c o n ­
stru c tio n te c h n iq u e w ith th e scalin g p r o p ­
e r t ie s o f t h e iv o r y c a r v i n g in fi g u r e 5.1 c a n
re v e a l irs u n d e r l y i n g s t r u c t u r e . T h e c a r v i n g
lias t h r e e i n t e r e s t i n g g eo uie lii> - f e a U u e s : 1 First, e a c h h e a d is la rg e r t h a n t h e o n e above!
) ir a n d f a c e s i n t h e o p p o s i t e d i r e c t i o n . S e c 1 o n d , e a c h h e a d is f r a m e d b y t w o l i n e s , o n e
f o r m e d by t h e j a w a n d o n e f o r m e d b y t h e
h a ir ; t h e s e li n e s i n t e r s e c t a t a p p r o x i m a t e l y
9 0 d e g re e s . T h i r d , t h e r e is a n a s y m m e t r y ;
t h e le ft s i d e s h o w s a d i s t i n c t a n g l e a b o u t
|j2o d e g r e e s fr o m t h e v e r t i c a l .
FIGURE 5 .4
M a n g e b e t u iv o ry sc u lp tu r e
(Transparency n o . 3 9 2 9 , p h o to g ra p h b y L y r u o n G a rd in e r ,
courtesy A m e r i c a n M u s e u m o f N a t u r a l H i s t o r y . )
65
FIGURE
5.5
Q e o m e t r i c a n a l y s i s o f a n ivory s c u l p t u r e
FIGURE 5 .6
(g e o m e tric r e l a t i o n s i n t h e M a n g b e t u ite r a tiv e s q u a r e s s t r u c t u r e
S i n c e 9[ a n d 02 a r e t h e a l t e r n a t e i n t e r i o r a n g l e s o f a t r a n s v e r s a l i n t e r s e c t i n g t w o p a r a l l e l lin e s ,
0 ,-6 ,.
A fr ic a n fractal m athem atics
68
A l l o f t h e s e f e a t u r e s c a n b e a c c o u n t e d fo r b y t h e s t r u c t u r e s h o w n in f i g '
u r e 5 .5 . T h i s s e q u e n c e o f s h r i n k i n g s q u a r e s c a n b e c o n s t r u c t e d by a n i t e r a t i v e
p rocess, b is e c tin g o n e s q u a re to c re a te t h e - l e n g t h o f th e sid e for t h e n e x t
s q u a r e , as i n d i c a t e d i n t h e d i a g r a m . W e w ill n e v e r k n o w fo r c e r t a i n if t h i s ite ra t i v e - s q u a r e s c o n s t r u c t i o n w as t h e c o n c e p t u n d e r l y i n g t h e s c u l p t u r e ’s d e s ig n , bu t.
it d o e s m a t c h t h e f e a t u r e s i d e n t i f i e d a b o v e . I n t h e iv o r y s c u l p t u r e , t h e le f t s id e
is a b o u t 2 0 d e g r e e s fr o m t h e v e r t i c a l . I n t h e i t e r a t i v e - s q u a r e s s t r u c t u r e , t h e le ft
s i d e is a b o u t 18 d e g r e e s f r o m t h e v e r t i c a l , as s h o w n i n fig u re 5 .6 . H e r e w e s e e s
/ t h a t t h e c o n s t r u c t i o n a l g o r i t h m c a n b e c o n t i n u e d i n d e f i n i t e l y , a n d t h e r e s u ltI in g s t r u c t u r e c a n b e a p p li e d to a w id e v a r i e ty o f m a t h t e a c h i n g a p p l i c a t i o n s , fro m /
\s im p le p ro c e d u ra l c o n s tr u c tio n to tr ig o n o m e try (E g iash 1998a).
Lusona
T h e C h o k w e p e o p le o f A n g o ja /r a d a tm d itiq n .q f c r e n tin g p a tte r m ^
li n e s c a l l e d j ' l u s o n a ” in t h e _ s a n d . G e r d e s ( 1 9 9 1 ) n o t e s t h a t t h e l u s o n a s a n d
d raw ings sh o w th e c o n s tr a in ts necessary to d efine w h a t m a th e m a tic ia n s c a ll an
“E u l e r i a n p a t h ” : t h e s ty lu s n e v e r le a v e s t h e s u r f a c e a n d n o l i n e is r e t r a c e d . T h e
lu s o n a also t e n d to use t h e s a m e p a t t e r n a t d i f f e r e n t scales, t h a t is, s u c c e s siv e i t e r ­
a t i o n s o f a s in g le g e o m e t r i c a l g o r i t h m . F ig u r e 5.7 s h o w s t h e first t h r e e i t e r a t i o n s
o f o n e o f t h e d o z e n s o f l u s o n a t h a t w e re r e c o r d e d by m i s s i o n a r i e s d u r i n g t h e n i n e ­
t e e n t h c e n t u r y , w h e n t h e lu s o n a t r a d i t i o n w a s s tiii i n t a c t .
A s in t h e c a s e o f t h e M a n g b e c u 4 5 - d e g r e e c o n s t r u c t i o n s , t h e r e s t r i c t io n t o
a n E u le ria n p a th pro v id es t h e C h o k w e w ith a m e a n s to c o m p a re designs w ith in
a single fram ew ork, a n d to sh o w h o w in c re a sin g c o m p l e x it y c a n b e a c h ie v e d w i t h i n
t h e s e c o n s t r a i n t s o f s p a c e a n d lo g ic. B u t u n l i k e t ’h e ' c o m p e t i r i v e ba sis fo i co m -p a r i s o n t h a t t h e M a n g b e t u d e s c r ib e , th e C h o k w e m a d e use o f th e s e figures t o c r e ­
a t e g ro u p id e n tity . T h e re p o r ts i n d i c a t e t h a t t h e l u s o n a w e re u se d in a n a g e -g r a d e
i n i t i a t i o n s y s te m ; r i tu a l s t h a t a l l o w e d e a c h m e m b e r t o a c h i e v e t h e s t a t u s o f
r e a c h i n g t h e n e x t , m o r e s e n i o r le v e l o f i d e n t it y . By u s i n g m o r e c o m p l e x l u s o n a ,
t h e i t e r a t i o n s o f s o c ia l k n o w l e d g e p a s s e d o n in t h e i n i t i a t i o n b e c o m e v is u a liz e d
by t h e g e o m e t r i c i t e r a t i o n s . In c h a p t e r 8 w e w ill s e e o t h e r e x a m p l e s o f i t e r a t i v e
scalin g p a tt e r n s in in i ti a ti o n rituals. T h i s tr a d it io n o f g ro u p id e n tity t h r o u g h k n o w l­
e d g e o f t h e lu s o n a w a s a ls o d e p l o y e d b y t h e C h o k w e as a w a y t o d e f l a te t h e e g o
o f o v e r c o n f i d e n t E u r o p e a n visito rs, w h o f o u n d t h e m s e l v e s u n a b l e t o r e p l ic a t e t h e
lu s o n a o f m a n y c h i l d r e n .
C o n c lu sio n
T h e s e t w o e x a m p l e s , t h e M a n g b e t u iv o ry c a r v i n g a n d t h e lu s o n a d r a w in g s , h e l p
us see t h a t A f r i c a n fractals are n o t ju s t t h e resu lt o f s p o n t a n e o u s i n t u i t i o n ; in so m e
G eom etric algorithms
cases th e y are c re a te d u n d e r r u le -b o u n d t e c h n i q u e s e q u iv a le n t to W e s te rn
m a t h e m a t i c s . A n d t h e i r c u l t u r a l s i g n i f i c a n c e m a k e s it c l e a r t h a t a ll m a t h e ­
m a t i c a l a c t i v i t y — n o m a t t e r in w h i c h s o c i e t y it i s j b u n d — is p r o d u c e d t h r o u g h
a n i n t e r a c t i o n b e t w e e n t h e f r e e d o m o f lo c a l h u m a n i n v e n t i o n a n d t h e u n i v e r ­
sal c o n s t r a i n t s w e d i s c o v e r in s p a c e a n d lo gic.
‘Myombo"— trees of the ancestors.
F I G U R E 5 .7
L usona
(a) These figures, “lusona," were traditionally drawn in sand by the Chokwe people of Angola.
Successive iterations of the same algorithm were sometimes used to produce similar patterns of
increasing size, (b) The first and third iterations of another lusona algorithm carved into a
wooden box lid.
(a, based on drawings in Gerdes 1995.)
69
A fr ic a n fra c ta l m a th em a tics
R e c a ll t h a t in b o t h e x a m p l e s t h e role o f “c o n s t r a i n t ” w as c ru c ia l t o t h e d e v e l ­
o p m e n t o f t h e i r sc a lin g g eo m etry . F o r t h e M a n g b e t u ’s d e s i g n it w as t h e c o n s t r a i n t s
o f s t r a i g h t - e d g e c o n s t r u c t i o n w i t h a n g l e s a t m u l t i p l e s o f 4 5 degrees'.--For t h e
C h o k w e ’s lu s o n a it w as t h e c o n s t r a i n t s o f a n E u l e r i a n p a t h . B u t in e a c h c a s e t h e
c h o i c e o f p a r t i c u l a r o b j e c t i v e c o n s t r a i n t s — d e c i d i n g w h i c h o f t h e i n f i n i t e laws
o f s p a c e a n d log ic w e a r e c o n c e r n e d w i t h — w a s e s t a b l i s h e d b y a n d fo r t h e s o c ia l
re la tio n s o f th e co m m u n ity . In th e case o f th e M a n g b e tu it w as a rtis tic c o m p e ­
t i t i o n , a n d in t h e c a s e o f t h e C h o k w e it w as a g e - g r a d e i d e n t i t y . I n o t h e r w o rd s,
t h e i n v e n t i o n a n d d is c o v e ry c o m p o n e n t s o f m a t h e m a t i c s a r e i n e x t r i c a b l y l i n k e d
t h r o u g h s o c i a l e x p r e s s io n .
P h i l o s o p h i c p e r s p e c t i v e s o n t h e r e l a t i o n o f c u l t u r e a n d m a t h e m a t i c s will
b e f u r t h e r d is c u s s e d in p a r t 11, b u t t o d o so w e n e e d a f u l le r p o r t r a i t o f A f r i c a n
f r a c ta l ge om e try . T h e n e x t c h a p t e r w ill e x a m i n e A f r i c a n c o n c e p t i o n s o f t h e m o s t
f u n d a m e n t a l c h a r a c t e r i s t i c o f fr a c ta ls : n o n l i n e a r s c a l in g .
CHAPTER
Scaling-
6
W e h a v e a lr e a d y s e e n m a n y e x a m p l e s o f s c a l i n g in A f r i c a n d esig ns. I n t h e s e t t l e ­
m e n t a r c h i te c t u r e o f c h a p t e r 2, for e x a m p l e , t h e c o m p u t e r sim u la tio n s c learly sh ow
t h a t w e c a n t h i n k a b o u t t h e s e p a t t e r n s in t e r m s o f f r a c ta l g e o m e try . H o w d o t h e
A f r i c a n a r t i s a n s t h i n k a b o u t s c a l in g ? Is it j u s t i n t u i t i o n , o r d o t h e y u se e x p l i c i t
m a t h e m a t i c a l p r a c t i c e s in t h i n k i n g a b o u t s i m i l a r i t y a t d i f f e r e n t sizes? By e x a m ­
in in g v a r i e t i e s o f d e s i g n s w i t h d i f f e r e n t s c a l i n g p r o p e r t i e s , a n d c o m p a r i n g th e s e
w ith t h e a r t i s a n s ’ d is c u s s io n s o f t h e p a t t e r n s , w e c a n g a i n s o m e i n s i g h t i n t o s c a l ­
ing as a m a t h e m a t i c a l c o n c e p t in A f r i c a n c u l t u r e s .
P ow er lo u ' s c a lin g in iv in d s c r e e n s fro m th e S a h e l
T h ^ S a h e l ^ s a b r o a d b a n d o f a r i d l a n d b e t w e e n t h e S a h a r a D e s e r t a n d t h e rest
o f s u b - S a h a r a n A f r i c a . S i n c e t h e r e a r e few t r e e s a n d a g r e a t d e a l o f m i l l e t c u l ­
ti v a ti o n , it is n o t s u r p r i s i n g t h a t a r t i s a n s u se m i l l e t s t a lk s to w e a v e f e n c e s , walls,
an d o t h e r c o n s t r u c t i o n s . B u t t h e c o n s i s t e n t use o f a n o n l i n e a r s c a l in g p a t t e r n in
these stra w s c r e e n s (fig. 6 . 1 a ) isja b j t o d d . R a t h e r t h a n u n if o r m l e n g t h s , t h e row s
o f m ille t s t r a w g e t s h o r t e r a n d s h o r t e r as t h e y go up. I n t h e U n i t e d S t a t e s w e are
used to t h e im a g e o f “ t h e w h i t e p i c k e t f e n c e ” as a s y m b o l o f u n c h a n g i n g , l i n e a r
r e p e t it io n , y e t h e r e t h e f e n c e s a re d i s t i n c t l y n o n l i n e a r . W h i l e 1 w as in M a l i o n
The straw windscreen in Niger.
FI GURE 6. 1
A n A f r i c a n w in d screen
(a) The diagonallengths of these rows from bottom to top-. L = 16 12 8 6 5 , 5 3 3 2 2
This pattern is quantitatively determined by the African artisans. Here we see how the bundles of
straw are first laid in long diagonal rows, then a row at the opposite angle is interlaced in back of
it. T he length of each diagonal tow— how high up you go before doing the interlace step—is
determined by counting a certain number o f diagonals to be crossed. In the first layer (c) we go
over eight, then six, then four, then three.
Each bundle is about 2 inches across the diagonal, which is why the lengths go as dovible the
number of crossings. The odd numbered lengths are created by splitting the bundles in two.
Why do the lengths repeat in pairs as we go toward the top? There is a discrete approximation to
the continuous nonlinear scale that the African artisans follow.
( a , p h o t o by p e r m i s s i o n o f G a r d i 1 9 7 3 . )
(f ig u re c o n t i n u e s )
Scaling
73
t h e o u t s k i r t s o f t h e c a p i t a l c it y o f B a m a k o , I h a d t h e o p p o r t u n i t y to i n t e r v i e w
s o m e o f t h e a r t i s a n s w h o c r e a t e t h e s e s c r e e n s a n d w as p r o v i d e d w i t h a s t r i k i n g
e x a m p le o f in d ig en o u s a p p lic a tio n o f th e scalin g c o n c e p t.
T h e a r t i s a n s b e g a n by e x p l a i t n n g th a .t.in .'T e rd le . a r e a s ” s u c h as t h e forests
o f t h e s o u t h , t h e s c r e e n s a re n o t m a d e w i t h s c a l i n g ro w s b u t r a t h e r w i t h ro w s o f
lo n g , u n i f o r m l e n g t h . T h i s is b e c a u s e t h e l o n g ro w s use less s t r a w a n d ta k e less
t i m e to m a k e . B u t h e r e in t h e S a h e l , t h e y s a id , w e h a v e s t r o n g w in d s a n d d u st.
T h e s h o r t e s t ro w s a re t h e o n e s t h a t k e e p o u t d u s t t h e b e s t, b e c a u s e th e y a re th e
t i g h t e s t w e a v e . B u t t h e y a ls o t a k e m o r e m a t e r i a l s a n d effort. “W e k n o w t h a t . t h e
w i n d b lo w s s t r o n g e r as y o u g o u p fr o m t h e g r o u n d , so we m a k e t h e w i n d s c r e e n
t o m a t c h — t h a t w ay w e o n l y u se t h e s t r a w n e e d e d a t e a c h l e v e l . ”
T h e r e a s o n in g t h e a r r i s a n s r e p o r t e d is e q u i v a l e n t t o w h a t a n e n g i n e e r
w o u ld c a ll a “c o s t - b e n e f i t ” a n a l ysis; d e v e l o p i n g t h e m a x i m u m in f u n c t i o n ( k e e p ­
in g o u t d u s t ) fo r a m i n i m u m o f c o s t ( e ff o rt a n d m a t e r i a l s ) . M y p r i m a r y i n t e r e s t
h e r e is in s h o w i n g t h a t t h e sc a l i n g c o n c e p t in A f r i c a c a n b e m u c h m o r e s o p h ist i c a t e d t h a n j u s t a n o b s e r v a t i o n , “th e s a m e t h i n g in d i f f e r e n t sizes." T h e c r e a t i o n
A ssum ing d e c re a se in wind
penetration is reciprocal of length:
a = I
(wind engineers:
a = 1/3 )
•
- 0 .4
.
-0 .6
.
< -0 .8
-
- 1 .0
.
^
♦
?-
»
♦
♦
♦
♦
L o g(H )
FIGURE 6 .1
(continued)
(d) T h e r e l a t i o n b e t w e e n w i n d s p e e d a n d v e r t i c a l h e i g h t as s h o w n in t h e W in d E n g in e e rin g
H a n d b o o k , ( e ) T h e A f r i c a n w i n d s c r e e n m a k e r s say t h a t t h e y h a v e s c a l e d t h e r o w s o f s tr a w to
m a t c h t h e c h a n g e o f w i n d s p e e d w i t h h e i g h t . I f w e a s s u m e , j u s t f o r s i m p l i c i t y , t h a t t h e d e c r e a s e in
w i n d p e n e t r a t i o n is t h e r e c i p r o c a l o f t h e l e n g t h , t h e n w e c a n g e t t h e A f r i c a n e s t i m a t e f o r a. b y
m easu rin g th e s lo p e o f ro w le n g t h v ersu s h e ig h t o n a lo g - lo g gra p h . T h is g iv e s a =
e n g i n e e r s u s e Ct =
16 —
1,
w h e r e a s the
n o t b a d for a b a llp a r k e s t im a t e .
N o t e t h a t t h e g r a p h is i n a v e r y s t r a i g h t l i n e , e x c e p t w h e r e t h e d i s c r e t e n a t u r e o f t h e s c r e e n
(the s c r e e n m a k e r s m u s t c o u n t in w h o l e n u m b e r u n it s d u e t o t h e s tr a w b u n d le s ) fo rce s a n a p p r o x i ­
m ation by re p e a tin g th e s a m e le n g th tw ic e .
A fr ic a n fra c ta l m athem atics
74
o f t h e w i n d s c r e e n as a n o p t i m a l d e s i g n r e q u i r e d m a t c h i n g t h e s c a l i n g v a r i a t i o n
o f w i n d s p e e d v e r s u s h e i g h t t o a s c a l i n g v a r i a t i o n in l e n g t h s o f straw . By t r a n s ­
ferring th is c o n c e p t b e tw e e n tw o c o m p le te ly d iffe re n t d o m a in s , th e - a r tis a n s
h a v e d e m o n s t r a t e d t h a t th e y u n d e r s t a n d sc a lin g in t h e a b s t r a c t ; i n d e e d , t h e d e sig n
e s s e n t i a l l y p l o t s t h e r e l a t i o n o f w i n d s p e e d to h e i g h t o n a s t r a w g r a p h .
A l t h o u g h 1 w as c o n c e r n e d o n l y w i t h t h e o v e r a l l r e l a t i o n o f s c a l i n g a n d
r e a s o n i n g , I m e a s u r e d t h e ro w s j u s t t o s e e h o w c l o s e t h e y c a m e t o w h a t a W e s t ­
e r n e n g i n e e r w o u ld d e v e l o p f o r a n o p t i m a l m a t c h w i t h w i n d s p e e d . If t h e s t r a w
s c r e e n h a d l i n e a r s c a l in g , t h e n e a c h ro w w o u ld d e c r e a s e in l e n g t h by t h e s a m e
a m o u n t (e.g., 12 in c h e s , 10 i n c h e s , 8 in c h e s , e t c . ) . B u t t h e ro w s d e c r e a s e less a n d
less w i t h h e i g h t ; it t u r n s o u t t h a t t h e s c r e e n d e s i g n s h o w s a c lo s e fit t o w h a t is
c a l l e d a " p o w e r l a w " — t h a t is, it s c a l e s a c c o r d i n g t o a n e x p o n e n t (fig . 6 . 1 c ) .
F ig u re 6 . 1 b , r e p r i n t e d fr o m t h e W in d E ng in eerin g H a n d b o o k , s h o w s t h e e q u a t i o n
o f w i n d s p e e d w i t h h e i g h t m o s t c o m m o n l y u s e d by e n g i n e e r s — a ls o a p o w e r law.
S o t h e S a h e l w i n d s c r e e n is n o t o n l y a p r a c t i c a l a p p l i c a t i o n o f t h e a b s t r a c t s c a l ­
in g c o n c e p t , it is also a fa irly a c c u r a t e o n e . O f c o u r s e , o n e m i g h t o b j e c t t h a t t h e
in d ig e n o u s e n g in e e rs d id n o t a c tu a lly se t up th e alg eb ra a n d p e rfo rm th e o p ti­
m i z in g c a l c u l a t i o n . B u t I a s k e d t h r e e A m e r i c a n m a t h e m a t i c i a n s h o w t h e y w o u ld
s e t u p th e s e e q u a ti o n s to d e t e r m i n e t h e o p t i m a l d esig n , a n d all t h r e e said t h e s a m e
th i n g : “ 1 w o u l d n ’t so lv e it a n a l y t i c a l l y , I ’d j u s t g r a p h t h e e q u a t i o n s o n t h e c o m ­
p u t e r a n d see w h e r e t h e f u n c t i o n s p e a k e d . " W h e t h e r w e m a k e o u r g r a p h s o n a
c o m p u t e r s c r e e n o r a stra w s c r e e n d o e s n ’t m a tte r, as lo n g as w e g e t t h e ri g h t answ er.
S tr e tc h in g s p a c e in k e n t e c lo th
If s o m e o n e in A m e r i c a w e re a sk ed to t h i n k o f a n A fr i c a n t e x t ile, k en te c lo t h w o u ld
b e t h e m o s t likely im ag e. Its c o m b i n a t i o n o f stro jig c o lo r s ,.h o ld d e s i g n s , a n d ja s so c i a t i o n s w i t h a n c i e r ^ kingdoms.of-.We.s.t-Africa h a s m a d e it a f a v o r i te fo r im p o rts.
B u t m o s t o f t h e i m p o r t e d k e n t e c l o t h is c r e a t e d by a u t o m a t e d m a c h i n e , a n d w h i l e
1 w o u l d fi e rc e l y d e f e n d it a s “a u t h e n t i c , ” t h e n e e d for p a t t e r n r e p e t i t i o n in
a u to m a tio n has e lim in a te d a w o n d erfu l scalin g tr a n sfo rm a tio n th a t c a n be seen
in t h e o l d e r p a t t e r n s c r e a t e d o n h a n d lo o m s (fig. 6 .2 a ) . T h e s c a l i n g - c h a n g e is n o t
ju s t s m a ll a n d large v e r s i o n s o f t h e s a m e t h i n g ; r a t h e r , it is as if t h e d e s i g n w as
d ra w n o n a ru b b e r sh e e t, w h ic h was h a lf s tr e tc h e d a n d h a lf c o n tr a c te d . In
G h a n a I tr a v e le d to th e villag e o f B o n w i r e , w h e r e h a n d - l o o m w e a v in g is still p r a c ­
ticed , a n d asked th e artisan s th e r e w h y th is scalin g tr a n s f o r m a tio n was c re a te d .
^
|
T h e w e a v e r s r e p l i e d t h a t t h e y t h i n k o f t h e c o m p r e s s e d v e r s i o n as t h e o rig in a l p a t t e r n , a n d said th e y call it “s p r e a d i n g " w h e n t h e y c r e a t e t h e s t r e t c h e d ver-
\ s i o n . T h e r e a s o n th e y g a v e for t h e s p r e a d i n g p a t t e r n c a n b e s t b e u n d e r s t o o d w i t h
I
I
\
FIGURE 6. 2
K e n te c lo th
(a) In this traditional kente cloth design, stretched and compressed versions of the same pattern
appear. The weavers call this “spreading” the pattern, (b) Why are weavers spreading the pattern?
They say that our eyes give "heavy looks” to the face, and only “light looks” to the rest of the body.
This is what neurobiologists call “saccadic” eye movements. Unlike “tracking” eye movements,
which are continuous, saccadic movements are discrete and tend to leap about. Since kente cloth
was traditionally worn as a toga over the shoulder, the part near the face was given a compressed
pattern, and the part along the body a stretched pattern, to match the scaling of the saccadic eye
movements, (c) T h e compression of space is used in mathematics to model scaling patterns, like
chat of the saccadic eye movements. Mathematicians call this a “contractive affine transformation."
76
A fric a n , fra c ta l ?natliernatics
t h e f o llo w in g e x p e r i m e n t . H o l d y o u r finger i n f r o n t o f y o u r face, a n d w i t h o u t m o v ­
in g y o u r h e a d , t r a c k t h e fin g er w i t h y o u r eyes as yo u m o v e it s lo w ly a c ro s s t h e
v is u a l field. N o w try t h e s a m e t h i n g a g a in , s m o o t h l y t r a c k i n g t h e v is u a l field, b u t
w i t h o u t t h e fin g er t o g u id e y o u r eyes. Y o u ’ll find t h a t it c a n ’t b e d o n e ! Y o u r eye
m o v e s i n v o l u n t a r i l y in l i t t l e j u m p s , c a l l e d “sa c c a d ic ” m o v e m e n t s . W h e n a p e r ­
s o n c o m e s i n t o y o u r v is u a l field, t h o s e s a m e s a c c a d i c m o v e m e n t s d e n s e l y c o v e r
t h e face, a n d t h e n m a k e a few g la n c e s a t t h e ' b o d y (fig. 6 . 2 b ) . T h e w e a v e rs in B o n w ire r e p o r t e d t h e s a m e idea: “W h e n y o u se e a p e r s o n y o u g iv e h e a v y lo o k s t o t h e
face , a n d l i g h t lo o k s t o t h e b o d y . ” T h e y e x p l a i n e d t h a t t h e p u r p o s e o f t h e s c a l - \
in g c h a n g e is t o m a t c h t h i s v is u a l s c a l in g : t h e c o m p r e s s e d p a r t o f t h e p a t t e r n is \
t h e c l o t h w o r n o v e r t h e s h o u l d e r , a n d t h e s t r e t c h e d p a r t is w o r n d o w n t h e
I
l e n g t h o f t h e body.
T h e m a t h e m a t i c a l t e r m for t h i s o p e r a t i o n is “c o n t r a c t i v e affin e t r a n s f o r ­
m a t i o n ” (fig. 6 .2 c ), w h i c h c a n be used for c r e a t i n g j r a c t a l s t h r o u g h a m e t h o d c a l led
“ i t e r a t e d f u n c t i o n s y s te m s ” ( s e e W a h l 1 9 9 5 , 1 5 6 - 1 5 7 ) . I n k e n t e c l o t h t h e r e is
n o i t e r a t i o n — t h e o p e r a t i o n is d o n e o n l y o n c e — b u t i t d o e s s h o w a c t i v e t h i n k ­
in g a b o u t a s c a l in g t r a n s f o r m a t i o n . A s in t h e case o f t h e w i n d s c r e e n , t h e w eav ers
are t a k i n g a r a t h e r a b s t r a c t o b s e r v a t i o n a b o u t a t i m e - v a r y i n g q u a n t i t y a n d m a p ­
pin g this m o d el in to a m a te ria l design.
L o g a r i t h m i c s p ira ls
I n c h a p t e r 3 (fig. 3 . 2 ) w e e x a m i n e d t h e c o n t r a s t b e t w e e n n o n l i n e a r c o n c e n t r i c
c ir c le s a n d l i n e a r c o n c e n t r i c c irc le s . I n t h e s a m e way, n o n l i n e a r s p ir a ls a re easy
to u n d e r s t a n d if w e * c o n tia * t t h e m w i t h lin e a i^ s p ira ls (fig. 6 . 3 a ) . T h e l i n e a r s p i ­
ral, a lso c a lle d a n A r c h e m e d e a n sp iral in h o n o r o f t h e G r e e k m a t h e m a t i c i a n w h o
fa v o re d it, is i n j h e s h a p e o f a c o il e d r o p e o r w a t c h sp r in g . E a c h r e v o l u t i o n b rin g s
y o u o u t by t h e s a m e d i s t a n c e ( j u s t as e a c h la y e r in t h e l i n e a r c o n c e n t r i c c i r c l e
w a s t h e s a m e t h i c k n e s s ) . F o r t h a t re a s o n , a l i n e a r s p i r a l o f a fi n it e d i a m e t e r c a n
h a v e o n ly a fin ite n u m b e r o f tu r n s . A n o n l i n e a r sp iral o f fi n it e d i a m e t e r c a n h a v e
a n infinite n u m b e r o f turns, b eca u se e v e n th o u g h t h e r e is less a n d less sp ace r e m a i n ­
in g as o n e g o e s t o w a r d t h e c e n t e r , t h e d i s t a n c e b e t w e e n e a c h r e v o l u t i o n c a n g e t
sm aller a n d s m a lle r,.
A g o o d e x a m p l e o f t h i s n o n l i n e a r s c a l i n g c a n b e s e e n in t h e l o g a r i t h m i c
sp ira l (fig. 6 . 3 b ) . L o g a r i t h m i c sp ira ls a r e ty p i c a l s t r u c t u r e s in t w o d i f f e r e n t c a t ­
e g o rie s o f natu ra l^ p h e n o m e n a . O n t h e o n e J m n d , t h e y a r e f o u n d in a s t o n i s h i n g »
v a r i e t i e s o f o r gaja.Lc--gr.o_w.th. T h e o d o r e C o o k ’s T h e C u r v e o f L ife ( 1 9 1 4 ) , fo r
\
J
e x a m p le , s h o w s d oz ens o f lo g a r ith m ic spirals from ev e ry b r a n c h o f t h e e v o l u t io n a r y
tree : s n a i l a n d n a u t i l u s sh e lls; t h e h o r n s o f ra m s a n d a n t e l o p e ; a lg a e , p i n e c o n e s ,
/
Scaling
77
FIGURE
6 .3
S pirals
( a ) I n t h e l i n e a r s p i r a l o f A r c h i m e d e s , t h e r e is a
c o n s ta n t d ista n c e b e tw e e n e a c h rev o lu tio n .
O n ly a f i n i t e n u m b e r o f t u r n s c a n fit i n t h i s
finite sp a c e , (b) In t h e lo g a r ith m ic spiral, th e re
is a n i n c r e a s i n g d i s t a n c e b e t w e e n e a c h
r e v o l u t i o n . A n i n f i n i t e n u m b e r o f t u r n s c a n fit
i n t h i s f i n i t e sp a c e .
j a n d su n flo w e rs; a n d e v e n a n a t o m i c a l p a r t s o f t h e h u m a n e a r a n d h e a r t . M a n y
I
r e s e a r c h e r s h a v e s p e c u l a t e d o n w h y t h i s is so; t h e i r a n s w e r is ty p ic a lly t h a t l i v ­
ing sy s tem s n e e d to k e e p t h e sa m e p r o p o r t i o n s as th e y grow, .and so a scaling.cu.rve
a ll o w s t h e s a m e f o r m to be m a i n t a i n e d . I p r e f e r t o t h i n k o f it as r e c u r s i o n : if we
\ l o o k a t t h e c h a m b e r e d n a u t i l u s , fo r e x a m p l e , w e c a n t h i n k o f each, n e w c h a m ­
b e r as t h e n e x t i t e r a t i o n t h r o u g h t h e s a m e s c a l i n g a l g o r i t h m .
O n t h e o t h e r h a n d , l o g a r i t h m i c sp ira ls a re a ls o f o u n d in flu id t u r b u l e n c e .
W e b e c o m e a w a re o f th is w h e n we w a t c h a h u r r i c a n e fr o m sp ac e, o r sim p ly a d m i re
th e swirls o f w a t e r a l o n g a r i v e r b a n k . E x p l a n a t i o n s for th e s e fluid c u rv e s aye m u c h
less s p e c u l a t i v e , s i n c e w e can _w rite e q u a t i o n s fo r t u r b u l e n c e a n d s h o w t h e m p r o ­
d u c i n g l o g a r i t h m i c s p irals in c o m p u t e r s i m u l a t i o n s (as w e w ill see in c h a p t e r 7).
B u t t h e E u r o - A m e r i c a n t r a d i t i o n is n o t t h e o n ly o n e i n t e r e s t e d in sim u la c ra . T h e
a r t is t s o f w h a t is n o w G h a n a — p a r t i c u l a r l y t h o s e o f t h e A k a n s o c i e ty — lo n g ag o
/
a b s t r a c t e d t h e l o g a r i t h m i c sp iral for p re c i s e l y t h e s e tw o c a te g o r ie s . T h e i r sym -\
b o ls for t h e life fo rce (fig. 6 .4 a ) a re c le a rly r e l a te d to t h e " c u r v e s o f life,” a n d icons
\
for T a n u , t h e r i v e r g o d (fig. 6 . 4 b ) , s h o w t h e l o g a r i t h m i c sw irls o f t u r b u l e n c e .
A fr ic a n fra c ta l m athem atics
78
a
b
FIGURE
6 .4
L o g a r i t h m i c spirals
(a)
S e v e r a l G h a n a i a n i c o n i c figure s, s u c h as t h i s g o l d w e i g h t , l i n k a s p i r i t u a l f o r c e w i t h t h e
s t r u c t u r e o f l i v i n g s y s t e m s t h r o u g h l o g a r i t h m i c s p i r a l s . T h i s e x a m p l e is p a r t i c u l a r l y s t r i k i n g s i n c e
it s h o w s h o w s p i r a l s c a n be c o m b i n e d w i t h b i l a t e r a l s y m m e t r y t o c r e a t e o t h e r s e l f - s i m i l a r s h a p e s
( t h e l a r g e d i a m o n d s h a p e c r e a t e d b y t h e m e e t i n g o f t h e la r g e s p i r a l a r m s is r e p e a t e d o n e i t h e r sid e
b y t h e s m a l l d i a m o n d a t t h e m e e t i n g o f t h e s m a l l s p i r a l a r m s ) , ( b ) T h i s figure, a g a i n b a s e d o n
l o g a r ith m ic spirals, a p p e a r s o n t h e t e m p le s o f T o n u , t h e r iv e r g o d , a n d lin k s th is s p ir itu a l fo rc e to
t h e g e o m e t r i c s t r u c t u r e o f flu id t u r b u l e n c e .
( a , pho to courtesy D o r a n Ross.)
^
A g a i n , we n e e d to a v o i d t h e a s s u m p t i o n t h a t t h e G h a n a i a n lo g s p i r a ls a r e ^
' s i m p l y m i m e t i c “r e f l e c t i o n s ” o f n a t u r e , a n d e x a m i n e h o w ..they, a r e used. .a n d
<
<
' d e s i g n e d . T h e A k a n a n d o t h e r s o c i e t i e s o f G h a n a c r e a t e d a c o l l e c t i o n o f specific
icons t h a t several resea rch ers h a v e c o m p a r e d to a w r it te n language . B u t r a t h e r t h a n
c o m p o s e d o f t h e v a s t n u m b e r o f s y m b o ls we c a ll “w o rd s ,” t h e G h a n a i a n s y m b o li c
v o c a b u l a r y is m u c h s m a lle r, a n d e a c h s y m b o l refers n o t t o a s in g le w o r d b u t a n
e n t i r e s o c ia l, re l ig i o u s ,.o r p h i l o s o p h i c a l c o n c e p t . M o r e o v e r , in m a n y c a s e s t h e
s t r u c t u r e o f t h e s y m b o l is n o t a r b i t r a r y (as G r e g o r y B a t e s o n said, “T h e r e is n o t h ­
in g ‘s e v e n i s h ’ a b o u t t h e n u m e r a l 7 "), b u t r a t h e r is s h a p e d so t h a t e a c h ic o n 's g e o ­
m e t r i c s t r u c t u r e r e c a lls t h e c o n c e p t it r e p r e s e n t s . In o c h e r w o rd s , t h e y a r e n o t
o n ly a b s tra c tio n s in t h e sense o f b e i n g stylized, b u t also g e n e ra liz a tio n s in t h e sense
o f t h e d e s i g n e r s ’ i n t e n t t o find a n u n d e r l y i n g s t r u c t u r e t h a t all e x a m p l e s h a v e in
c o m m o n . F o r th is r e a s o n w e c a n a c c u r a t e l y d e s c r ib e t h e G h a n a i a n log sp iral .icons
as g e o m e t r i c m o d e l s f o r .th e p h e n o m e n a o f o r g a n i c g r o w t h a n d fluid t u r b u l e n c e .
S o m e a s p e c ts o f t h e s e d e s i g n s illu s tr a r e a c o n s c i o u s r e f l e c t i o n o n t h e i r g e o ­
m e tric pro pertie s. Figure 6 .4 a, for e x a m p l e , n o t o n ly displays t h e log s p ira l’s Euclid-
Scaling
79
ear* s y m m e t r y — for we c a n see h o w c l o c k w i s e a n d c o u n t e r c l o c k w i s e spirals c o m ­
p a r e — b u t a ls o e x p e r i m e n t s w i t h o t h e r k i n d s o f s c a l i n g s y m m e t r y : n o t e t h a t t h e
large d i a m o n d s h a p e c r e a t e d by t h e m e e t i n g o f t h e la r g e s p ira l a r m s is r e p e a t e d
’**>v
o n b o t h s id e s by t h e s m a ll d i a m o n d a t t h e ^ m e e t i n g o f t h e s m a ll s p ira l a rm s . C a n
th is s c a l in g b e c o n t i n u e d in f u r t h e r i t e r a t i o n s ? 1 w ill l e a v e t h a t q u e s t i o n as a n
e x e r c i s e fo r t h e read ers.
' T h e r e are h in t s th a t th e p r e c o lo n ia l G h a n a i a n desig n ers w ere h e a d e d
t o w a r d a q u a n t i t a t i v e a p p r o a c h i n t h e i r lo g s p i r a l d e s i g n s . F i g u r e 6 . 5 a s h o w s
t h e s c u l p t u r e o f a w a t e r b u f f a l o in w h i c h t h e y h a v e i n s c r i b e d u n i f o r m d i s c r e t e
s te p s . 1 d o n ’t t h i n k t h i s w as m o t i v a t e d b y n u m e r i c m e a s u r e s , b u t r a t h e r t h e
r e v e r se . By c u t t i n g t h e s e s te p s w e c a n c l e a r l y g a u g e t h e n o n l i n e a r n a t u r e o f t h e
spiral— th e w ay steps o f a c o n s t a n t i n c r e m e n t sh o w a n in c re a sin g a m o u n t of
c u r v e g e n e r a t e d — a n d t h i s p r a c t i c e c o u l d h a v e le d t o q u a n t i t a t i v e m e a s u r e s .
A n o t h e r m o v e in t h a t d i r e c t i o n w o u l d g e n e r a l i z e s u c h d i s c r e t i z e d l o g a r i t h m i c
f ig u r e
6.5
L o g a rith m ic sc a lin g
in Q h a n a ia n d e sig n s
(a) L o g a r i t h m i c s c a l i n g c a n b e d e m o n s t r a t e d
in a t h r e e - d i m e n s i o n a l c u r v e b y s h o w i n g
how d isc re te step s o f th e sam e vertical
in c re m e n t lead to rap id ly in c re a sin g area.
(b) O v e r h e a d v i e w o f p y r a m i d - s h a p e d
goldw eight. (c ) L o g a r ith m ic p lo t of
goldw eight tria n g le le n g th s.
(a, p h o i o f r o m the M e t r o p o l i t a n M u s e u m o f A r t .
b, photo
co urte sy G e o r g e A rt h u r, M a rsh a ll
‘
University.,)
no
e
-a ■
£
step o f pyram id
FIGURE 6 . 6
A d a p t i v e s c a l i n g - w i t h tr ia n g le s
(a) A n t e l o p e h e a d d r e s s c r e a t e d b y t h e K r u m b a o f B u r k i n a F a s o , ( b ) M a s k s o l d in A c c r a , G h a n a ,
'
b a s e d o n d e s i g n u se d in t h e S n k a r a - B m m o u r e l i g i o u s d a n c e s - ( c ) R e p r e s e n t a t i o n o f t h e w a t e r s p i r i t
c r e a t e d b y t h e B a g a o f G u i n e a , ( d ) S c u l p t u r e fr o m t h e C o n g o , ( e ) A K i k u y u w o o d e n s h i e l d .
T h e w o o d h a s a n o n l i n e a r c u r v e t o w a r d t h e c e n t e r , a n d t h e t r i a n g l e s a re s c a l e d t o m a t c h .
(a,
courtesy Musee
M u s e u m o f A r t . c,
<!e I ' H o m m e . c , M e t r o p o l i t a n M u s e u m o f A r t ; />liofo b y E l io t E l i s n / o n . d , D e t r o i t
British
M u s e u m ; fr o m Z aslavsky 1 9 7 3 .)
Scaling
s c a l i n g to fo r m s o t h e r t h a n s p ira ls , a n d t h a t d i d i n d e e d o c c u r , as w e c a n s e e in
figure 6 .5b , o n e o f t h e A k a n g o ld w e ig h ts . A p l o t o f t h e l e n g t h o f th e s e tr ia n g l e s
(fig. 6 . 5 c ) i n d i c a t e s t h a t r e a s o n a b l e a c c u r a c y w as a c h i e v e d in t h i s i n d i g e n o u s
lo g arith m ic_ scajin g p ra c tic e .
'
A d a p tiv e sc a lin g
S o far this c h a p t e r h a s focused o n q u e s tio n s o f intentional.ity, precision, a n d m a t h e ­
m a t i c a l r e a s o n i n g in A f r i c a n s c a l i n g d e s i g n s . A d a p t i v e s c a l i n g h a s l i t t l e m a t h e ­
m a t ic a l s o p h i s ti c a ti o n , b u t it to o is a n i m p o r t a n t p a r t o f t h e A f r i c a n fra c ta l de sign
\ t h e m e . By a d a p t i n g t h e s c a l e o f a p a t t e r n t o fit v a r i o u s fo rm s, a n u m b e r o i
{ e s t h e t i c a n d p r a c t i c a l effects c a n b e a c h i e v e d . T h e s e e x a m p l e s fall i n t o tw o c a t-^
egories. In c o n f o r m a l m a p p in g , t h e p a t t e r n sim p ly fits alo n g t h e c o n t o u r s o f a c o n ­
c r e t e , p r e e x i s t i n g s t r u c t u r e . I n g l o b a l m a p p i n g , t h e p a t t e r n is d i s t o r t e d by
c o m p r e s s i o n o r e x p a n s i o n — as w e sa w h a p p e n a l o n g o n e d i m e n s i o n in k e n t e
c l o t h — a j x o r d i n g t q a m o r e u n i v e r s a l, a b s t r a c t t r a n s f o r m a t i o n .
F ig u re 6 .6 s h o w s s e v e r a l e x a m p l e s o f c o n f o r m a l m a p p i n g o n t r ia n g l e s . .My
s e a r c h o f t h e f a c ia l m a r k i n g s o f a n t e l o p e o f t h e w e s t e r n S u d a n d id n o t t u r n u p
a n y t h i n g lik e t h e s c a l i n g p a t t e r n o f fig u r e 6 .6 a ; t h e s e t r i a n g l e s a r e d e c o r a t i v e
a d d itio n s , sized to fit in to t h e s h a p e o f t h e s c u l p tu re . O t h e r e x a m p le s (fig. 6 . 6 b - e )
s h o w a s e r ie s o f t r i a n g l e s c o n f o r m i n g t o t h e s c a l i n g c o n t o u r s o f a m a s k , a s i n ­
u o u s c u r v e , a c a r v e d h u m a n fig u re , a n d a s h i e l d . F i g u r e 6 . 7 a s h o w s c o n f o r m a l
m a p p i n g in t h e h a i r s t y l e A m e r i c a n s c a ll “c o r n - r o w i n g ”; its s i m u l a t i o n is s h o w n
in fig u re 6 . 7 b . T h e Y o ru b a n a m e f o r t h i s s t y l e is ipako e le d e , w h i c h m e a n s t h e
n a p e o f t h e n e c k ofTi_boar— b e c a u s e t h e b o a r ’s b ristle s s h o w a s i m i l a r n o n l i n e a r
s c a l in g . F ig u r e 6 . 7 c s h o w s a h a i r s t y l e t h a t c o m b i n e s c o n f o r m a l m a p p i n g w i t h
i t e r a t i o n . A d a p t i v e s c a l i n g o f c i r c l e s c a n b e s e e n in t h e S e n e g a l e s e t e x t i l e in
fig u re 6 . yd.
A p r a c t i c a l a p p l i c a t i o n o f c o n f o r m a l m a p p i n g a p p e a r s in figure 6 . ye, a n
a e r i a l p h o t o o f t h e N k o n g - m o n d o q u a r t e r i n t h e c i t y o f E d e a in s o u t h e r n
C a m e r o o m j v h e r e w e s e e a s c a l i n g s e r ie s o f h o u s e s . A s e x p l a i n e d t o m e by o n e
m e m b e r o f t h e n e ig h b o r h o o d , Mr. S o ss o , t h e h o u s e s w ere c o n s t r u c t e d a lo n g a n a r ­
ro w in g rid g e , a n d t h e s c a l i n g w as s i m p l y c o n f o r m i n g to t h e n a t u r a l la n d s c a p e .
H o w e v e r , t h e o l d e s t i n h a b i t a n t o f t h i s B assa n e i g h b o r h o o d , M r. B e l l m b o c k ,
to ld m e t h a t t h e p a t t e r n w as c r e a t e d b e c a u s e p e o p l e w a n t e d neighbors.o(_a..sim ­
ilar e c o n o m i c class n e x t d o o r , so t h a t t h e r a n g e in h o u s e size re f le c te d a n e c o ­
n o m i c g r a d i e n t , fr o m p o o r e s p t p _ w e a l t h i e s t . M r. B e l l m b o c k liv e d in t h e s m a ll e s t]
h o u s e , a n d Mr. S o s s o in t h e la r g e s t, s o I w o u ld n o t d i s c o u n t t h e p o s s ib ility t h a t /
t h e r e was a n e c o n o m i c s c a l i n g as w ell.
81
FIGURE 6 . 7
A d a p tiv e s c a lin g b a s e d on v a r io u s s h a p e s
(a, b ) A Y o r u b a h a ir sty le , l p a k o E le d e , a d a p t s th e s c a lin g o f th e bra id s to t h e n o n l i n e a r c o n t o u r s o f
t h e h e a d , (c) T h i s h a ir s t y le b e g in s b y b r a id in g a sm a ll h o r s e s h o e sh a p e in t h e 't o p c e n t e r , a n d th e n
t r a c in g th e c o n t o u r in in c r e a s in g p e r im e te r s — a c o m b i n a t i o n o f a d a p t iv e s c a l i n g a n d ite r a t io n .
( d ) F i t t i n g c i r c le s b e t w e e n i n t e r s e c t i n g c u r v e s c r e a t e s a s c a l i n g s e r ie s in t h is t e x t i l e d e s i g n fro m
G u i n e a , ( e ) A n a e r ia l p h o t o o f t h e N k o n g - m o n d o q u a r t e r in t h e c i t y o f E d e a in s o u t h e r n
C a m e r o o n , w h e re w e see a sc a lin g series o f houses,
f a , fr o m Sagay 1 9 8 3 . c , fr o m Sagay 1 9 8 3 . d , p h o to c o u rt e s y I F A N , D a k a r .)
Scaling
+ infinity
- in f in it y
FIGURE 6 . 8
M a p p in g f r o m th e p la n e
to a s p h e r ic a l s u r fa c e
(a) M apping bars of infinite
length from the plane to a sphere,
( l i ) A Y o i n h a h a i r s t y l e , Koioba
(“hurker”) ......................... ......
(b,/nmi Sagay j 983.)
It is p o s s ib le t o m i s r e a d th e s e e x a m p l e s o f c o n f o r m a l m a p p i n g as b e i n g th e
p r o d u c t o f a r t i s a n s w h o a r e s t r o n g l y g u i d e d by c o n c r e t e f o r m s r a t h e r t h a n
a b s tr a c t th o u g h t . B u t a d a p tiv e sc a lin g c a n a ls o b e s e e n in m o re a b s tra c t
e x a m p le s : g lo b a l t r a n s f o r m a t i o n s in w h i c h s p a c e it s e l f is d is t o r t e d . T h i s is a c o m ­
m o n o p e ra tio n in W e s te rn geom etry, th e m o st fre q u e n t e x a m p le b e in g a m a p ­
p i n g b e t w e e n t h e p l a n e a n d a s p h e r e (fig. 6 . 8 a ) . F ig u r e 6 . 8 b s h o w s a h a i r s t y l e
t h a t a p p e a r s t o h a v e a p l a n a r d e s i g n m a p p e d o n t o a s p h e r i c a l su rfac e. F ig u re 6 .9
p ro v id e s a n e v e n m o r e a b s t r a c t illu s tratio n , th e in ve rse o f t h e p re v io u s m a p p i n g —
n o w g o in g fr o m s p h e r i c a l to r e c t a n g u l a r — a n d u tiliz in g t h r e e d i m e n s i o n s in s te a d
83
A fr ic a n fra c ta l m athem atics
84
o f t w o . I n t h i s C h o k w e s c u l p t u r e , t h e e n t i r e h u m a n fig u re is d i s t o r t e d as if its
sp h erical v o lu m e h a d b e e n m a p p e d to a c u b ic v o lu m e; th e re s u ltin g n o n lin e a r
s c a l i n g is d r a m a t i c a l l y i l l u s t r a t e d by t h e d i s c r e t e s t e p s in t h e h e a d d r e s s . A r t h i s ­
to r ia n W illia m Fagg (1 9 5 5 ) m a d e a sim ila r su g g e stio n a b o u t o t h e r A fric a n
d e s ig n s , w h i c h h e c o m p a r e d to t h e d r a w i n g s o f n a t u r a l g r o w t h by b i o l o g i s t
D ’A r c y T h o m p s o n : “I b e l i e v e t h a t t h e m o r p h o l o g y o f A f r i c a n s c u l p t u r e m a y b e
usefully s t u d ie d . . . by r e f e r e n c e to m a t h e m a t i c s . . . . F o r e x a m p l e in c e r t a i n m asks
• , FIGURE 6 .9
M a p p in g fr o m a s p h e r ic a l v o lu m e
to a r e c ta n g u la r , v o lu m e
(a ) B astin (1 9 9 2 , 6 8 ) sh o w s th a t th is C h o k w e cro w n ,
t h e C i p e n y a - M u t w e , is m a d e u p o f l i n e a r b a n d s i n r e a l
life. T h e n o n l i n e a r s c a l i n g w e s e e in t h i s s c u l p t u r e c a n
b e e x p l a i n e d as t h e i n v e r s e o f t h e t r a n s f o r m a t i o n w e saw
i n figure 6 .8 a . R a t h e r t h a n s h r i n k i n g a s w e m o v e fro m
th e c e n te r to th e m arg in s, th e in v erse m a p p in g causes
e x p a n s i o n f r o m c e n t e r t o m a r g i n s . T h i s is n o t o n l y t h e
in v e rs e o f t h e p re v io u s m a p p in g , b u t also o p e ra te s o n
th re e -d im e n s io n a l v o lu m e ra th e r th a n surface. S im ilar
t r a n s f o r m a t i o n s a r e u s e d in n e u r o s c i e n c e t o m o d e ! t h e
ways th a t ta ctile receptors are m a p p e d from bo d y to
^
b r a i n , s i n c e t h e r e is a t n u c h g r e a t e r d e n s i t y o f s e n s o r y
n e u r o n s a t t h e e x t r e m e s , ( b ) T h e r e a s o n f o r t h i s t r a n s f o r m a t i o n is t o i n v o k e t h e i m p r e s s i o n o f
p o w e r a n d s t a b ilit y ( C h a n d a
1 9 9 3 ). T h e m e a n in g h a s n o t h in g in p a r t ic u la r to d o w it h g e o m e t r ic
m a p p i n g , o t h e r t h a n a c h i e v i n g t h e d e s ir e d e f fe c t , b u t it is i n t e r e s t i n g t o n o t e t h a t t h e t r a n s f o r ­
m a t i o n is u n i f o r m l y a p p l i e d t o n l! e x t e r n a l a r e a s , e v e n t o t h e e x t e n t o f d e f o r m i n g t h e f o r e h e a d .
( a , c o u r t e s y J a c q u e s K e r c h a c h e a n d M u s e u m o f M a n k i n d , L o n d o n , b , c o u r t e s y M u s e u m o f the P h i l a d e l p h i a
C ivic C e n te r .)
Scaling
f o r t h e G e l d e s o c i e t y t h e n a t u r a l . . . p h y s i o g n o m y is ‘b l o w n u p , ’ so t o s p e a k , in
a w ay w h ic h c o u ld b e p lo tte d o n a set o f flarin g e x p o n e n ti a l c o o rd in a te s .”
( 1 9 1 7 . 4 3 )-
C o n c lu sio n
T h e e x a m p l e s o f s c a l i n g d e s i g n s in t h i s c h a p t e r v a ry g r e a t l y in p u r p o s e , p a t t e r n ,
a n d m e t h o d . W h i l e it is n o t difficult to i n v e n t e x p l a n a t i o n s b a sed o n u n c o n s c i o u s
s o c ia l fo r c e s— fo r e x a m p l e , t h e f l e x ib ility in c o n f o r m i n g d e s i g n s to m a t e r i a l s u r ­
fa c e s as e x p r e s s i o n s o f so c ia l f l e x ib ility — I d o n o t t h i n k t h a t a n y s u c h e x p l a n a ­
t i o n c a n a c c o u n t fo r t h i s d iv e rsity . F r o m o p t i m i z a t i o n e n g i n e e r i n g , t o m o d e l i n g
o r g a n i c life, t o m a p p i n g b e t w e e n d if fe re n t s p a tia l s tru c tu r e s , A f r i c a n a rtis a n s h a v e
d e v e l o p e d a w id e r a n g e o f t o o l s , t e c h n i q u e s , a n d d e s i g n p r a c t i c e s b a s e d o n t h e
c o n s c i o u s a p p l i c a t i o n o f s c a l in g g e o m e t r y . I n t h e n e x t c h a p t e r , w e w ill see t h a t
A f r i c a n n u m e r i c s y s te m s a ls o s h a r e m a n y f r a c t a l c h a r a c t e r i s t i c s .
85
C H A P T E R
N um eric
-systems-
S o far vve h a v e fo cu s ed o n g e o m e t r i c s t r u c t u r e s r a t h e r t h a n n u m e r i c sy stem s. T h e
o n l y e x c e p t i o n w as in t h e w i n d s c r e e n , w h e r e t h e n o n l i n e a r s c a l i n g w a s c r e a t e d
by c o u n t i n g a s p e c ific s e q u e n c e o f d i a g o n a l s t r a w ro w s . B u t t h e r e a r e m a n y
o t h e r i n s t a n c e s in w h i c h t h e A f r i c a n a p p r o a c h t o f r a c t a l g e o m e t r y m a k e s use o f
num bers* '
...................... ...........................
N o n lin e a r a d d itiv e se rie s in A f r i c a
T h e c o u n t i n g n u m b e r s ( 1 ,2 ,3 • • •) c a n b e t h o u g h t o f as a k i n d o f i t e r a t i o n , b u t
o n l y in t h e m o s t tr iv ia l w a y .1 I t is t r u e t h a t w e c o u l d p r o d u c e t h e c o u n t i n g n u m ­
b e r s fr o m a r e c u r s i v e lo o p , t h a t is, a f u n c t i o n in w h i c h t h e o u t p u t a t o n e sta g e
b e c o m e s t h e i n p u t for t h e n e x t : X n+( = X n + 1 . B u t t h i s is a s t r i c t l y l i n e a r series,
i n c r e a s i n g by t h e s a m e a m o u n t e a c h t i m e — t h e n u m e r i c j e q u i v a l e n t o f w h a t we
sa w in t h e li n e a r c o n c e n t r i c c ir c le a n d l i n e a r s p i r a l.'- A d d i tio n e a n , h o w e v e r , prod u c e n o n l i n e a r se r ie s ,2__and t h e r e a r e a t l e a s t tw o e x a m p l e s o f n o n l i n e a r a d d i t i v e
se ries in A f r i c a n c u l t u r e s . T h e t r i a n g u l a r n u m b e r s ( 1 , 3 , 6 , 1 0 , 1 5 . . .) a r e u se d in
a g a m e c a l l e d " t a r u m b e t a " in e a s t A f r i c a (Z a s la v s k y j 9 7 3 , 1 1 1). F ig u r e 7.1 s h o w s
h o w th e s e n u m b e r s a re d e r i v e d from t h e s h a p e o f t r i a n g l e s o f in c r e a s i n g size, a n d
h o w t h e n u m e r i c series c a n b e c r e a t e d by a r e c u r s i v e lo o p . A s in t h e c a s e o f c er-
N u m e ric systems
87
num ber
o f s to n e s:
num ber
o f it e r a t io n s :
A g a m e c a l l e d ‘' t a r u m b e t a " i n E a s t A f r i c a m a k e s u s e o f t h e t r i a n g u l a r n u m b e r s , s t a r t i n g w i t h 3
( 3 , 6 , 1 0 , 15
In th is g a m e , o n e p la y e r calls o u t a c o u n t as h e re m o v e s s to n e s co n se c u tiv e ly ,
l e ft t o r i g h t a n d b o t t o m t o t o p , w h i l e t h e o t h e r p l a y e r , w i t h h i s b a c k t u r n e d , m u s t s i g n a l w h e n e v e r
t h e f ir st s t o n e i n a r o w h a s b e e n r e m o v e d .
T h e s t o n e s i n e a c h t r i a n g u l a r a r r a y c a n b e b u i l t u p i n a n i t e r a t i v e f a s h i o n , t h a t is, t h e n e x t
tria n g le c a n be c r e a t e d by a d d in g a n o t h e r la y e r to a n y s id e o f t h e p r e v io u s tria n g le . T h e n u m b e r to
b e a d d e d i n e a c h a d d i t i o n a l l a y e r is s i m p l y t h e n u m b e r o f i t e r a t i o n s . F o r e a c h i t e r a t i o n i, a n d t o t a l
n u m b er o f sto n es N , we have:
Ni*i = N , + 1
1 =
+.1
3 =
+ 2
“ '6 =
+ 3
10
=
+ 4
15 =
10 + 5
(sta rtin g w ith N 0 = 0 )
(a triv ia l array, n o t u s e d in t h e g a m e )
I n o t h e r w o r d s , t h e n e x t n u m b e r w i l l b e g iv e n
b y t h e la s t n u m b e r p lu s t h e it e r a t io n c o u n t :
FIGURE 7.1
T h e tr ia n g u la r n u m b e r s in a n E a s t A fr ic a n g a m e
t a i n f o r m a l a g e - g r a d e i n i t i a t i o n p r a c t i c e s (s e e c h a p t e r s 5 a n d 8 ) , che sim p le
ve rsion s a re u sed by s m a l l e r c h i l d r e n , a n d t h e h i g h e r i t e r a t i o n s a re p i c k e d u p w ith
increasing age. W h i l e th e r e is n o in d i c a ti o n o f a form al r e l a ti o n s h ip in this in stance,
t h e r e is s till a n u n d e r l y i n g p a r a l l e l b e t w e e n t h e .ite ra tiv e c o n c e p t o f a g i n g c o m ­
m o n t o m a n y A f r i c a n c u h u r e s — e a c h i n d i v i d u a l p a s s in g t h r o u g h m u l t i p l e t u r n s
o f t h e “h f e c ycle.*.’.— a n d t h e k e r a t i v e n a t u r e o f t h e t r i a n g u l a r n u m b e r series.
A n o t h e r n o n l i n e a r a d d it iv e series was fo u n d in a r c h a e o l o g ic a l e v id e n c e from
N o r t h A fric a . B ada w y ( 1 9 6 5 ) n o t e d w h a t a p p e a rs t o .b e use o f t h e F ib o n a c c i series
in t h e la y o u t o f t h e t e m p l e s o f a n c i e n t E gypt. U s i n g a s l ig h t ly d if fe re n t a p p r o a c h ,
T h e Fibonacci series
.(I, 1 , 2 , 3 , 5 , 8 , 13 . . . ) was
found by Badawy (1965)
in his study o f the layout
o f th e temples of Egypt.
H is analysis was quite
complex, but it ismot
difficult to create a simple
visualization. Here we see
th e series in the successive
c ham be rs of the tem ple of
Karnak.
T h e Fibonacci series is
p roduced by adding the
previous n u m b e r to the
curren t n u m b e r to get the
n ex t number, starting with
1 + 1 = 2 . For each iteration
i, th e n u m b er N in th e series
is given by:
G ray rectangles added
fo r m easurem ent 9
N i +1 = N i
t h a t is,
N n e x t = H cu rten t + ^previous
1 + 1 =
2
1 + 2 =
3
2 + 3 =
5
=
8
3 + 5
5 + 8 = 13
next
N previous
current
FIGURE 7 . 2
T h e F ib o n a c c i series in a n c ie n t E g y p t
N um eric systems
I f o u n d a v is u a lly d i s t i n c t e x a m p l e o f t h i s se r ie s in t h e s u c c e s s iv e c h a m b e r s o f
t h e t e m p l e o f K a r n a k , as s h o w n in figure 7 .2 a. F ig u re 7 .2 b s h o w s h o w th e s e n u m ­
b e r s c a n b e g e n e r a t e d u s i n g a r e c u r s i v e lo o p . T h i s f o r m a l s c a l in g p l a n m a y h a v e
•
b e e n d e r i v e d f r o m t h e n o n n u m e r i c v e r s i o n s o f s c a l i n g a r c h i t e c t u r e w e see
t h r o u g h o u t A f r i c a . A n a n c i e n t s e t o f b a l a n c e w e ig h ts , a p p a r e n t l y u se d in E gypt,
S y ria, a n d P a l e s t i n e c irca 1200 b .c .e ., also a p p e a r to e m p lo y a F ib o n a c c i s e q u e n c e
( P e t r u s o 1 9 8 5 ) . T h i s is a p a r t i c u l a r l y i n t e r e s t i n g use, s i n c e o n e o f t h e s t r i k i n g
m a t h e m a t i c a l p r o p e r t i e s o f t h e se r ie s is t h a t o n e c a n c r e a t e a n y p o s i t i v e in t e g e r
t h r o u g h a d d i t i o n o f s e l e c t e d m e m b e r s — a p r o p e r t y t h a t m a k e s it id e al for a p p l i ­
c a t i o n to b a l a n c e m e a s u r e m e n t s ( H o g g a t t 1 9 6 9 , 7 6 ) . T h e r e is n o e v i d e n c e t h a t
a n c i e n t G r e e k m a t h e m a t i c i a n s k n e w o f t h e F i b o n a c c i series. T h e r e w as u se o f
t h e F i b o n a c c i se rie s in M i n o a n d e s i g n , b u t P re zio si ( 1 9 6 8 ) c i t e s e v i d e n c e i n d i ­
c a t i n g t h a t it c o u l d h a v e b e e n b r o u g h t f r o m E g y p t by M i n o a n a r c h i t e c t u r a l
w orkers e m p lo y e d a t K ahun.
D o u b lin g se ries in A f r i c a
S o m e a c c o u n t s r e p o r t t h a t A f r i c a n s u se a “ p r i m i t i v e ” n u m b e r s y s te m in w h i c h
th e y c o u n t b y m u l t i p l e s o f tw o . It is tr u e t h a t m a n y c ase s o f A f r i c a n a r i t h m e t i c
a re b a s e d o n m u l t i p l e s o f tw o , b u t as w e w ill s e e , b a s e - 2 s y s te m s a re n o t c r u d e
a r t if a c ts fr o m a f o r g o t t e n past. T h e y h a v e s u r p r is in g m a t h e m a t i c a l s ig n if i c a n c e ,
n o t o n ly in r e l a t i o n t o A f r i c a n fractals, b u t to t h e W e s t e r n h is to ry o f m a t h e m a t i c s
a n d c o m p u t i n g as well.
T h e p re s e n c e o f d o u b lin g as a c u ltu ral t h e m e occurs in m a n y different A fric a n
so cieties a n d in m a n y d ifferen t social d o m a i n s , c o n n e c t i n g th e sa c re d n e ss o f tw ins,
' s p i r i i 'd o u b i e s , a n d d o u b l e v is io n w i t h m a t e r i a l o b j e c t s , s u c h a s t h e b l a c k s m i t h 's
tw in b e llo w s a n d t h e d o u b l e i r o n h o e g i v e n in b r i d e w e a l t h (fig. 7 .3 ). F ig ure 7 .4 a
s h o w s t h e I s h a n g o b o n e , w h i c h is a r o u n d 8 , 0 0 0 y e a r s o ld a n d a p p e a r s t o s h o w a
d o u b l i n g s e q u e n c e . j D o u b l i n g is f u n d a m e n t a l t o m a n y o f th e c o u n t i n g s y s te m s o f
A f r i c a in m o d e r n t i m e s as well. It is c o m m o n , fo r e x a m p l e , t o h a v e t h e w o rd for
. .* a n ' e v e n n u m b e r 2 N m e a n “N p lu s N ” (e .g ., t h e n u m b e r 8 in t h e S h a m b a a l a n ­
guage o f T a n z a n i a is “n e n a n e , ” literally “fo u r a n d f o u r ”). A s im ilar d o u b li n g takes
p lac e fo r t h e p re c i s e l y a r t i c u l a t e d s y s te m o f n u m b e r h a n d g e s tu re s (fig. 7 . 4 b ) , for
ex am ple, Ilfour” re p r e s e n te d by t w o gro ups o f tw o fingers, a n d “e i g h t ’’ by tw o g roups
o f four. P e t i t t o ( 1 9 8 2 ) f o u n d t h a t d o u b l i n g w as u s e d in m u l t i p l i c a t i o n a n d
d iv i s i o n t e c h n i q u e s in W e s t A f r i c a (fig. 7 . 4 c ) . G i l l i n g s ( 1 9 7 2 ) d e t a i l s t h e p e r ­
s i s t e n t use o f p o w e r s o f t w o in a n c i e n t E g y p t i a n m a t h e m a t i c s as w e l l , a n d
Z aslav sky ( 1 9 7 3 ) s h o w s a r c h a e o l o g i c a l e v i d e n c e s u g g e s tin g t h a t a n c i e n t E g y p t ’s
use o f b a s e - 2 c a l c u l a t i o n s d e r i v e d f r o m t h e use o f b a s e - 2 in s u b - S a h a r a n A f r i c a .
A frican fractal m athematics
90
D o u b l i n g practices; w e r e a ls o u se d by A f r i c a n d e s c e n d a n t s in cine A m e r i ­
cas. B e n j a m i n B a n n e k e r , for e x a m p l e , m a d e u n u s u a l u se o f d o u h l i n g in b is c a lc u l a t t o n s , w h i c h m a y h a v e d e r i v e d fr o m t h e t e a c h i n g s o f h i s A fric a iV T a th e r a n d
g r a n d f a t h e r ( E g la s h 1 9 9 7 c ) . G a t e s ( 1 9 8 8 ) e x a m i n e d t h e c u l t u r a l s i g n i f i c a n c e o f
d ou b.ling in W e s t A f r i c a n re l ig i o n s s u c h as v o d u n a n d its t r a n s f e r t o “v o o d o o ”
in t h e A m e r i c a s . In t h e r e l i g i o n o f S h a n g o , for e x a m p l e , t h e v o d u n g o d o f t h u n ­
d e r a n d l i g h t n i n g is r e p r e s e n t e d by a d o u b l e - b l a d e d a x e (fig. 7 . 5 a ) , u s e d by
S h a n g o d e v o te e s in t h e n e w w o rld as well (R. T h o m p s o n 198,3). F igure 7.51-) show s
figure 7 .3
D o u b lin g in A fr ic a n so c ia l p r a c tic e s
( a ) T h i s f i g u r e is u s e d b y w o m e n i n G h a n a t o e n c o u r a g e t h e
b i r t h o f t w i n s , ( b ) A d o u b l e i r o n h o e is s o m e t i m e s u s e d a s p a r t
o f th e b rid e p rice c ere m o n y , (c) T h e d o u b le b ello w s o f th e
b la c k s m ith , (d) D o u b le v is io n ; a c o m m o n th e m e .in sev eral
A fric a n sp iritu a l p ra c tic es, o fte n im p ly in g th a t o n e c a n see
b o t h t h e m a te r ia l w o rld a n d t h e s p irit w orld.
( b , M a r c a n d E v e l y n B e r n h e i m f r o m R a / d io G i u / l i o n c u e ; c o u r t e s y
o f U g a n d a N a t io n a l M u se u m ,
c , Jihoio c o u r t e s y I F A N , D a k a r .
d , fro n B e rjo n n e a u a n d Som tery 1 9 8 7 .)
N um eric systems
9i
(a) T h e Ishango bone, estimated to be over 8, 000 years old, shows
wh at appears to be use of doubling: 3 + 3 = 6, 4 + 4 = 8, 10 = 5 + 5 .
10
(b) Even numbers are
typically represented by
doubli ng in t h e precisely
articulated system of
Afri can h a n d gestures.
(c)
D o u b l i n g w a s t r a d i t i o n a l l y u s e d b y t a i l o r s in W e s t A f r i c a w h e n d o i n g la r g e m e n t a l
m u l t i p l i c a t i o n s ; it is e s s e n t i a l l y b a s e d o n w h a t w e w o u l d c a l l f a c t o r i n g .
For e x a m p l e , 3 x 2 7 3 ( “ 3 t a k e n 2 7 3 t i m e s ” ) w o u l d b e c a l c u l a t e d b y s u c c e s s i v e l y
d o u b l in g 3 ( 6 , 1 2, 2 4 . . . ) w h i l e k e e p i n g c r a c k o f t h e c o u n t e r p a r t in p o w e r s o f t w o ( 2 , 4 , 8 . . . ).
W h e n th e n e x t p ow er o f tw o w o u ld o v e r s h o o t 2 7 3 , h e t h e n h as to m em orize th e n u m b e r
re a ch e d so fa r t h r o u g h d o u b l i n g s o f 3 ( 7 6 8 ) . w h i l e s u b t r a c t i n g t h e p o w e r o f t w o t h a t w a s
r e a c h e d ( 2 7 3 - 2 5 6 = 1 7 ) . T h e n h e s t a r t s a g a i n , d o u b l i n g 3, a n d k e e p i n g t r a c k o f t h e p o w e r s
o f tw o. W h e n t h e n e x t p o w e r o f t w o w o u l d o v e r s h o o t 1 7 , h e a g a i n m e m o r i z e s t h e n u m b e r
re a ch e d t h r o u g h d o u b l i n g s o f 3 ( 4 8 ) a n d s u b t r a c t s t h e p o w e r o f t w o ( 1 7 — 1 6 — 1 ) . S i n c e o n e
is left o v e r , h e j u s t n e e d s t o a d d a n a d d i t i o n a l . 3 . T h e a n s w e r is t h e n g i v e n b y . t h e s u m o f t h e
u n d e r l in e d t e rm s : 7 6 8 + 4 8 + 3 = 8 1 9 .
D e s p it e t h e c o m p l e x i t y o f t h e m e t h o d , t h e t a i l o r s w e r e q u i t e fast a t p e r f o r m i n g t h e s e s i l e n t
m e n ia l o p e r a t i o n s .
FI GURE 7.4
D o u b l i n g in A f r i c a n a r i t h m e t i c
f t i i n t l b, f r o m F r o m Z a s l a v s k y 1 9 7 3 .)
t he use o f a d o u b l i n g s e q u e n c e in t h e ser uccure o f a S h a n g o t e m p l e a n d in r e l i ­
gi ous c e r e m o n i e s ( r i t u a l c h o r e o g r a p h y a l i g n i n g t w o p r i es t s , f o u r c h i l d r e n ,
eight legs). A c u r a t o r a t t h e M u s e e E t h n o g r a p h i q u e in P o r t o N o v o , B e n i n , w h o
speci ali zed in S h a n g o e x p l a i n e d t o m e t h a t t h e s e d o u b l i n g s t r u c t u r e s w e r e u s e d
I9.ecause.the g o d o f l i g h t n i n g r e q u i r e d a p o r t r a i t o f t h e f o r k e d s t r u c t u r e oFaTigft tni ng bol t . T h e m o d e l is p a r t i c u l a r l y i n t e r e s t i n g in t h a t t h e l e n g t h s o f e a c h i t e r ­
ation are s h o r t e n e d , so t h a t o n e c o u ld h a v e in fin ite d o u b li n g s in a finite
(a) S h an g o , th e god o f
li gh tnin g, is p a rt of the vodun
religion o f B enin an d was one
of t h e im p o rta n t c o m p o n en ts
in th^, c reatio n of the voodoo
religion in th e N e w World.
Here we see th e double-bladed
" t h u n d e r ax e ,” w ith a n o th e r
double blade w ith in each side.
a
b
(b) S h ang o temple arid initiation. H e re we see
th e doubling sequence carried o u t further,
using the bilateral symmetry of t h e h u m a n body
itself in tbe last iteration. T h is is used to symbolize
the bifurcating p attern of the lig h tn in g bolt.
FIGURE
7 .5
D o itb lin g in th e re lig io n o f S h a n g o
fa, courtesy / F AN, Dakar, b: hath center j>ho(os, courtesy / F A N , Dakar; lower right, courtesy Dave
C ro w ley, xoxuw.storinguy.cnm.)
N u m e r ic system s
sp a c e — a tr u e fra c ta l. T h e self-sim ilar s tru c tu r e o f lig h tn in g h a s b e e n a favorite
e x a m p le fo r fr a c ta l g e o m e try te x ts (see M a n d e lb r o t 1977 ). T h e d o u b lin g
s e q u e n c e u s e d t o m o d e l t h e f r a c t a l s t r u c t u r e o U i g h t n i n g in S h a n g o w o u ld n o t
. g iv e a n a c c u r a t e v a l u e fo r t h e e m p i r i c a l ' f r a c t a l d i m e n s i o n — re a l l i g h t n i n g t e n d s
to b r a n c h m u c h m o r e t h a n d o u b l i n g a llo w s for— b u t i t ’s e n o u g h to k n o w t h a t t h e
•
v o d u n r e p r e s e n t a t i o n offers a t e s t a b l e ^ i u a n t i .t a t i v e . m o d e l .
T h e m o s t m a t h e m a t i c a l ly s ig n i f i c a n t a s p e c t o f d o u b l i n g in A f r i c a n relig io n o c c u r s in t h e d i v i n a t i o n ( “f o r t u n e t e l l i n g ’^) t e c h n i q u e s o f v o d u n a n d its r e l i­
gious re la tiv e s (E g la s h 19 9 7 b ). T h e f a m o u ^ j f a d j v i n a t i o n sy stem (fig. 7.6) is b ased
o n to ssin g pairs o f flat sh e lls o r see d s s p l it in tw o. E a c h l a n d s o p e n - s i d e o r c losed side (lik e “h e a d s o r t a i ls ” in a c o i n to s s ). T h e y a re c o n n e c t e d by a d o u b l e d c h a i n
to m a k e f o u r p airs. E a c h g r o u p o f fo u r p a ir s g iv e s o n e o f t h e 16 d i v i n a t i o n s y m ­
b ols, w h i c h te ll t h e f u t u r e o f t h e d i v i n e r ’s c l i e n t . T h e Ifa s y s t e m is w h a t a
m a t h e m a t i c i a n w o u ld c a ll “s t o c h a s t i c , ” t h a t is, i t o p e r a t e s by p u r e c h a n c e . B u t
a clo sely r e l a t e d d i v i n a t i o n s y s te m , C e d e n a, h a s a n o n s t o c h a s t i c e l e m e n t— it is
c lo s e r t o w h a t m a t h e m a t i c i a n s c a ll “d e t e r m i n i s t i c c h a o s . ” .
M y i n t r o d u c t i o n t o c e d e n a , o r s a n d d i v i n a t i o n , t o o k p l a c e in D a k a r, S e n e ­
gal, w h e r e t h e lo c a l Is la m ic c u l t u r e c r e d i t s t h e B a m a n a (a ls o k n o w n as B a m b a r a )
w i t h a p o t e n t p a g a n m y s ti c is m . A l m o s t all d i v i n e r s h a d s o m e k i n d o f p h y s ic a l
d e f o r m i t y — “ t h e p r i c e p a id for t h e i r p o w e r . ” -5 O n e d i v i n e r s e e m e d q u i t e w ill in g
to t e a c h m e a b o u t t h e s y s te m , s u g g e s tin g t h a t it “w o u ld b e ju s t lik e s c h o o l . ” T h e
first few se s s io n s w e n t s m o o t h l y , w i t h t h e d i v i n e r s h o w i n g m e a s y m b o l i c c o d e
in w h i c h e a c h s y m b o l, r e p r e s e n t e d by a s e t o f f o u r v e r t i c a l d a s h e d li n e s d r a w n
in t h e s a n d , s t o o d for s o m e a r c h e t y p i c a l c o n c e p t ( t r a v e l , d esire, h e a l t h , e t c . ) w ith
w h i c h h e a s s e m b l e d n a r r a t i v e s a b o u t t h e f u t u r e . B u t w h e n I fin ally a s k e d h o w
• he d eriv ed th e sym bols
in p a r t i c u l a r , t h e m e a n i n g o f s o m e o f t h e p a t t e r n s
d r a w n p r i o r t o t h e s y m b o l w r i t i n g — t h e y all l a u g h e d a t m e a n d s h o o k t h e i r
h e a d s . “T h a t ’s t h e s e c r e t ! ” M y o ff e rs o f i n c r e a s i n g l y h i g h p a y m e n t s w e r e m e t
w ith d is in t e r e s t . Finally, 1 tr ie d to e x p l a i n t h e so c ia l sig n ific a n c e o f c ro s s -c u ltu ra l
m a t h e m a t i c s . I h a p p e n e d to h a v e a c o p y o f L i n d a G a r c i a ’s F ra cta l E x p lo r e r w ith
m e a n d b e g a n by s h o w i n g a g r a p h o f t h e C a n t o r s e t, e x p l a i n i n g its re c u r s iv e c o n ­
s t r u c t i o n . T h e h e a d d iv i n e r , w i t h a n e x p r e s s i o n o f e x c i t e m e n t , s u d d e n l y s t o p p e d
m e , s n a p p e d t h e b o o k s h u t , a n d s a i d “s h o w - h i m w h a t h e w a n t s ! ”
A s it t u r n s o u t , t h e re c u r s iv e c o n s t r u c t i o n o.fJihe.,.Can.tQ.r_set w as j u s t t h e
rig h t t h i n g to sh o w , b e c a u s e t h e B a m a n a d i v i n a t i o n is a ls o b a s e d o n r e c u r s io n
Xs (fig- l - l ) - T h e d i v i n a t i o n b e g in s w i t h f o u r h o r i z o n t a l d a s h e d lin es, d r a w n rapidly,
/
so th a t th e r e is so m e r a n d o m v a r i a ti o n in t h e n u m b e r o f dashes in e a c h . T h e dashes
\
are t h e n c o n n e c t e d in p airs, s u c h t h a t e a c h o f t h e fo u r lin e s is left w i t h e i t h e r
\
o n e s i n g le d a s h ( i n t h e case o f a n o d d n u m b e r ) o r n o d a s h e s (a ll p a irs, t h e case
b
c
FIGURE
7 .6
B i n a r y codes in d i v i n a t i o n
(a) This Nigerian priest is telling the future by Ifa divination, in which pairs of flat shells or seeds
split in two are tossed with each landing open-side or closed-side. They are connected hy a doubled
chain to make four pairs, giving a total of 16 divination symbols. In this version of lfa (used in the
Abigba region of Nigeria) they use two doubled chains and consider the cast more accurate if there
is a correlation between the two sets, (b) Here we see a chain using split seeds. Each half lands
eititer “closed" (meaning we see the rounded outside) or "open" (meaning we see tire interior).
By using open to represent 0 (double lines), and closed to represent 1 (single line), we can see how
the divination symbol is obtained, (c) The divination chain is interpreted as pairs summing to odd
(one stroke) or even (two strokes).
(a, j ) h o t o b y E . M. McClelland, courtesy R o y a l A n t h r o p o l o g i c a l I n s t i t u t e )
N u n ie r ic sy ste m s
f o i a n e v e n n u m b e r ) . T h e n a r r a t i v e s y m b o l is t h e n c o n s t r u c t e d as a c o l u m n o f four
I v e r t i c a l m a r k s , w i t h d o u b l e v e r t i c a l li n e s r e p r e s e n t i n g a n e v e n n u m b e r o f d a s h e s
i
a n d s i n g l e li n e s R e p r e s e n t in g a n o d d n u m b e r . A t t h i s p o i n t t h e sy s te m is s im ila r
^ t o t h e f a m o u s I f m d i v i n a t i o n : t h e r e a re t w o p o s s ib le m a r k s in fo u r p o s i t i o n s , so
16 p o s s ib le s y m b o ls . U n l i k e lfa, h o w e v e r , t h e r a n d o m s y m b o l p r o d u c t i o n is
r e p e a t e d fo u r ti m e s r a t h e r t h a n tw o . T h e d i f f e r e n c e is q u i t e s i g n ifi c a n t. E a c h o f
t h e lfa s y m b o l p a irs a re i n t e r p r e t e d as o n e o f 2 5 6 p o s s ib le O d u , o r verses. T h e
lfa d i v i n e r m u s t m e m o r i z e t h e O d u ; h e n c e , f o u r s y m b o ls w o u ld be t o o c u m b e r ­
s o m e ( 6 5 , 5 3 6 p o s s ib le v e rs e s ). B u t t h e B a m a n a d i v i n a t i o n d o e s n o t r e q u i r e a n y
verse m e m o riz a tio n ; as w e will see, its use o f re c u r s io n allows for verse self-assembly.
A s in t h e a d d i t i v e s e q u e n c e s w e e x a m i n e d , t h e d i v i n a t i o n c o d e is g e n e r ­
a t e d b y a n i t e r a t iv e lo o p in w h i c h t h e o u t p u t o f t h e o p e r a t i o n is used as t h e i n p u t
for t h e n e x t stage. I n t h i s c a se , t h e o p e r a t i o n is a d d i t i o n m o d u l o 2 ( “m o d 2” for
shortTT w h i c h sim p ly g iv e s t h e r e m a i n d e r a f t e r d i v i s i o n by tw o. T h i s is t h e s a m e
e v e n / o d d d i s t i n c t i o n u se d in t h e p a r i t y b it o p e r a t i o n t h a t c h e c k s for e rro rs o n
c o n t e m p o r a r y c o m p u t e r sy s te m s . T h e r e is n o t h i n g p a r t i c u l a r l y c o m p l e x a b o u t
m o d 2; in f a c t , 1 w as q u i t e d i s a p p o i n t e d a t f i rs t b e c a u s e its r e a p p l i c a t i o n
d e s t r o y e d t h e p o t e n t i a l f o r a b i n a r y p l a c e h o l d e r r e p r e s e n t a t i o n in t h e B a m a n a
d i v i n a t i o n . R a t h e r t h a n i n t e r p r e t e a c h p o s i t i o n in che c o l u m n as h a v i n g s o m e
m e a n i n g (as w o u ld o u r b i n a r y n u m b e r 1 0 1 1, w h i c h m e a n s o n e 1, o n e 2, zero 4s,
a n d o n e 8 ) , t h e d i v i n e r s r e a p p l i e d m o d 2. t o e a c h ro w o f t h e first tw o .s y m b o l s
a n d t o e a c h ro w o f t h e last t w o . T h e r e s u l t s w e r e t h e n a s s e m b le d i n t o tw o n e w
s y m b o ls , a n d m o d 2 w as a p p l i e d a g a i n t o g e n e r a t e a t h i r d s y m b o l. A n o t h e r fo u r
s y m b o ls w e r e c r e a t e d by r e a d i n g t h e ro w s o f t h e o r i g i n a l f o u r 'a s c o l u m n s , a n d
m o d 2 w a s a g a i n r e c u r s i v e l y a p p l i e d t o g e n e r a t e a n o t h e r t h r e e s y m b o ls . .
T h e use o f a n i t e r a t i v e l o o p , p a s s i n g o u t p u t s o f a n o p e r a t i o n b a c k as
in p u t s fo r t h e n e x t s ta g e , w as a s h o c k t o m e ; 1 w a s a t le a s t as t a k e n a b a c k by t h e
s a n d s y m b o ls as t h e d i v i n e r s h a d b e e n b y t h e C a n t o r sec. It w o u ld b e n a i v e to
c l a i m t h a t th i s w as s o m e h o w a l e a p o u t s i d e o f o u r c u l t u r a l b a r r ie r s a n d p o w e r
d i f f e r e n c e s — in f a c t , t h a t ’s ju s t t h e s o r t o f p r e t e n s i o n t h a t t h e last t w o d e c a d e s
o f r e f l e x i v e a n t h r o p o l o g y h a s b e e n d e d i c a t e d a g a i n s t — b u t it w o u ld a ls o be
e t h n o c e n t r i c to ru le o u t th o s e a s p e c t s t h a t w o u ld b e a t t r i b u t e d to m a t h e m a t i c a l
c o l l a b o r a t i o n e l s e w h e r e in che w o r l d ; t h e m titiia]_ _d eh gh t in t w o r e c u r s i o n
f a n a t i c s d i s c o v e r i n g e a c h o t h e r . A n d t h e a p p e a r a n c e o f t h e sy m b o JsJ U k L o iit in
tw o g r o u p s o f " s e v e n — t h e R o s i c r u c i a n ’s m y s t i c n u m b e r — a d d e d s o m e n u m e r olo g ica T icT n g o r T t h e c a k e .
T h e f o l l o w i n g d ay 1 f o u n d t h a t t h e p r e s e n t a t i o n h a d n o t b e e n c o m p l e t e :
a n a d d it io n a l tw o sym bo ls w ere left o u t. T h e s e w e re also g e n e ra t e d by m o d 2 re c u r ­
s i o n u s i n g t h e t w o b o t t o m s y m b o l s t o c r e a t e a f i f t e e n t h , a n d u s i n g t h a t last
A fr ic a n fra c ta l m athem atics
s y m b o l w i t h t h e first s y m b o l t o c r e a t e a s i x t e e n t h ( b r i n g i n g t h e t o t a l d e p t h o f
r e c u r s i o n t o five i t e r a t i o n s ) . T h e f i f t e e n t h s y m b o l is c a l l e d “t h i s w o r l d , " a n d t h e
s i x t e e n t h is “t h e n e x t w o r l d , ” so t h e r e w as g o o d r e a s o n t o s e p a r a t e t h e m fro m
t h e o t h e r s . T h e final p a r t o f t h e s y s te m — c r e a t i n g a n a r r a t i v e f r o m t h e s y m b o ls —
w a s s till u n c l e a r , b u t I w as a s s u re d t h a t it c o u l d b e l e a r n e d if I c a r e f u l l y fo l lo w e d
t h e i r i n s t r u c t i o n s . 1 w as t o g i v e s e v e n c o i n s t o s e v e n le p e rs, p l a c e a k o l a n u t o n
b
1
—
I
1
< —
1
< —
^ —
1
1
1
1
1
1
I I
1
—
^
1
I
1
I I
—
^
1
1 1
1 1
—
>
1
1
1
1
—
>
1
1
I
1
1
1
1
c
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
I I
1 1
1 1
1
1
(e ) A f te r th is , th e o rig in al four
a re read sid ew ay s to c re a te four
1
1
1
1
d
1
>
1
----- >
1 1
1 1
m o re sy m b o ls, a n d th e e n tire
N..
1
-
-
1
1
p r o c e s s is r e p e a t e d , p r o d u c i n g
1
---
1
----- ^
1
a n o th e r g ro u p o f sev en . In th e
fin a l s t e p , t h e first a n d la s t f r o m
e a c h g ro u p d f s e v e n a re p a ire d off
to g e n e r a t e t h e final tw o s y m b o ls.
FIGURE 7 . 7
(a)
F o u r sets o f ra n d o m d a sh e s
B a m a n a s a n d d iv in a tio n
a r e d r a w n , (b) E a c h o f t h e d a s h e s
is p a i r e d , a n d t h e o d d / e v e n r e s u l t s
a r e r e c o r d e d , ( c ) T h e p r o c e s s is r e p e a t e d f o u r t i m e s , r e s u l t i n g i n f o u r s y m b o l s . E a c h r o w o f t h e firs t
tw o s y m b o ls a n d t h e last t w o s y m b o ls a re p a ire d o ff to g e n e r a t e t w o n e w s y m b o ls , (d ) T h e tw o
n e w ly g e n e r a t e d s y m b o ls, n o w p l a c e d b e lo w t h e o r ig i n a l four, a re a g a i n p a i r e d o ff t o g e n e r a t e a
s e v e n th sy m b o l.
N u m e ric system s
a p ile o f s a n d n e x t t o m y b e d a t n i g h t , a n d in t h e m o r n i n g b r i n g a w h i t e c o c k , 1^
w h i c h w o u ld h a v e t o b e s a crificed t o c o m p e n s a t e fo r t h e h a r m f u l e n e rg y re l e a s e d I
in t h e t e l l i n g o f t h e s e c r e t. I f o l lo w e d all t h e in s t r u c t i o n s , a n d t h e n e x t m o r n - i
in g b o u g h t a large w h i t e c o c k a t t h e m a r k e t . T h e y h e l d t h e c h i c k e n o v e r t h e d iv - '
i n a t i o n s a n d , a n d I w as t o l d t o e a t t h e b i t t e r k o l a n u t as th e y m a r k e d d i v i n a t i o n
s y m b o ls o n its f e e t w i t h a n i n k p e n . A l i t t l e s a n d w as r h r o w n in its m o u t h , a n d
i
t h e n 1 w as to ld t o h o l d it d o w n as p ra y e r s w e r e c h a n t e d . T h e r e w as n o a c t i o n o n j
t h e p a r t o f t h e d i v i n e r ; t h e c h i c k e n s i m p l y d i e d in m y h a n d s .
W h i l e still a b i t s h a k e n by t h e c h i c k e n ’s d e m i s e (as w ell as e x p e r i e n c i n g
a re s p e c ta b le buzz fr o m t h e k o la n u t ) , I was to l d t h e r e m a in i n g mystery. E a c h sy m ­
b o l h a s a “ h o u s e ” in w h i c h it b e lo n g s — fo r e x a m p l e , t h e p o s i ti o n o f t h e s i x t e e n t h
s y m b o l is “ t n e n e x t w o r l d " — b u t in a n y g i v e n d i v i n a t i o n m o s t s y m b o ls w ill n o t
b e l o c a t e d in t h e i r o w n h o u s e . T h u s t h e s i x t e e n t h s y m b o l g e n e r a t e d m i g h t be
“d e sire ,” so we w o u ld h a v e d e s ire in t h e h o u s e o f t h e n e x t w orld , a n d so o n . O b v i ­
o usly t h i s still l e a v e s r o o m f o r c r e a t i v e n a r r a t i o n o n t h e p a r t o f t h e d i v i n e r , b u t
t h e b e a u t y o f t h e s y s te m is t h a t n o v e rs e s n e e d to b e m e m o r i z e d o r b o o k s c o n ­
s u l te d ; t h e sy s te m c r e a t e s its o w n c o m p l e x variety.
T h e m o s t e l e g a n t p a r t o f t h e m e t h o d is t h a t it r e q u i r e s o n l y f o u r r a n d o m
d r a w i n g s ; a f t e r t h a t t h e e n t i r e s y m b o l i c a r r a y is q u i c k l y s e l f - g e n e r a t ed- S e lf g e n e r a t e d v a r i ety is i m p o r t a n t i n j n o c l e r n c o m p u t i n g , w h e r e it is c a l l e d “p s e u d o ­
r a n d o m n u m b e r g e n e r a t i o n ” (fig. 7 . 8 ) . T h e s e a l g o r i t h m s t a k e l i t t l e m e m o r y ,
b u t c a n g e n e r a t e v e r y l o n g li s t s o f w h a t a p p e a r t o b e r a n d o m n u m b e r s ,
a l t h o u g h t h e list w ill e v e n t u a l l y s t a r t o v e r a g a i n ( t h i s l e n g t h is c a l l e d t h e
“p e r i o d ” o f t h e a l g o r i t h m ) . A l t h o u g h t h e B a m a n a o n l y r e q u i r e a n a d d i t i o n a l
12 s y m b o l s to b e g e n e r a t e d in t h i s f a s h i o n , a m a x i m u m - l e n g t h p s e u d o r a n d o m
n u m b e r g e n e r a t o r u s i n g t h e i r i n i t i a l fo u r s y m b o l s w ill p r o d u c e 6 5 . 5 3 5 s y m b o ls
b e f o r e it b e g i n s t o r e p e a t .
A s i m i l a r s y s te m f o r s e l f - g e n e r a t e d v a r i e t y w as d e v e l o p e d as a m o d e l for
t h e “c h a o s ” o f n o n l i n e a r d y n a m i c s by M a r s t o n M o r s e ( 1 8 9 2 - 1 9 7 7 ) . B e fo re th e
1970s, m a t h e m a t i c i a n s h a d a s s u m e d t h a t , b e s i d e s a few e s o t e r i c e x c e p t i o n s (che
a l g o r i t h m s for p r o d u c i n g i r r a t i o n a l n u m b e r s s u c h as V 2), t h e o u t p u c o f a n e q u a ­
t i o n w o u l d e v e n t u a l l y s t a r t r e p e a t i n g . T h a t a s s u m p t i o n w as p a r t ly b a s e d o n
E u r o p e a n c u l t u r a l id e a s a b o u t fr e e w il U - c o m p l e x b e h a v i o r c o u l d n o t b e t h e
result o f p r e d e t e r m i n e d sy ste m s (see P o r t e r 1 9 8 6 ). It was n o t u n t i l r h d ^ i Q h o s ^ o s j
t h a t m a t h e m a t i c i a n s realized t h a t e v e n sim ple, c o m m o n eq u a tio n s d escrib in g 'th in g s
lik e p o p u l a t i o n g r o w t h o r.flu id flow c o u l d r e s u l t in w h d t t h e y c a l l e d “d e t e r m i n ­
istic c h a o s ”— a n o u t p u t t h a t n e v e r re p e a ts, g iv i n g th e a p p e a r a n c e o f r a n d o m n u m ­
bers fr o m a n o n r a n d o m ( d e t e r m i n i s t i c ) e q u a t i o n . M o r s e d e v e l o p e d t h e m i n i m a l
case for s u c h b e h a v io r .
^
A fr ic a n fra c ta l mathematics
98
I 1 1I
0111
001 1
0001
1000
0 100
0010
1001
0110
1 100
01 I 0
1011
0101
1010
1 101
1110
FIGURE
7 .8
P seu d o ra n d o m n u m b e r genera tio n
f r o m s h ift r e g is t e r c ir c u its
(a) If we think of the two-strokes as zero and
single stroke as one, the Bamana divination,
system is almost identical to the process of
pseudorandom number generation used by digital
circuits called “shift registers.” Here the circuit
cakes mod 2 of die last two bits in the register
and places the result in the first position. T h e
other bits are shifted to the right, with the last
discarded.
This four-bit shift register will only produce
15 binary words before the cycle starts over, but
the period of the cycle increases with more bits
( 2 n - 1). For .the entire 16 bits (four symbols of
four bits each) that begin the Bamana
divination, 65,535 binary words can be produced
before repeating the cycle.
(b ) Electrical circuit representation of a four-bit
shift register combined with exclusive-or to
perform the mod 2 operation. W hile school­
teachers are making increasing use of African
culture, in the mathematics classroom, few have
explored the potential applications to
technology education.
T h e c o n s t r u c t i o n o f t h e M o r s e se q u e n c e b e g i n s b y c o u n t i n g f r o m z e r o j n
b in ary n o ta tio n : 0 0 0 ,
oot
, 01 o , 0 1 1 . . . . I t t h e n t a k e s t h e s u m o f t h e d ig i ts in
e a c h n u m b e r - -o + o + o = o , n . + o + 1 = i , e t c . — a n d f i n a l l y m o d 2 o f e a c h
s u m . T h e r e s u l t is a s e q u e n c e w i t h m a n y r e c u r s i v e p r o p e r t i e s , ^ b u t o f e n d l e s s
v a r i e t y . M o r s e d id t h e s a m e “ m i s r e a d i n g ” o f t h e b i n a r y n u m b e r a s d i d t h e \
' B a m a n a — a lth o u g h h e did n o t h a v e a n a n th r o p o lo g is t sc o w lin g a t h im for
•! i g n o r i n g p l a c e v a l u e — a n d h e d i d it for t h e s a m e r e a s o n : c o m b i n e d w i t h t h e j
• m o d 2 o p e r a t i o n , i t m a x i m i z e s v a r i e ty .
I n m y r e a d i n g o f d i v i n a t i o n l i t e r a t u r e 1 e v e n t u a l l y c a m e a c ro s s t h e d u p l i ­
c a t e o f t h e B a m a n a t e c h n i q u e 5 , 0 0 0 m i le s t o t h e e a s t in M a l a g a s y sik id y ( S u s s m a n a n d S u s s m a n 1 9 7 7 ) , w h i c h i n s p i r e d a s tu d y o f t h e h i s t o r y o f its d if fu s io n .
T h e s tr o n g jj r n i la v i ty o f b o t h s y r n b o lic t e c h n i q u e a n d s e m a n t i c c a t e g o r i e s { a . w h a t
E u r o p e a n s t e r m e d “g e o m a n c y ” w a s first n o t e d by F l a c o u r t { r 6 6 r ), b u t it w as n o t
u n t i l T r a u t m a n n ( 1 9 3 9 ) t h a t a s e r io u s c l a i m w a s m a d e for a c o m m o n s o u r c e fo r
th is A r a b ic , E u r o p e a n , W e s t A f r i c a n , a n d East A f r i c a n d i v i n a t i o n t e c h n i q u e . T h e
c o m m o n a l i t y w as c o n f i r m e d in a d e t a i l e d f o r m a l a n a ly s is by J a u l i n ( 7 9 6 6 ) . B u t
w h e t e d id it o r i g i n a t e ?
j
N u m e ric system s
( S k i n n e r ( i q 8 o ) provides a w e ll- d o c u m e n t e d his tory o f t h e diffusion e v id e n c e , i
from cPie'first specific w r it te n r e c o r d — a n i n t h - c e n t u r y Jew ish c o m m e n ta r y by A r a n '
b e n J o s e p h — to its m o d e r n use in A l e i s t e r Crq.yvley’s L ib e r 7 7 7 . T h e o l d e s t A r a - [
bic d o c u m e n t s ( t h o s e o f a z - Z a n t i in t h e t h i r t e e n t h c e n t u r y ) c la i m t h e o r ig in o f
g e o m a n c y (i!m aLrcim l, “t h e s c i e n c e o f s a n d ” ) t h r o u g h t h e E gy p tian go d Idris ( H e r - ;
m e s T ris m e g i s tu s ) ; w h ile we n e e d n o t t a k e t h a t as a n y t h i n g m o r e t h a n a c l a i m !
to a n t i q u i t y , a N i l o t i c i n f l u e n c e is n o t u n r e a s o n a b l e . B u d g e ( 1 9 6 1 ) a t t e m p t s t o j
c o n n e c t t h e use o f s a n d in a n c i e n t E g y p t i a n r i tu a l s to A f r i c a n g e o m a n c y , b u t it 1
is h a r d to see th is as u n i q u e . M a t h e m a t i c a l l y , h o w e v e r , g e o m a n c y is s trik in g ly out./
o f p l a c e in n o n - A f r i c a n sy s te m s .
L ik e o t h e r li n g u is ti c c o d e s , n u m b e r b a s e s t e n d j o h a v e a n j x t r . e m e l y lo n g
h is to r i c a l p e r s is t e n c e . E v e n u n d e r P l a t o n i c r a t i o n a l i s m , t h e a n c i e n t G r e e k s h e l d
10 t o b e t h e m o s t s a c r e d o f all n u m b e r s ; t h e K a b b a l a h ’s A y in S o f e m a n a t e s by
10 S e firo t, a n d t h e C h r i s t i a n W e s t c o u n t s o n its “H i n d u - A r a b i c ” d e c i m a l n o t a ­
t i o n . I n ^ ' f r i c gjl, o n t h e o t h e r h a n d , b a se -2 c a l c u l a t i o n w as u b i q u i t o u s, e v e n for
m u l t i p l i c a t i o n a n d d iv is io n . A n d it is h e r e t h a t we find t h e c u lt u ra l c o n n o t a t i o n s
o f d o u b l i n g t h a t g r o u n d t h e d i v i n a t i o n p r a c t i c e in its re lig io u s sig n i f i c a n c e.
T h e im p lic atio n s o f this trajecto ry — fro m s u b - S a h a r a n A frica to N o r t h A frica
to E u r o p e — a re q u i t e s i g n if i c a n t fo r t h e h i s t o r y o f m a t h e m a t i c s . F o l l o w i n g t h e
i n t r o d u c t i o n o f g e o m a n c y to E u ro p e by H u g o o f S a n ta l la in tw e lf th - c e n tu r y S p a in ,
it w as t a k e n u p w i t h g r e a t i n t e r e s t b y t h e p r e - s c i e n c e m y s tic s o f t h o s e t i m e s —
a lc h e m is ts , h e r m e t i c i s t s , a n d R o s i c r u c i a n s (fig. 7 .9 ). B u t th e s e E u r o p e a n g eom a n c e r s — R a y m o n d L u ll , R o b e r t F l u d d , d e P e r u c h i o , H e n r y d e P isis, a n d
o th e r s — p e rs iste n tly re p la c e d t h e d e t e r m i n i s t i c asp e c ts o f t h e system w i t h c h a n c e .
By m o u n t i n g t h e 16 figures o n a w h e e l a n d s p i n n i n g it, t h e y m a i n t a i n e d t h e i r
s o c i e t y ’s e x c l u s i o n o f a n y c o n n e c t i o n s b e t w e e n d e t e r m i n i s m a n d u n p r e d i c t a b i l ­
ity. T h e A fric a n s , o n t h e o t h e r h a n d , se e m t o h a v e e m p h a s iz e d s u c h c o n n e c t i o n s .
In c h a p t e r 10 we will e x p l o r e o n e s o u r c e of t h i s d if f e r e n c e : t h e A f r i c a n c o n c e p t
o f a “t r i c k s t e r " g o d , o n e w h o is b o t h d e t e r m i n i s t i c a n d .u.nor_e d i c t a b l e •
O n a v id eo reco rd in g 1 m a d e o f th e B a m a n a d iv in a tio n , I n o tic e d th a t th e
p r a c t i t i o n e r s h a d u se d a s h o r t c u t m e t h o d in s o m e d e m o n s t r a t i o n s ( t h i s m a y
h a v e b e e n a p a r t in g gift, as th e v id e o was s h o t o n my last day). A s th e y first ta u g h t
m e, w h e n th e y c o u n t off t h e p airs o f r a n d o m d a s h e s , t h e y li n k t h e m by d r a w i n g
sh o r t cu rves. T h e s h o r t c u t methodI t h e n links t h q s e c u rv e s witly.larger cu rv es , a n d
^th o s e b e lo w w i t h e v e n la r g e r c u r v e s . T h i s u p s i d e - d o w n C a n t o r s e t s h o w s t h a t
th e y are n o t s i m p l y a p p l y i n g m o d 2 a g a i n a n d a g a i n in a m i n d l e s s f a s h i o n . T h e
s e lf-s im ila r p h y s i c a l s t r u c t u r e o f t h e s h o r t c u t m e t h o d v iv i d ly il lu s t r a t e s a r e c u r ­
sive p r o c e s s ^ a n d as a n g m .tra d itio n a l i n v e n t i q n ( t h e r e is n o r e c o r d o f its use e l s e ­
w h e re ) it sh o w s a c t i v e m a t h e m a t i c a l p r a c t ic e . O t h e r A f r i c a n d i v i n a t i o n p ra c tic e s
A fr ic a n fra c ta l m athem atics
IOO
l i i d in f i
i*i
;)c
c o lc m
'{ iH h n n i YDi f i c f m
.p iitn r o ucT .m m
po
oo
xjo
oo
.
/• '
J^ ? Y ) b i t R b m a - c o i t r a t . Of)
Hjb nimcrficiipetfifi? 0 .9n hMrQ'Jr Afrufantrtbftfc
° ° A nQ°
c£«wii gunXrnprc urtra
n 'J ^ r ^ ^ r n b c o v t t m
-
(ttJtnbicurnimrptti®
nr.
u
Q
Ctrvwtca opm tdurbu oo QJ> <WGMubiaa*afcufk
m ru a h ta tu
o
0 cuuaiw rncabtrfaK u
0
Oft- V -
n
q
{c>orcnramficrr filn .
n
o
m fltiu n rrp tiitw lurt
0im
0 n An
o ficwr r a r nHumbert*
y w i n r v bu tf
u
00 or
V&Vw n tittu ffta ftiu .
00
oo
ro am uhcgMctfum
n u r r r c B iim M
00
? y to « b w w A tttfrte u
c o ty S ir tf- lu r te m a f tf
v d imuftfl mnutesriBU
f tu V r w u : '
S
FIGURE
q
utram fK tum
0 5? h i r m i f & ftraocrafi
00 00 o u r uroU m m nftnis
0 0 ftp.tT#jBj,ia?j{*ta,._
^
Oa -O
00 0 0
w jj
(V
•
w tcw iito iflttitin icr* *
cutiufottm ttftnt O j*»
< V o ....................
7.9
Q eom ancy
African divination was taken up under the name “geomancy” by European mystics. This chart
drawn for King Richard u in 1391.
( F r o m S k in n e r 1 9 8 0 .)
c a n b e lin k e d to r e c u r s io n as w ell; for e x a m p l e D e v i s c h ( 1 9 9 1 ) d e s c r ib e s t h e Yaka
d i v i n e r s ’ “s e l f - g e n e r a t i v e ” i n i t i a t i o n a n d u t e r i n e s y m b o l i s m .
Before lea v in g d i v i n a t i o n , t h e r e is o n e m o r e i m p o r t a n t c o n n e c t i o n t o m a t h e ­
m a t i c a l h is to ry . W h i l e R a y m o n d L u ll, l i k e o t h e r E u r o p e a n a l c h e m i s t s , c r e a t e d
w h e e l s w i t h s i x t e e n d i v i n a t i o n fig ures, h i s p r i m a r y i n t e r e s t w a s in t h e c o m b i ­
n a t o r i a l p o s s ib il it ie s o f f e r e d by b a s e - 2 d i v i s i o n s . L u l l ’s w o r k w as c l o s e ly e x a m ­
i n e d by G e r m a n m a t h e m a t i c i a n G o t t f r i e d L e i b n i z , w h o s e D iss erta tio d e arte
c o m b in a to r ia , p u b l i s h e d in 1 6 6 6 w h e n h e w a s t w e n t y , a c k n o w l e d g e s L u l l ’s w o r k
as a p c e c u rs o r. F u r t h e r e x p l o r a t i o n le d L e ib n i z t o i n t r o d u c e a b a s e - 2 c o u n t i n g
s y s te m , c r e a t i n g w h a t we n o w call, t h e b i n a r y c o d e . W h i l e t h e r e w e r e m a n y o t h e r
N u m e r ic sy ste m s
i n f l u e n c e s i n t h e li v e s o f L u ll a n d L e i b n i z , it is n o t f a r - f e t c h e d t o see a h i s t o r ­
ic a l p a t h fo r b a s e - 2 c a l c u l a t i o n t h a t b e g i n s w i t h A f r i c a n d i v i n a t i o n , r u n s
th r o u g h t h e g e o m a n c y o f E u r o p e a n a lc h e m is ts , a n d is finally tr a n s la t e d i n t o binary
c a l c u l a t i o n , w h e r e i t is -n o w a p p l i e d inl,e v e r y d i g i t a l c i r c u i t f r o m a l a r m c l o c k s
to su p erco m p u te rs.
I n a 1 9 9 5 i n t e r v i e w i n W ir e d m a g a z i n e , t e c h n o - p o p m u s i c i a n B r i a n E n o
c la i m e d t h a t t h e p r o b l e m w i t h c o m p u t e r s is t h a t “th e y d o n ’t h a v e e n o u g h A f r i c a n
in t h e m . ” E n o w as, n o d o u b t , t r y i n g t o b e c o m p l i m e n t a r y , s a y i n g t h a t t h e r e is
s o m e i n t u i t i v e q u a l i t y t h a t is a v a l u a b l e a t t r i b u t e o f A f r i c a n c u lt u r e . B u t in d o i n g
so h e o b s c u r e d t h e c u l t u r a l o r i g i n s o f d i g i t a l c o m p u t i n g a n d d id a n i n j u s t i c e
to th e very c o n c e p t h e was try in g to c o n v e y .
D iscrete se lf-o rg a n iz a tio n in O w a r i
F ig u r e 7 .1 0 a s h o w s a b o a r d g a m e t h a t is p l a y e d t h r o u g h o u t A f r i c a in m a n y d if ­
f e r e n t v e r s io n s v a r i o u s ly t e r m e d a y o , b a o , g iu th i, lela, m a n ca la , omi v eso , ow a ri, lei,
a n d songo ( a m o n g m a n y o t h e r n a m e s ) . B o a r d s t h a t w e re c u t i n t o s t o n e s , s o m e
o f e x t r e m e a n ti q u it y , h a v e b e e n f o u n d f r o m Z i m b a b w e t o E t h i o p i a (see Z aslavsky
1 97 3, fig. 1 1 -6 ) . T h e g a m e is p l a y e d b y s c o o p i n g p e b b l e o r s e e d c o u n t e r s fro m
o n e cup, a n d placin g o n e o f th o s e c o u n te r s in to e a c h cup, s ta rtin g jv itJ x th e cup
t o t h e r i g h t o f t h e s c o o p . T h e g o a l is t o H a v e t h e last c o u n t e r l a n d in a c u p t h a t
h a s o n l y o n e o r t w o c o u n t e r s a l r e a d y in it, w h i c h a ll o w s t h e p l a y e r t o c a p t u r e
t h e s e c o u n t e r s . I n t h e G h a n a i a n g a m e o f o w a r i , p la y e r s a r e k n o w n fo r u ti li z in g
a se r ie s o f m o v e s t h e y c a ll a “ m a r c h i n g g r o u p . ” T h e y n o t e t h a t if t h e n u m b e r
o f c o u n t e r s in a se r ie s o f c u p s e a c h d e c r e a s e s b y o n e (e .g ., 4 - 3 - 2 - ! ) , t h e e n t i r e
p a t t e r n c a n - b e r e p l i c a t e d w k h a rig h c -s h ift by s c o o p i n g fr o m t h e largest c u p , a n d
t h a t if th e p a t t e r n is left u n i n t e r r u p t e d it c a n p ro p a g a te in th is way as far as n e e d e d
for a w i n n i n g m o v e (fig. 7 . 1 0 b ) . A s s i m p l e as it s e e m s , t h i s c o n c e p t o f a selfr e p l i c a t i n g p a t t e r n is a t t h e h e a r t o f s o m e s o p h i s t i c a t e d m a t h e m a t i c a l c o n c e p t s .
J o h n v o n N e u m a n n , w h o p l a y e d a p i v o t a l r o l e in t h e d e v e l o p m e n t o f
t h e m o d e r n d i g i t a l c o m p u t e r , w a s a ls o a f o u n d e r o f t h e m a t h e m a t i c a l t h e o r y
o f s e l f - o r g a n i z i n g s y s te m s . I n i t i a l l y , v o n N e u m a n n ’s t h e o r y was t o b e b a s e d o n
se lf-rep ro d u c in g ph y sical ro b o ts. W h y w ork on a th e o ry o f s e lf-re p ro d u c in g
m a c h i n e s ? 1 b e l i e v e t h e a n s w e r c a n b e f o u n d in v o n N e u m a n n ’s s o c i a l o u t ­
look. H e i m s ’s ( 1 9 8 4 ) b i o g r a p h y e m p h a s iz e s h o w t h e d is o r d e r o f v o n N e u m a n n ’s
p r e c a r io u s y o u t h as a H u n g a r i a n Je w w a s r e f l e c t e d in h is a d u l t e ffo rts t o i m p o s e
a s t r i c t m a t h e m a t i c a l o r d e r o n v a r i o u s a s p e c t s o f t h e w o rld . I n v o n N e u m a n n ’s
a p p l i c a t i o n o f g a m e t h e o r y to s o c i a l s c i e n c e , fo r e x a m p l e , H e i m s w r it e s t h a t his
“H o b b e s i a n ” a s s u m p t i o n s w e r e “c o n d i t i o n e d b y t h e h a r s h p o l i t i c a l r e a l i t i e s o f
102
A fr ic a n fra c ta l mathematics
FIGURE 7 - 1 0
O w ari
(;i) T h e owari board has i 2 cups, plus one cup on each side for captured counters. This board is
hinged in the center, with a beautifully carved cover (see fig. 7.14). (b) Scoop from the first cup,
and plant one counter in each succeeding cup. (c) The Marching Group is replicated with a
right-shift. Repeated application will allow it to propagate around the board.
.his H u n g a r i a n e x i s t e n c e . ” H is e n t h u s i a s m for t h e use o f n u c l e a r w e a p o n s a g a in s t
t h e S o v i e t U n i o n is a ls o a t t r i b u t e d t o th i s e x p e r i e n c e .
D u r i n g t h e H i x o n S y m p o s i u m ( v o n N e u m a n n 195 1) h e w a s a s k e d if c o m ­
p u t i n g m a c h i n e s c o u l d b e b u i l t s u c h t h a t th e y c o u l d r e p a i r t h e m s e l v e s if “d a rn a g e d in a i r r a i d s ,” a n d h e r e p l i e d t h a t “ t h e r e is n o d o u b t t h a t o n e c a n d e s i g n
m a c h i n e s w h i c h , u n d e r s u i t a b l e c i r c u m s t a n c e s , w ill r e p a i r t h e m s e l v e s . " H is
w o r k o n n u c l e a r r a d i a t i o n t o l e r a n c e for t h e A t o m i c E n e r g y C o m m i s s i o n in
1 9 5 4 - 1 9 5 5 i n c l u d e d b i o l o g i c a l e ff e c ts as w e ll as m a c h i n e o p e r a t i o n . T u t t i n g
t h e s e fa c ts to g e t h e r , 1 c a n n o t e s c a p e t h e c r e e p y c o n c l u s i o n t h a t v o n N e u m a n n ’s
i n t e r e s t in s e l f - r e p r o d u c i n g a u t o m a t a o r i g i n a t e d i n f a n t a s i e s a b o u t h a v i n g a
m o r e p e r f e c t m e c h a n i c a l p r o g e n y s u r v i v e t h e n u c l e a r p u r g i n g o f o r g a n i c life
o n th is p lan et.
M o d e l s for p h y s i c a l r o b o t s t u r n e d o u t to b e t o o c o m p l e x , a n d a t t h e s u g ­
g e s t i o n o f h is c o l l e a g u e S t a n i s l a w U l a m , v o n N e u m a n n s e t t l e d for a g r a p h i c a b ­
s t r a c ti o n : “c e llu la r a u t o m a t a .” as t h e y c a m e to b e c a lle d . In th is m o d e l (fig. 7.11 a),
e a c h s q u a r e in a grid is s a i d t o b e e i t h e r a l i v e o r d e a d ( t h a t is, in o n e o f t w o p o s ­
sible s t a te s ) . T h e i t e r a t i v e rules for c h a n g i n g t h e s t a t e o f a n y o n e s q u a r e are based
■
In the cellular automaton called “the game of life,’’ each cell in the grid is in one
of two states: live or dead. Here we see a live ceil in the center, surrounded by dead
h
cells in its eight nearest neighbors. The state of each cell in the next iteration is
B [
determined by a set of rules. In "classic” life (the rules first proposed by John Horton
Conway), a dead cell becomes a live celf'Tf it has three live nearest neighbors, and a
cell dies unless it has two or three live neighbors.
BBBBBBBB
B f l f lf l B B B f l
■■■■■■■■
■■■ ■ ■■ ■
■■■
■■■
■■ BBBBB
BBBBBBBB
BBBBBBBB
—
>■
■■■■■■■■
■■■■■■■■
BBBBBBBB.
■■■■■■■■
■■■■■■■■
BBBBBBBB
■■■
um n
■■ ■ ■■■
BB
■ ■ ■ — > ■ ■ B B B B B — >■ ■ ■
■■■ ■■■■
■■
BBB
■■■■■■■■
B B B B B B B fl
BBBBBBBB
BBBBBBBB
B B f lB f l f l B f l.
BBBBBBBB
BBBBBBBB
■■■
B ■ ■ — =►
■■■
BBB
BBBB BBS
■
B B B B B B B fl
_
B f lflflfll
BBBBBI
BBBBBI
■■■
i
■■ ■■
■ ■ BB
BBB
I
BBBBBI
This initial condition produces a fixed pattern after four iterations. The patterns occurring before
it settles down to stability are called the “transient.”
BB■
BBBBBBBBB
■BB BBBBB
■■ ■ BBBB
■BBBBBBBB
■
B B B BB
BBBBB B
■ ■ B B B BB
■■B
BBBB
■■■■■■■■■
BBBBBBBB
BBBBBBBB
BBBB BBB
BBBB BBB
BBBB BBB
BBBBBBBB
BBBBBBBB
BBBBBBBBB
BBB BBBBB
BBB BBBBB
B
BBBBB
fl
BBBBBB
a
aa
This stable pattern flips back and forch between these
two states. This is called a “period-2” pattern.
■
BBB
BBB
BBB
BBB
BBBBBBBBB
BBBBBBBBB
BBBB
BBB
B B B BBBB.
fl B B B B B B
BB BBB ' B
—>- B B B B B B B B B '
BB BB B BBB
BBBBBBBBB
BBBBBBBBB
BBB
BBB
BBB
BBB
B
BB
fl
BBBBBB
B ^
BBBBB BBB
BBBBB BBB
BBBBBBBBB
A period-4.pattern. Periods of any length can be produced, as we saw in the previous examples
of pseudorandom number generation. Deterministic chaos, in which the pattern never repeats
(i.e., a pcrioJ-infinity pattern, like the Morse sequence), is also possible.
Ite ratio n 49
Iteration 133
I t e r a t io n 182.
A constam-growth pattern, shown in high resolution, looks similar to the cross-section of an
internal organ. The rules: a dead cell becomes a live cell if it has three live nearest neighbors, and
a cell dies only if it has seven or eight live neighbors.
FIGURE
7. 1 I
C ellu la r a u to m a ta
A fr ic a n fracial mathematics
104
o n t h e e i g h t n e a r e s t n e i g h b o r s (e.g ., if t h r e e o r m o r e n e a r e s t - n e i g h b o r s a r e full,
t h e c e l l b e c o m e s full in t h e n e x t i t e r a t i o n ) . A t first, r e s e a r c h e r s e a r n e d o u t o n
t h e s e c e l l u l a r a u t o m a t a e x p e r i m e n t s o n c h e c k e r e d t a b l e c l o t h s w i t h p o k e r c h ip s
a n d d o z e n s o f h u m a n h e lp e r s (M ay er-K ress, pers. c o m m . ) , b u t by 1 9 7 0 it h a d b e e n
d e v e l o p e d i n t o a sim p le c o m p u t e r p r o g r a m ( C o n w a y ’s “ g a m e o f life” ), w h i c h was
d e s c r i b e d by; M a r t i n G a r d n e r i n h is f a m o u s “M
a t i-------c a l G a m-e s ” c o l u m n in
v- a t h e m--/
S cien tific A m erica n . T h e “g a m e o f life" story was a n in s t a n t h i t , a n d c o m p u t e r screens
a ll o v e r t h e w o r ld b e g a n to - p u l s a t e w i t h a b izarre a r r a y o f p a t t e r n s (fig. 7.1 i b ) .
A s t h e s e a c ti v it ie s d re w in c r e a s i n g p ro f e ssio n a l a t t e n t i o n , a w id e r a n g e o f m a t h e ­
m a t i c a l l y o r i e n t e d s c i e n t i s t s b e g a n to re alize t h a t t h e s p o n t a n e o u s e m e r g e n c e
o f s e l f - s u s t a i n i n g p a t t e r n s c r e a t e d in. c e r t a i n , c ejlul.ar a . m o m a t a - ^ e r e e x c e l l e n t
m o d e l s for t h e k i n d s o f s e lf-o r g a n i zin g p a t t e r n s t h a t . l m d b e e n _sj?„elu.siyeJ.n s t u d ­
ies o f fluid flow a n d b io l o g i c a l g r o w t h .
S i n c e s c a l i n g s t r u c t u r e s a re o n e o f t h e h a l l m a r k s o f b o t h flu id t u r b u l e n c e
a n d b io l o g i c a l g r o w t h , t h e o c c u r r e n c e o f f r a c ta l p a t t e r n s i n c e l l u l a r a u t o m a t a
a ttra c te d a g reat deal o f in terest. But a m o re sim p le sc a lin g stru c tu re , th e log­
a r i t h m i c s p ira l (fig. 7 -1 2 ), h a s g a r n e r e d m u c h o f t h e a t t e n t i o n . E v e n b a c k i n t h e
i 9 ^ o s m a t h e m a t i c i a n A l a n T ux lng , w h o s e t h e o r y - o f c o m p u t a t i o n p r o v i d e d v o n
N e u m a n n w i t h ' t h e i n s p i r a t i o n f o r t h e first d i g i t a l c o m p u t e r , b e g a n h i s r e s e a r c h
o n “ b io l o g i .c a l x n o r p h o g e n e s i s ” w i t h a n a n a ly s is o f l o g a r i t h m i c s p ira ls in g r o w t h
p a t t e r n s . ( M a r k y s ( 1 9 9 1 ) n o t e s t h a t t h e a p p l i c a O o n a j e a s fox,cN^
m o d e l s o f s p ira l w a v e s i n c l u d e n e r v e a x o n s , t h e r e t i n a , t h e s u r f a c e o f fe r tiliz e d
eggs, t h e c e r e b r a l c o r t e x , h e a r t tissu e, a n d a g g r e g a t i n g s l i m e m o l d s . I n t h e t e x t
for
c a l a b
,
t h e first c o m p r e h e n s i v e s o f t w a r e fo r e x p e r i m e n t i n g w i t h c e l l u l a r
a u t o m a t a , m a t h e m a t i c i a n R u d y R u c k e r ( 1 9 8 9 , 1 6 8 ) refers t o s y s te m s t h a t p r o ­
d u c e p a ir e d log spirals as “Z h a b o t i n s k y C A s , " a f t e r t h e c h e m i s t w h o first o b s e r v e d
s u c h s e lf-o r g a n iz i n g p a t t e r n s in a r t ifi c ia l m e d i a : “W h e n y o u l o o k a t Z h a b o t i n ­
sky C A s , y o u a r e s e e i n g v e r y s t r i k i n g t h r e e d i m e n s i o n a l s t r u c t u r e s ; t h i n g s like
p a i r e d v o r t e x s h e e t s in t h e su rfa c e o f a r i v e r b e l o w a d a m , t h e s c r o ll p a i r s t r e t c h ­
i n g all t h e w a y d o w n t o t h e r i v e r b o t t o m . . . . In t h r e e d i m e n s i o n s , a Z h a b o t i n ­
sky r e a c t i o n w o u ld b e like tw o p a ir e d n a u t i l u s s h ells, fa c in g e a c h o t h e r w i t h t h e i r
lips b l e n d i n g . T h e s u cce s siv e layers o f s u c h a g r o w i n g p a t t e r n w o u ld b u il d u p v ery
li k e a f e t u s ! ”
/
F ig u r e 7 .1 3 s h o w s h o w t h e o w a r i m a r c h i n g - g r o u p s y s te m c a n b e u s e d as a
I o n e -d im e n s io n a l cellu lar a u to m a to n to d e m o n s tr a te m a n y o f th e d y n a m ic phe-
( n o m e n a p r o d u c e d o n t w o - d i m e n s i o n a l s y s te m s .- ’ E a r l i e r w e n o t e d t h a t t h e
A k a n a n d o t h e r G h a n a i a n s o c i e t i e s h a d a r e m a r k a b l e p r e c o l o n i a l u s e o f lo g a ­
r i t h m i c sp ira ls in i c o n i c r e p r e s e n t a t i o n s fo r l i v i n g s y s te m s . T h e G h a n a i a n f o u r ­
fo ld s p i r a l (fig. 6 . 4 a ) a n d t h e f o u r - a r m e d c o m p u t e r g r a p h i c in fig u re 7 . 1 2 b a r e
(a) Paired spirals emerge from a three-state cellular automation. Black cells are live, white cells are dead,
aiu! gray cells are in a refractory or f‘ghosc" state. The rules: Any dead nearest neighbors of a live cell
become live in the next iteration, and any live cell goes into the ghost state in the next iteration. The
refractory layer acts as a memory, providing the directed growth (i.e., the breaking of symmetry) needed
to create a spiral pattern.
(b) This four-armed logarithmic spiral from Markus (1991) was produced by a
six-state cellular automaton in which a sequence of ghost states corresponds
to increasingly dark shades of gray. The system makes use of a very highresolution grid as well as some random noise to prevent the tendency for
the patterns to follow the grid shape (as in the square contours of the spiral
above). Compare with che Ghanaian fourfold spiral in figure 6.4a.
• Bivalve shell.
(From H a e c k e l 1904.)
Mushroom cut in half.
North African sheep.
(From Cook 1914.)
(c) Paired logarithmic spirals often occur in natural growth forms.
(J) Recursive line replacement, as we saw for other fractal generations, can also produce such paired spirals.
FIGURE 7 . 1 2
S p ira ls in c e llu la r a u to m a ta
W e c a n v i e w t h e o w a r i b o a r d as a o n e - d i m e n s i o n a l c e l l u l a r a u t o m a t o n . O n e
d i m e n s i o n is n o t n e c e s s a r i l y a d i s a d v a n t a g e ; in f a c t , m o s t o f t h e p r o f e s s i o n a l
m a t h e m a t ic s o n c e llu la r a u t o m a t a (see W o lfr a m 19 8 4 , 1 9 8 6 ) h a v e b e e n d o n e on
o n e - d i m e n s i o n a l v e r s i o n s , b e c a u s e it is e a s i e r t o k e e p t r a c k o f t h e r e s u l t s . T h f e y c a n
sh ow a ll th e d y n a m ic s o f tw o dim en sion s.
T h e p a ttern s n o t e d by tra d itio n a l o w a r i players offer a g re a t d ea l o f in s ig h t in to
s e l f - o r g a n i z i n g b e h a v i o r . T h e i r o b s e r v a t i o n o f a c la s s o f s e l f - p r o p a g a t i n g p a t t e r n s ,
t h e ‘' m a r c h i n g g r o u p , " p r o v i d e s a n e x c e l l e n t s t a r t i n g p o i n t .
3 4 2 1 —> 5 3 2 —> 4 3 1 1 1 —> 4 2 2 2 —> 3 3 3 1 ~ » 4 4 2 —> 5 3 1 1 —> 4 2 2 1 1 —> 3 3 2 2 —> 4 3 3 —> 4 4 1 1 ~ » 4 5 5 2 —> 3 3 2 1 1 —>4321
T h e m a r c h i n g g r o u p is a n e x a m p l e o f a c o n s t a n t p a t t e r n . H e r e w e s e e c o u n t e r s in
th e in itial se q u e n c e 3421 c o n v e rg e o n th e ir in a rc h in g f o rm a tio n sim ply by re p e a tin g
t h e “s c o o p f r o m t h e left c u p " r u l e t h r o u g h 13 i t e r a t i o n s .
J u s t as w e s a w in t w o - d i m e n s i o n a l c e l l u l a r a u t o m a t a , t r a n s i e n t s o f m a n y d i f f e r e n t
l e n g t h s c a n b e p r o d u c e d . T r a n s i e n t s o f m a x i m u m l e n g t h a r e u s e d as a n e n d g a m e t a c t i c
by i n d i g e n o u s G h a n a i a n p l a y e r s , w h o c a l l it " s l o w m o t i o n ”— a c c u m u l a t i n g p i e c e s o n
y o u r side to p r e v e n t your o p p o n e n t from c a p tu r in g t h e m .In n o n l i n e a r d y n a m ic s , th e
c o n s t a n t p a t t e r n is c a l l e d a “ p o i n t a t t r a c t o r , ” a n d th e . t r a n s i e n t s w o u l d b e s a i d t o fie
in th e "b a sin of a t t r a c t i o n .”
T h e m a r c h i n g g ro u p rule c a n also p r o d u c e p e r io d ic b e h a v i o r (a " l im it c y c le " o r
“ p e r i o d i c a t t r a c t o r ” i n n o n l i n e a r d y n a m i c s t e r m s ) . H e r e is a p e r i o d - 3 s y s t e m u s i n g
only four counters:
2 U - » 22-* 31-» 2 1 J
W h i c h lead s to m a r c h i n g groups, a n d w h i c h o n e s iead to p e r i o d i c cycles?
T o ta l n u m b e r
of counters
T h e n u m b e r s w h i c h le a d t o m a r c h i n g g r o u p s —
1, 3, 6 , 10 , 1 5 • - • — s h o u l d l o o k f a m i l i a r t o re a de rs :
it ’s t h e t r i a n g u l a r n u m b e r s w e s a w in t a r u m b e r a !
T h e p e r i o d o f c y c l e s in b e t w e e n e a c h m a r c h i n g
g r o u p is g i v e n b y o n e p lu s t h e i t e r a t i o n l e v e l o f t h e
prev iou s trian gular n u m b e r rea ch ed .
( N o t e : S o m e s e q u e n c e s w ill b e t r u n c a t e d fo r
1 3 , 1 4, a n d 1 5 s i n c e t h e r e a re m o r e c o u n t e r s
th a n h oles.)
FIGURE
1
2
3
4
5.
6
B ehavior
(afrer tran sien ts)
....................M a r c h i n g
................ P e r io d 2
.............. . . M a r c h i n g
................ P e r i o d 3
.
P e r io d 3
....................M a r c h i n g
7.........................P e r io d
4
8...... ................ P e r io d 4
9 ...................... P e r io d 4
10 ....................M a r c h i n g
11
P e r io d 5
12
................. P e r i o d 5
13
................ P e r i o d 5
J4
................ P e r i o d 5
15 ....................M a r c h i n g
7 .1 3
O w ari as o n e-d im en sio n a l ce llu la r a u to m a to n
N um eric systems
107
q u i r e d i s t a n t in te r m s o f t h e t e c h n o l o g i e s t h a t p r o d u c e d t h e m , b u t t h e r e m a y
w ell b e s o m e s u b t l e c o n n e c t i o n s b e t w e e n t h e tw o . S i n c e c e l l u l a r a u t o m a t a
m o d e l t h e e m e r g e n c e o f s u c h p a t t e r n s in m o d e r n . s c i e n t i f i c s t u d i e s o f li v i n g sys­
tem s,, a n d c e r t a i n G h a n a i a n lo g s p i r a h ic o .n s w e r e a ls o i n t e n d e d as g e n e r a l i z e d
m o d e l s fo r o r g a n i c g r o w t h , it is n o t u n r e a s o n a b l e t o c o n s i d e r t h e p o s s ib ility t h a t
t h e s e l f - o r g a n iz i n g d y n a m i c s o b s e r v a b l e in o w a r i .w ere a ls o l i n k e d t o c o n c e p t s
o f b i o l o g i c a l m o r p h o g e n e s i s in t r a d i t i o n a l G h a n a i a n k n o w l e d g e sy s tem s .
R a t t r a y ’s c lassic v o l u m e o n t h e A s a n t e c u l t u r e o f G h a n a i n c l u d e s a c h a p ­
te r o n ow a ri, b u t u n f o r t u n a t e l y it o n ly c o v e r s t h e rules a n d stra te g ie s o f t h e gam e.
R e c e n t l y Kofi A g u d o a w u ( 1 9 9 1 ) o f G h a n a h a s w r i t t e n a b o o k l e t o n o w a r i “d e d ­
ic a te d to A f r i c a n s w h o a r e e n g a g e d in t h e f o r m id a b l e task o f r e c l a im in g t h e i r h e r ­
it a g e ,” a n d h e d o e s n o t e its a s s o c i a t i o n w i t h r e p r o d u c t i o n : w ari in t h e G h a n a i a n
n
lan g u a g e Tw i m e a n s “h e / s h e m a r r ie s .” H e r s k o v i t s ( 1 9 3 0 ) , n o t i n g t h a t t h e " a w a r i ”
1
F I GURE 7 . I 4
L ogarith.7nie cttrv es a n d oxvari
T he cover of die hinged owari board
we saw in figure 7.10 shows concentric
circles emanating from the Adinkra
icon for the power of god, “Gye
Nyame.” A similar icon; without the
logarithmic curves, is attributed to a
closed fist as a symbol of power. The
Gye Nyame symbol thus appears to be
a pair of logarithmic curves held in a
fist: God Holding the power of life.
A fr ic a n fra c ta l matfiemntics
io 8
g a m e p la y e d by t h e d e s c e n d a n t s o f A f r i c a n slave s in t h e N e w W o r l d h a d r e t a i n e d
som e o f th e p re co lo n ial cu ltu ral a sso ciatio n s from A frica, re p o rts t h a t aw ari h ad
a d i s t i n c t “ s a c r e d c h a r a c t e r 1' t o it, p a r t i c u l a r l y i n v o l v i n g t h e c a r v i n g o f t h e
b o a r d . O w a r i b o a r d s w i t h c a r v i n g s o f l o g a r i t h m i c s p ira ls (fig.^7.14) c a n b e c o m •of
m o n l y f o u n d in G h a n a to d a y , s u g g e s tin g t h a t W e s t e r n s c i e n t i s t s m a y n o t b e t h e
o n ly o n e s w h o d e v e l o p e d a n a s s o c i a t i o n b e t w e e n d i s c r e t e s e lf-o r g a n iz in g p a t t e r n s
a n d b io l o g ic a l r e p r o d u c t i o n . I t is a b i t v i n d i c t i v e , b u t I c a n ’t h e l p b u t e n j o y t h e
th o u g h t o f v o n N e u m a n n , ap o stle o f a m e c h a n is tic N e w W o rld O r d e r th a t
w o u ld w ip e o u t t h e i r r a t i o n a l c a c o p h o n y o f l i v in g s y s te m s , s p i n n i n g i n h i s g ra v e
e v e ry t i m e we w a t c h a c e l l u l a r a u t o m a t o n — w h e t h e r in p i x e l s o r o w a r i c u p s —
b r i n g f o r t h c h a o s in t h e g a m e s o f life.
C o n c lu sio n
-B oth t a r u m b e t a a n d o w a ri's m a r c J ) i n j : g ro up _d yn a,m ics a r e g o v e r n e d by t h e t r i ­
a n g u l a r n u m bers. T h e r e is n o t h i n g s p e c i a l a b p u t t h e t r i a n g u l a r n u m b e r s e r ie s —
s i m i l a r n o nHnjear^gtowth^prQ pe.r.t.ie^_can b e f o u n d i n t h e n u m b e r s t h a t fo r m
su c c e s s iv e ly la r g e r r ^ a j y g l e s ^ p e j y t a g o n s ^ o r o t h e j shap.es. N o r is t h e r e a n y t h i n g
s p e c i a l a b o u t t h e p o w e r s o f t w o w e f o u n d in d i v i n a t i o n — s i m i l a r a p e r i o d i c p r o p ­
e r t ie s c a n b e p r o d u c e d b y a p p l i c a t i o n s o f m o d 3 , m o d 4 , e t c . W h a t is s p e c i a l is
t h e u n d e r l y i n g c o n c e p t o f r e c u r s i o n — t h e w av s i n w h i c h a k i n d o f m a t h e M a t i c a l .
f e e d b a c k l o o p c a n g e n e r a t e n e w s t r u c t u r e s in s p a c e a n d ^ n e w d y n a m i c s i n tim e .
In t h e n e x t c h a p te r , w e w ill see h o w t h i s u n d e r l y in g p ro c e s s is f o u n d in b o t h p r a c ­
tical a p p lic a tio n s a n d a b stra c t sym bolics o f A fric a n cu ltu res.
CHAPTER
•Recursion-
_
8
_
R e c u r s i o n j s t h e m o t o r o f f r a c t a l ge o m e try; k j s j h e r e t h a t j h e , 1b as.ic,..transfor'
m a t i o n s — w h e t h e r n u m e r i c o r s p a t i a l — a re s p u n i n t o w h o l e c l o t h , a n d t h e p a tt e rn s t h a t e m e r g e o f t e n r e l l j h e s t o r y o f t h e i r w h i r l i n g b i r t h . W e w ill b e g i n by
d e f i n in g t h r e e ty p es o f re c u r sio n * 1 W h i l e it is p o ssib le to c a te g o riz e t h e e x a m p l e s
in t h i s c h a p t e r s o le ly o n t h e b asis o f t h e s e t h r e e ty p e s , it is m o r e i l l u m i n a t i n g
to c o m b i n e t h e a n a ly s is w i t h c u lt u r a l , c a t e g o r i e s . It is in e x a m i n i n g t h e i n t e r ­
a c t i o n b e t w e e n t h e t w o t h a t t h e use o f f r a c ta l g e o m e t r y as a k n o w l e d g e s y s te m ,
a n d n o t ju s t u n c o n s c i o u s s o c i a l d y n a m i c s , b e c o m e s e v i d e n t . T h e c u l t u r a l c a t ­
e g o rie s b e g i n w i t h t h e c o n c r e t e i n s t a n c e s o f re c u r s iv e c o n s t r u c t i o n t e c h n i q u e s
a n d g r a d u a l l y m o v e t o w a r d t h e a b s t r a c t i o n s o f r e c u r s i o n , . a s . s y m b o l i z e d in
A f r i c a n ic o n o g r a p h y .
T hree types o f r e c u r s io n
T h e le a s t p o w e rfu l o f t h e t h r e e , j s c a s c a d e r e c u r s i o n , in w h i c h t h e r e is a p r e - 1
d e t e r m i n e d s e q u e n c e o f s i m i l a r p ro c e s s e s . F o r e x a m p l e , t h e r e is a c h i l d r e n ’s
story in w h i c h a m a n buys a C h r i s t m a s t r e e , , b u t d is c o v e r s it is t o o tall for his
c e ilin g a n d c u ts o ff t h e t o p . H is do g s fin d t h e d i s c a r d e d t o p , a n d p u t it in t h e i r
d o g h o u s e , b u t t h e y t o o d i s c o v e r it is t o o t a l l , a n d c u t o ff t h e to p . F i n a l l y t h e
A fr ic a n fractal m athem atics
n o
m i c e d r a g t h i s t i n y t o p i n t o t h e i r h o l e , w h e r e it fits j u s t f i n e — t h e r e c u r s i o n
“ b o t t o m s o u t . ” N o t e t h a t t h e s e w e r e all i n d e p e n d e n t t r a n s f o r m a t i o n s ; it is o n ly
by c o in c i d e n c e , so to s p e a k , t h a t th e y h a p p e n e d t o b e t h e s a m e . F ig u r e 8. i a s h ow s
t h e n u m e r i c v e r s i o n o f c a s c a d e r e c u r s i o n , in w h i c h w e d i v t d e a n u m b e r by tw o
in e a c h p a r t o f t h e s e q u e n c e . TTiis is n o t a v ery p o w e r f u l ty p e o f r e c u r s i o n , fo,r
t w o T e n o n s . ' F i r s t , ' i t r e q u i r e s t h a t w e k n o w h o w m a n y t r a n s f o r m a t i o n s we w a n t
a h e a d o f t i m e — a n d t h a t is n o t a lw a y s p o ss ib le . I f t h e m o u s e w a s in c h a r g e , h e
w p.,uj,dhave sa id “j u s t k e e p d i v i d i n g u n t i l i t ’s s m a l l e n o u g h t o fit in m y . h o le . "
. S e c o n d ; w e h a v e to k n o w w h a t t r a n s f o r m a t i o n t o m a k e a h e a d o f t i m e , a n d t h a t
is n o t a l w a y s p o s s i b l e , e i t h e r . R e c a l l , fo r e x a m p l e , t h e g e n e r a t i o n o f t h e
F i b o n a c c i series w e sa w in c h a p t e r 7 (fig. 8 .1 b ) . A l t h o u g h t h e g e n e r a t i o n is ju s t
u s i n g a d d i t i o n , it c a n n o t b e c r e a t e d b y a r e c u r s i v e c a s c a d e , b e c a u s e t h e
a m o u n t t o be a d d e d i n e a c h t r a n s f o r m a t i o n c h a n g e s i n r e l a t i o n t o p r e v i o u s
re s u lts . G e n e r a t i n g t h e F i b o n a c c i s e r ie s r e q u i r e s a f e e d b a c k l o o p o r, as m a t h e ­
m a t i c i a n s c a ll it, i t e r a t i o n .
0
I n i t e r a t i o n , t h e r e is o n l y o n e t r a n s f o r m a t i o n . p r o c e s s , b u t e a c h t i m e t h e
p ro c e s s c r e a t e ^ a n o u t p u t ^ it u s e s t h i s r e s u l t as t h e i n p u t f o r t h e n e x t i t e r a t i o n ,
as w e ’v e s e e n in g e n e r a t i n g f r a c ta l s . A p a r t i c u l a r l y i m p o r t a n t v a r i e t y o f i t e r a ­
t i o n is “n e s t i n g , ” w h i c h m a k e s u se o f lo o p s w i t h i n lo o p s . H o f s t a d t e r ( 1 9 8 0 ,
1 0 3 - 1 2 9 ) n i c e l y i l l u s t r a t e s n e s t i n g w i t h a s t o r y in w h i c h o n e o f t h e c h a r a c t e r s
s ta rts to tell a story, a n d w i t h i n chat s t o r y a c h a r a c t e r s t a r ts t o re a d a passag e fro m
a b o o k . B u t a t t h a t p o i n t t h e r e c u r s i o n “b o t t o m s o u t " : t h e b o o k p a s s a g e g e ts
f i n i s h e d a n d w e s t a r t t o r a s c e n d b a c k u p t h e s t o r i e s . N e s t e d lo o p s a r e v e ry
c o m m o n in c o m p u t e r p r o g r a m m i n g , a n d w e c a n i l l u s t r a t e t h i s w i t h a p r o g r a m
fo r d r a w i n g t h e a r c h i t e c t u r e j ^ f . M f l k o u l e k . (fig. 8 . 1 c ) , w e ex.arni.ned. in . c h a p t e r . .2....
T h e B a -ila a r c h i t e c t u r e we saw in c h a p t e r 2 c a n a ls o b e s i m u l a t e d t h i s way, using
o n e lo o p for th e r i n g s - w it h in - r in g s , a n d a n o t h e r for t h e f r o n t- b a c k s c a lin g
g r a d i e n t t h a t m a k e s u p e a c h o f t h o s e rin g s. In c h a p t e r 6 t h e first c o r n - r o w h a i r ­
s t y le (ipciko e le d e ) s h o w e d b r a i d i n g as a n i t e r a t i v e lo o p ; t h e s e c o n d c o r n - r o w
e x a m p l e a d d e d a n o t h e r i t e r a t i v e lo o p o f s u c c e s s iv e p e r i m e t e r s o f b r a i d s . 2 U is
c o m m o n for c o m p u t e r p r o g r a m s to d o s u c h n e s t i n g s e v e r a l layers d e e p , a n d k e e p ­
i n g t r a c k o f all t h o s e lo o p s w i t h i n l o o p s c a n . b e q u i t e a .c h o re .:T h e t h i r d ty p e o f r e c u r s i o n is “s e l f - r e f e r e n c e . ” W e a r e all f a m i l i a r w i t h t h e
w a y t h a t sy m b ols o r i c o n s c a n re fe r t o s o m e t h i n g : t h e s t a r s a n d s t r ip e s flag refers
t o A m e r i c a , t h e s k u l l - a n d - c r o s s - b o n e s la b e l re fe rs t o p o i s o n , t h e g r o u p o f l e t ­
ters c - a - t refers to a n a n i m a l . B u t i t ’s a ls o p o s s ib le for a s y m b o l t o re f e r t o itself.
K e l l o g g ’s c o r n f l a k e s , fo r e x a m p l e , o n c e c a m e in a b o x t h a t f e a t u r e d a p i c t u r e o f
a fa m il y s i t t i n g d o w n to b r e a k f a s t . In t h i s p i c t u r e y o u c o u l d se e t h a t t h e fa m ily
h a d a b o x o f K e llo g g ’s c o r n f l a k e s o n t h e i r h re a k f a s t ta b l e , a n d y ou c o u l d see t h a t
Recursion
ill
th is b o x s h o w e d che s a m e p i c t u r e o f t h e family, w i t h t h e s a m e b o x o n t h e i r t a b l e ,
a n d so o n t o i n f i n i t y ( o r a t le a s t to as s m a l l as t h e K e llo g g c o m p a n y ’s a r t i s a n s
c o u ld draw ).
S e l f - r e f e r e n c e is b e s t . k n o w n fo r its r q l e j n Jp g ic a ] p a r a d o x . If, fo r e x a m p l e ,
yo u w ere to a c c u s e s o m e o n e o f lying, it w o u ld b e a n o r d i n a r y statement".”£ u t s u p ­
pose you a c c u s e y ou rself o f lying? T h i s is t h e p a r a d o x o f E p im e n id e s o f C r e t e , w h o
d e c l a r e d t h a t “ all C r e t a n s a re liars.” If h e ’s t e l l i n g t h e t r u t h , h e m u s t b e lying,
b u t if h e ’s lying, t h e n h e ’s t e l l i n g t h e t r u t h . T h e r o l e o f s e l f - r e f e r e n c e in lo g ic a l
input
X
2
Nnr
e
previous
4
2
X
2
X
output
2
Nncxt
w h ile
e -co u n t < 4 do:
• d raw en closu re
w h ile
g-cou n t < 1 2 do:
• draw granery
• rocate to w a rd c e n te r
• s h r in k g r a n e r y size
• in crea se g -c o u n t by 1
end
o f g - c o u n c ’s l o o p
• reset g - c o u n t to 0
• rotate tow ard c e n t e r
"
• s h r in k e n c lo s u r e size
-
• in crease e -c o u n t b y 1
end
of e - c o u n t ’s loop.
FIGURE 8.1
R ecizrsiv e c a sc a d e v e r s u s ite r a tio n
(a) A r e c u r s i v e c a s c a d e , i n w h i c h t h e s a m e t r a n s f o r m a t i o n ( d i v i s i o n b y t w o ) h a p p e n s t o b e u s e d
m e a c h p a r t o f a s e q u e n c e . T h i s r e q u i r e s k n o w i n g h o w m a n y t i m e s ch e t r a n s f o r m a t i n s h o u l d
h ap p e n a h e a d o f t i m e . It a l s o r e q u i r e s t h a t t h e t r a n s f o r m a t i o n is i n d e p e n d e n t o f p r e v i o u s r e s u l t s .
{b) T h e F i b o n a c c i s e q u e n c e is p r o d u c e d by a d d i n g t h e p r e v i o u s n u m b e r t o t h e c u r r e n t n u m b e r t o
gee t h e n e x t n u m b e r , s t a r t i n g w i t h 1 + 1 = 2 . I n t h e F i b o n a c c i s e q u e n c e w e a d d a d i f f e r e n t a m o u n t
ia e a c h i t e r a t i o n — w e c o u l d n o t k n o w h o w m u c h e a c h t r a n s f o r m a t i o n s h o u l d a d d a h e a d o f t i m e ,
s a a r e c u r s i v e c a s c a d e w o u l d n o t d o t h e j o b . ( c ) I n s o m e c a s e s it is n e c e s s a r y t o p u t a n i t e r a t i v e
loop in sid e a n o t h e r i t e r a t i v e l o o p ( " n e s t i n g " ) . H e r e is a n e x a m p l e o f n e s t i n g i n a c o m p u t e r
program for d r a w i n g t h e a r c h i t e c t u r e o f M o k o u l e k w e e x a m i n e d i n c h a p t e r 2. It is w r i t t e n i n w h a t
p ro g ram m ers c a ll “ p s e u d o c o d e , " a m i x t u r e o f a p r o g r a m m i n g l a n g u a g e a n d o r d i n a r y E n g l i s h . T h e
first lo o p d r a w s t h r e e la r g e e n c l o s u r e s , a n d t h e i n n e r l o o p d r a w s 1 2 g r a n e r i e s i n s i d e e a c h e n c l o s u r e .
Variable “e - c o u n t ” t r a c k s t h e n u m b e r o f e n c l o s u r e s , a n d g - c o u n t t r a c k s t h e n u m b e r o f g r a n e r i e s .
A fr ic a n fra c ta l m athem atics
112
p a r a d o x h a s b e e n i m p o r t a n t for m a t h e m a t i c a l th e o r y , b u t i t h a s a l s o b e e n p u t to
prac tical use in c o m p u t e r p ro g ra m m in g . M o s t p r o g r a m m i n g h a s little ro u tin e s called
“ p r o c e d u r e s , ” a n d o fte n , a p r o c e d u r e w ill n e e d t o c a ll o t h e r p r o c e d u r e s . I n selfr e f e r e n t i a l p r o g r a m m i n g t h e p r o c e d u r e .calls itself.
P r a c t i c a l f r a c t a l s : recu rsio n in c o n s t r u c t i o n t e c h n i q u e s
f I n h is d is cu ssio n o f t h e m e t a l - w o r k i n g t e c h n i q u e s o f A f r i c a , D e n i s W i l l i a m s g iv e s \
| a p o e t i c d e s c r i p t i o n o f re c u r s iv e c a s c a d e in t h e e d a n b ra s s s c u l p t u r e s o f t h e I
V Y oruba: “T h e im a g e p r o l i f e r a t e s li k e l i g h t s in a b u b b l e : o n e e d a n b e a r s in its lap
a n o t h e r , s m a ll e r v e r s io n o f itself, w h i c h b e a rs in t u r n a s m a l l e r in its la p , a n d th is
b e a rs a n o t h e r in its lap , e t c . — a s o r t o f s c u l p t u r a l re la y r a c e ” ( 1 9 7 4 , 2 4 5 ) . W h i l e
t h e e d a n s c u l p t u r e s a r e u n i q u e t o t h e Y o ru b a , r e c u r s i v e c o n s t r u c t i o n t e c h n i q u e s
a re q u i t e c o m m o n in A f r i c a . For e x a m p l e , W i l l i a m s g o e s o n t o n o t e t h a t m u c h
A f r i c a n m e t a lw o rk , u n l i k e E u r o p e a n i n v e s t m e n t c a s t in g , uses a “spiral t e c h n i q u e ”
t o b u i l d u p s t r u c t u r e s fr o m s in g le s t r a n d s ( w h e t h e r b e f o r e c a s t i n g , as in t h e lost
w ax te c h n iq u e ,
o r a f t e r w a r d s as w ir e ) , r e s u l t in g in “h e l i c a l c o il s f o r m e d fro m
s m a l l e r h e l i c a l c o i l s .” A w ig m a d e f r o m m e t a l w ir e s (fig. 8 . 2 a ) s h o w s a s i m i l a r
i t e r a t i v e c o n s t r u c t i o n u s i n g c o i l s m a d e o f c o ils . I n c h a p t e r 6 w e s a w s o m e
e x a m p l e s o f A f r i c a n h a i r s t y l e s in w h i c h e i t h e r a d a p t a t i o n t o c o n t o u r s o r
ab stra c t sp atial tr a n s fo rm a tio n resu lted in a scalin g p a tte r n . T h e fractal braids
; s h o w n in fig u re 8 . 2 b h a v e n o t h i n g to d o w i t h t h e s h a p e o f t h e h e a d ; t h e y a re
r a th e r th e result o f su ccessiv e ite ra tio n s t h a t c o m b i n e s tr a n d s o f h a ir in to
b ra id s , b r a i d s i n t o b r a i d s o f b r a i d s , a n d so o n . F i g u r e 8 . 2 c s h o w s a n o t h e r w ig,
th i s o n e fo r a s c u l p t u r e , t h a t f e a t u r e s b r a i d s o f m a n y sc a le s.
T h i s c o l l e c t i o n o f s c u l p t u r e , m e t a l w o r k , a n d h a i r s t y l i n g s o u n d s like., a
m o t l e y a s s o r t m e n t , b u t o n c e w e . s t a r t .l o o k i n g f o r r e c u r s i o n w e see a c lo s e r e l a ­
t i o n : a ll e x a m p l e s u s e d a s in g le t r a n s f o r m a t i o n — s t a c k i n g , b r a i d i n g , c o i l i n g —
t h a t w a s a p p l i e d s e v e r a l t i m e s . L o o k i n g a t t h e R e la tio n b e t w e e n t h e b a s i c
t r a n s f o r m a t i o n a n d its final o u t c o m e c a n h e l p us d is ti n g u is h m n o n g j d i/ f e r e n t types
cT r e c u r s i o n . T h e b r a i d i n g p a t t e r n o f fig u re 8 .2 b , fo r e x a m p l e , is b a s e d o n i t e r ­
a t i o n , b e c a u s e t h e w a y e a c h s t a g e is b r a i d e d d e p e n d s o n t h e b r a i d s p r o d u c e d in
p r e v i o u s sta g e s; t h e y a r e b r a i d s o f b r a i d s . T h e b r a i d s in fi g u r e 8 . 2 c , o n t h e
o t h e r h a n d , a r e o f d i f f e r e n t s c a l e s s i m p l y b e c a u s e e a c h s t a g e u ses d i f f e r e n t
a m o u n ts o f sin g le-h air strands— a cascade o f p re d e te rm in e d tran sfo rm atio n s.
S i m il a r l y , t h e c o i l s o f c o il s i n d i c a t e i t e r a t i o n , b e c a u s e t h e o u t p u t o f o n e s t a g e
b e c o m e s t h e i n p u t for t h e n e x t .
R e c u r s i v e c o n s t r u c t i o n .. t e c h n i q u e s a r e a l s o u s e d . f o r t h e d e c o x a t i v e
d e s i g n s o f A f r i c a n a r t i s a n s . In o u r d i s c u s s i o n o f t h e f r a c t a l e s t h e t i c in cha p'-
Recursion
t e r 4, w e e x a m i n e d d e c o r a t i v e p a t t e r n s w h i c h d i d n o t p r o v i d e e v i d e n c e fo r a
fo r m a l g e o m e t r i c m e t h o d . T h a t d o e s n ’t m e a n n o f o r m a l m e t h o d c o u l d p ossibly
e x is t; i t ’s j u s t t h a t n o n e c o u l d b e r e a d i l y d i s c e r n e d f r o m t h e d e s i g n itse lf, a n d
th e a r t is a n s d id n o t r e p o r t a n y t h i n g b e y o n d i n t u i t i o n o r e s t h e t i c ta s te . B u t th e r e
a re s o m e d e s i g n s t h a t d o i n d i c a t e a n e x p l i c i t re c u r s iv e t e c h n i q u e fr o m t h e p a t ­
t e r n itself. F ig u r e 8 . 2 e s h o w s a M a u r i t a n i a n t e x t i l e w i t h t w o s u c h s c a l i n g p a t ­
t e r n s . I n t e n t i o n a l a p p l i c a t i o n o f i t e r a t i o n a s a c o n s t r u c t i o n t e c h n i q u e is
i n d i c a t e d by t h e w a y t h e X f r a c t a l ’s s e e d s h a p e is s h o w n o n e i t h e r s i d e , a n d
by h a v i n g i t e r a t i o n c a r r i e d o u t o n t w o c o m p l e t e l y d i f f e r e n t s e e d s h a p e s in t h e
s a m e p i e c e . T h e t r i a n g l e f r a c t a l ( c l o s e t o w h a t m a t h e m a t i c i a n s c a l l t h e “S ie r p i n s k i g a s k e t ” ) is a ls o f o u n d in M a u ' r i t a n i a n - S t o n e w o r k (fig . 8 . 2 f ) . A t h r e e d i m e n s i o n a l v e r s i o n f r o m G h a n a (fig. 8 . 2 h ) m a y h a v e b e e n i n s p i r e d b y t h e s e
d e s ig n s .
B o t h o f t h e a b o v e a re e x a m p l e s o f a d d i t i v e c o n s t r u c t i o n , as w e s a w i n th e
K o c h c u r v e o f c h a p t e r i , b u t s u b t r a c t i v e i t e r a t i o n s , as w e s a w fo r t h e C a n t o r
s e t, a r e a ls o f o u n d i n _ A f r i c a n d e c o r a t i v e f r a c t a l s (fig. 8 . 2 i ) . C a r v i n g d e s i g n s
in clu d e a p p lic a tio n s o f ite ra tiv e c o n s tru c tio n , p a rtic u la rly for c a la b a sh d e c o ­
r a t i o n s (fig. 8 . 2 I ) . A g e o m e t r i c a l g o r i t h m fo r p r o d u c i n g n o n l i n e a r s c a l i n g
t h r o u g h f o l d i n g w as i n v e n t e d b y t h e Y o r u b a a r t i s a n s w h o p r o d u c e d t h e a d ir e
c l o t h o f fig u re 8 . 2 n . It is n o t m e r e l y a m e t a p h o r t o re f e r t o a s p e c i f i e d se rie s
o f fo ld s as a l g o r i t h m i c ; in f a c t , o n e o f t h e c la s s ic f r a c ta l s , t h e “ d r a g o n c u r v e , ”
w as d i s c o v e r e d in i 9 6 0 w h e n p h y s i c i s t J o h n H e i g h w a y e x p e r i m e n t e d w i t h
i t e r a t i v e p a p e r f o l d i n g ( G a r d n e r 1 9 6 7 ) . T h e a d i r e c l o t h a l sp .s h e w -s -th g .a p p lic a t i o n o f r e f l e c t i o n ^ yjTuj i e t r y a t e v e ry-scale f r o m s i n g l e - s t i t c h j p w s , w h i c h are
r e f l e c t e d o n e i t h e r s i d e o f t h e fo ld e d g e s , t o t K F e n t i r e fa b r ic , w h i c h is c r e a t e d
by t h e j o i n i n g o f t w o m i r r o r i m a g e c l o t h s .
S o far w e h a v e o n ly discussed t h e t e c h n i c a l m e t h o d e m p l o y e d , b u t o f course
c id tiira l m e a n i n g is_.ofl.en a t t a c h ed.tp-t-hegfi. t e c h n i q u e s a s w e ll. R e c u r s i v e h a i r ­
styles, for e x a m p l e , e m b e d layers o f s o c i a l l a b o r w i t h e a c h i t e r a t i o n , a w a y t o
invest p h y s i c a l a d o r n m e n t w i t h socJal_.me a n i n g (s u c h as f r i e n d s h i p b e t w e e n sty l­
ist a n d s t y l e e ) . F ig u r e 8 . 3 a s h o w s a F u l a n i w e d d i n g b l a n k e t , i n w h i c h s p i r i t u a l
e n e rg y is e m b e d d e d in t h e p a t t e r n t h r o u g h its i t e r a t i v e c o n s t r u c t i o n 1'. P r e s tig e ^
c a n a ls o b e a s s o c i a t e d w i t h i n c r e a s i n g i t e r a t i o n s , as w e f i n d f o r b ra s s c a s t i n g
a n d b e a d w o r k in t h e g ra s s la n d a r e a s o f C a m e r o o n (fig. 8 - 3 b ,c ) . T h e s c a l in g iter' a ti o n s in o n e o f t h e b ra s s s c u l p t u r e s (fig . 8 . 3 d ) w as r e p o r t e d t o b e s y m b o l i c as
well: it s h o w e d t h r e e g e n e r a t i o n s o f ro y a l ty . B u t k i n s h i p g r o u p s a r e n o t j u s t
st a ti c e n t i t i e s ; t h e y c h a n g e a c ro s s t i m e , a n d in t h e f o l l o w i n g tw o s e c t i o n s we
will see t h a t A f r i c a n r e p r e s e n t a t i o n s o f s u c h t e m p o r a l p r o c e s s e s o f t e n i n v o l v e
're c u rs io n .
I
.e, -
FIGURE 8 . 2
R e cu rsiv e co n stritctio n te c h m q u e s
(a) Coils of coils are used to create this metal wig from Senegal, (b) A scaling cascade of
a mask from the Dan societies of Liberinand Cote d'Ivoire, (c) Iterative braiding in tiiis
from Yaounde, Cameroon, la t r e s s e cle f l , can be simulated by fractal graphics- (d) Three if
of the t r e s s e d e f l simulation.
fb, f r o m Barbier-Muelfer 1988.)
(/'.far
F IG U R E
8 .2
(c o n tin u e d )
I te r a t iv e c o n s tr u c tio n
in M a u r i t a n i a n d e c o ra tio n
(e) Recursive construction with triangles and
X-shapes in'Tuareg leatherwojk. The X-slvape
is related to the quincunx discussed in chapter 4.
(f) Designs using several iterations of triangles
can also be found in Mauritanian stonework.
(g) T he use of triangles in this nomadic
architecture from Mauritania may be one
reason for the popularity of the design. Unlike
rectangles, triangles can create a rigid frame
using flexible joints— an important feature in
a landscape where long poles are scarce and
lashing is the most common joinery, (h) A single
iteration of a three-dimensional version of the
recursive triangle construction, created by Akan
artists in Ghana.
(e, f r o m Jefferson 1973 ;• f and g, photos courtesy
/FAN, D a k a r ; h, f r o m P h i lli p s 1 9 9 5 , f i g 5. J03 )
(fig u re c o n tin u e s )
A fr ic a n fra c ta l m a th e m a tic s
FIGURE 8 -2 (continued)
S c a lin g p a t t e r n fro m s u b tr a c tiv e i t e r a tio n
(i) A
F a n te w o m a n p o s in g in fr o n t o f a p a in t e d s t u d io b a c k d r o p , C a p e C o a s t , G h a n a , i 8 6 0 .
(j) T h e Fante pattern can be th o u g h t of as tw o iteratio ns o f scaling su btraction ( t h a t is, erasing).
Strips are erased from an ail-black background. W h e r e th e th ick strips intersect, we get large
squares, and where the r.hin strips intersect we get small squares.
(i, photo f r o m the Notional Museum of African Art, Smit/istmfan Institution.)
-
( fig u r e .c o n ijv .u
R e p r e s e n t i n g r e c u r s io n a s a p ro c e s s in tim e : p a r t I , l u c k a n d age
A sim p le e x a m p l e o f A fr i c a n rep rese n t a t i o n for re c u rs io n as a tim e - v n ry ing.orDc.ess
is s h o w n in fig u re 8 .4 , w h e r e w e_gee t h r e e d e s i g n s t h a t d e p i c t w is h e s f o r c a t c h e s
o f e v e r l a r g e r fish. S i n c e t h e e x p e r i e n c e o f b a d l u c k o r g o o d l u c k in f i s h i n g c a n
o c c u r o n a d a ily b asis, it is easy t o s e e h o w a b ig fish c o u l d b e c o m e a n i c o n fo r
g o o d luck . B u t in th e s e d e s i g n s t h e a r t i s a n s t a k e t h e c o n c e p t a s t e p f u r th e r. G o o d
f o r t u n e is n o t in t e r m s o f a s i n g u l a r c h a n c e e v e n t , as o n e se e s in che m y t h s o f
t h e N a t i v e A m e r i c a n t r ic k s t e r .'' T h e w is h is f o r a n i t e r a t i v e p r o c e s s — t h a t e a c h
fisiTfs to h e s u c c e s s iv e l y , .larger..t.han t h e j a s t o n e.
W h i l e th e s e g o o d lu c k ic o n s a re o f t e n a m o r e in f o r m a l p a r t o f c u l t u r a l p r a c ­
tic e , o t h e r re c u r s iv e p ro c e s s e s a r e t a k e n m u c h m o r e serio u sly . A n t h r o p o l o g i s t s
Seed sh iipe, w i t h a c t i v e
li ne s in gra y.
F ourth iteration en larged, w ith ad a p tiv e scalin g
( m a p p i n g fro m a sph e re to a p la n e ) a p p lie d to
m a tc h the a d a p tiv e s ca lin g o f th e calab ash design.
F I G U R E 8 . 2 (continued)
Itera tio n in c a r v in g s
j& O t) T h e B a k u b a o f Z a i r e c r e a t e d s e v e r a l c a r v i n g s t h a t f e a t u r e a s e l f - s i m i l a r d e s i g n . T i t is B a k u b a
,'fcooJen b o t t l e m a k e s u s e o f h e x a g o n s o f h e x a g o n s a s w e l l a s a d a p t i v e s c a l i n g a s it n a r r o w s i n t o c h e
n c c L ( I) C h a p p e l ( 1 9 7 7 ) r e c o r d s a w i d e v a r i e t y o f c a l a b a s h d e s i g n s , m a n y w i t h s c a l i n g a t t r ib u t e s .
<,T h i s is p r o b a b l y t h e b e s t e x a m p l e o f i t e r a t i v e c o n s t r u c t i o n i n t h e s e c a r v i n g s . T h e d e s i g n
J iin iu la t io n n o t o n l y r e q u ir e s r e c u r s i o n b u t a d a p t i v e s c a l i n g a s w e ll - ( m ) S e e d s h a p e a n d f o u r t h
I t e r a t i o n ; f o u r t h i t e r a t i o n e n la r g e d , w i t h a d a p t i v e s c a l i n g a p p lie d .
ifk, c o u r te s y
M u s e e R o y a l d c I A f r i q u e C e n t r a l , B e l g iu m . )
(figure c o n tin u e s )
F I G U R E 8 . 2 (continued)
A d i r e c l o t h : s c a l i n g fr o m i t e r a t i v e f o l d i n g
(n ) T h is Yoruba adire clo th is actually two separate pieces a tta c h e d along th e horizontal midline.
T h e dye pattern is created by sewing alon g folds before dye is applied an d th e n rem oving the
th reads so th a t th e w h ite lines are left where th e dye did n o t p e n e tra te , (o) T h e folding m etho d is
based o n reflection symmetry across a diagonal. It is easiest to understand by malting a paper model.
T h e adire artisans have n o t only
developed an algorithm fot generating
this n o n lin ear scaling series, b u t have
d on e so in a way th a t maximizes efficient
production: all folds fall along th e same
two edges, so only two edges need be
sewn. Your paper m odel c a n im itate this
effect by ru n n in g a heavy felt marker
along the two edges, so th a t the ink
bleeds th rough all th e layers (you can
c h eat by inking e a c h fold as you unfold
it). N o te th a t th e w hite lines in th e adire
are triple— this, too, is created by a
reflection symmetry, sewing n ex t to the
fold to create the tw o outer lines (on e on
each side of the fold), and sewing right
on the edge of th e fold to create the
ce n te r line.
(n, photo from Picton and Mack 1979.)
First,
cu t o u t
a p a p e r rectan g le w ith w id th tw ice d ie h e ig h t,
a n d f o ld it in h a lf , m a k i n g a s q u a re .
S e c o n d , fold t h e s q u a re a lo n g
d ia g o n a l, m a k in g a triangle.
T hird,
m ark po in ts
at
Zz
a n d Id o f
t h e o u t e r sides
o f th e triangle.
T h e s e p o in ts c a n
be
d e t e r m i n e d h y f o l d i n g , if
o n e w ish es tn m a i n t a i n th e
o rig am i e q u iv a le n t o f co m p ass a n d
F o u r th , fold f ro m t h e c o r n e r s o u
F in ally , f o ld
S traig h t-ed g e c o n s tru c tio n , h u t d o in g
o p p o s ite sides a lo n g t h e lin e b e tw e e n
o v erlap p in g c o rn e r on
it hy e y e b a l l w o r k s j u s t fine.
the
Zz
rind
Vt
m aiks.
side.
in
th e sm n
FIG.URE 8 .3
M a k i n g m e a n in g
th r o u g h ite r a tiv e
c o n s tr u c tio n
(a) T h is Fulani
w e d d i n g b l a n k e t f ro m
M a l i is b a s e d o n
d i a m o n d s t h a t s c a le
f r o m e i t h e r s i d e as
we m ove tow ard the
c e n t e r ; a p a t t e r n t h a t is
eas ily s i m u l a t e d u s i n g a
f rac ta l' ( s e e " d i a g r a m ) : " "
T h e w eavers w ho
c r e a t e d it r e p o r t t h a t
s p i r i t u a l e n e r g y is
w oven in to th e pattern,
m d t h a t e a c h s u c c e s s i v e i t e r a t i o n s h o w s a n i n c r e a s e in
h i s e n e r g y . R e l e a s i n g t h i s s p i r i t u a l e n e r g y is d a n g e r o u s ,
i n d if t h e w e a v e r s w e r e t o s t o p in t h e m i d d l e t h e y w o u l d
isk d e a t h . T h e e n g a g e d c o u p l e m u s t b r i n g t h e w e a v e r
o o d a n d k o l a n u t s t o k e e p h i m a w a k e u n t i l it is f i n is h e d ,
b) T h e prestige b ro n ze o f F o u m h a n , C a m e r o o n , o ften
n a k e s u se o f s e l f - s i m i l a r i t e r a t i o n s , ( c ) P r e s t i g e is als o
y m b o l i z e d by t h e l a b o r a n d a r t i s t r y r e q u i r e d t o p r o d u c e
th e m a n y ite ra tio n s of b ead p a tte n s for this e le p h a n t
m ask, (d ) A c c o rd in g to S alefou M b e tu k o m , th e lead in g
ca s to r of F o u m b a n , this sc u lp tu re show s th e successio n of
k i n g s in t h e r o y a l fam ily .
(c,/ro m
rig en ce H o a - Q u i . )
A fr ic a n fra c ta l mac/iematics
120
h a v e alw a ys b e e n i n t e r e s t e d in t h e c o n t r a s t b e t w e e n t h e e l a b o r a t e p o l i t i c a l a n d
e c o n o m i c h i e r a r c h y o f E u r o p e a n s o c i e t i e s a n d t h e r e l a t i v e l y “c las sles s" ( s o m e ­
t i m e s e v e n r i d e rle s s ) s t r u c t u r e o f m a n y p r e c o l o n i a l A f r i c a n s o c i e t i e s ^ J f it is n o t
p o litic a l a n d e c o n o m i c s tr u c tu r e t h a t g o v e rn s th e i r so ciety , t h e n w h a t does?
O n e p a r t o f t h e a n s w e r is ag e. A l l h u m a n c u l t u r e s d i f f e r e n t i a t e b e t w e e n c h i l -
c
f i g u r e 8 .4
I f w is h e s w e r e fish e s
(a) Scaling scales: this Bamana tattoo,
created with henna, is said to represent the
scales of fish, h is good luck, .signifying everlarger fish catches, (b) This is an "abbia,” a
carved gambling chip from Cameroon.
Given the high stakes of the game, it could
be a more aggressive symbolism than just
luck, e.g., "just as you have swallowed others,
! will swallow you.” O ther chips appear to
carry the iteration out several more levels,
although they are less recognizable as fish (c).
(d) This print with four iterations of fish is
from northern Ghana. It was reported to be
a fertility symbol.
( d , p/iofo courtesy o f Traci Roberts a n d
A n n C am pbell.)
d
Recursion
121
c lr e n , adj.iJcs,jind e ld e r s , b u t in m a n y A f r i c a n s o c i e t i e s t h e d iv i s i o n s a r e m u c h
m o r e e l a b o r a t e a n d s t r u c t u r e d . I n t h e s e a g e - g r a d e sy s te m s , aH c o m m u n i t y m e m ­
b e r s b o r n w i t h i n a g i v e n n u m b e r o f y e a rs w ill m o v e t o g e t h e r t h r o u g h a se r ie s
o f r i t u a l i n i t i a t i o n s . In c h a p t e r 5 w e saw ,.one e x a m p l e in w h i c h t h e s e i n i t i a t i o n
sta g e s a p p e a r e d ro be a c c o m p a n i e d by a n i t e r a t i v e s c a l in g g e o m e try , t h e lu s o n a .
Figure 8 .5 a show s a n o t h e r g e o m e t r i c v is u a liz a tio n o f ag e-g ra de in itia tio n : a h e x a g ­
o n a l m a s k c r e a t e d by che Bassari o f t h e S e n e g a m b i a n a n d G u i n e a - B i s s a u region .
A l t h o u g h t h e niask is o n ly a l i n e a r - c o n c e n t r i c sc a lin g o f h e x a g o n s , a n d th u s
n o t a fra ctal, it d o e s suggest a n i t e r a t i v e pro cess, a n d w e m i g h t well s u s p e c t a lin k
b e t w e e n sta g e s in a g e -g r a d e a n d s t a g e s in i t e r a t i o n . T h e i n i t i a t i o n p r o c e s s is a
c lo sely g u a r d e d s e c r e t, so it is n o t s i m p l y a m a t t e r o f a s k i n g Bassari e x p e r t s , b u t
d u r i n g m y v is it w i t h t h e B assari in 1 9 9 4 1 f o u n d t h a t t h e m e a n i n g o f o t h e r
m a t h e m a t ic a l p a tte rn s in Bassari c u ltu re c a n b e used to m a k e so m e ed u c a te d guesses
a b o u t t h e m e a n i n g o f t h e m as k . D e s p i t e t h e e x t e n s i v e m i g r a t i o n s fr o m t h e v i l ­
lages to c it ie s ( N o l a n 1 9 8 6 ) , t h e r e is s till s t r o n g p a r t i c i p a t i o n in t h e age g ro u p s
a n d t r a n s i t i o n rituals. T h e “fo r e st s p i r i t ” A n n a k u d i , fo r e x a m p l e , se e m s t o be
u n d a u n t e d by t h e c ity o f T a m b a c o u n d a , w h e r e a lo c al age g ro u p h o s t e d h i m a t
a w e l l - a tt e n d e d d a n c e d u r i n g m y stay. I n d e e d , I fo u n d t h e s te re o ty p e o f tr a d i t i o nal
e ld ers a n d i r r e v e r e n t y o u t h s to b e s o m e w h a t r e v e r s e d ( w h i c h w as e x p l a i n e d to
FIGURE
S c a lin g h e x a g o n s
in
8 .5
a Bassari m a sk
|.U) T h e B a s s a r i i n i t i a t i o n m a s k s f r e q u e n t l y f e a t u r e s c a l i n g h e x a g o n s in t h e c e n t e r . T h i s a p p e a r s t o
a lin e a r s c a l i n g , ( b ) O n e o f t h e B a s s a r i e l d e r s d e m o n s t r a t e s t h e t r a d i t i o n a l s t T i n g t a lle y s , w i t h
5ls
in g r o u p s o f s ix .
I,p fioto f r o m a g e n c e H o a - Q u i/ M ic h e / R e n n n d e m i■)
A fr ic a n fractal m athem atics
122
m e as a n e ff e c t o f t h e s t r o n g h i e r a r c h y o f s e c r e t k n o w l e d g e : t h e y o u t h a r e o f t e n
m o r e w a ry a b o u t b r e a k i n g t a b o o s b e c a u s e t h e y a re less c e r t a i n a b o u t b o u n d a r i e s
a n d c o n s e q u e n c e s ) . T h i s is n o t t o say t h a t t h e r e is a n y o v e r t p r e s e n c e o f fear. In
« '1
f a c t , it is t h e p o s i t i v e a s p e c t s o f t h e s e c r e ts t h a t a re stre s s e d , as b e c a m e o b v i o u s
\
w h e n e l d e r s g l e e f u ll y re f u s e d m y q u e s t i o n s w h i l e e m p h a s i z i n g t h e w o n d e r f u l
'. n a t u r e o f th e in f o r m a tio n th e y c o u ld n o t div u lg e.
T h e n u m b er.-S.lx is a p r o m i n a n t f e a t u r e o f B assari m a t h e m a t i c s in m a n y
a re a s o f t h e i r life. T h e y h a v e a p o p u l a r g a m e , for e x a m p l e , p l a y e d w i t h p e b b l e s
o n a s a n d p a t t e r n , w h i c h m a k e s u se o f tw o a x e s w i t h six h o l e s in e a c h li n e . In
t h e i r t r a d i t i o n a l c a l e n d a r t h e r e a r e six m o n t h s p e r year, e a c h o f 3 0 ( 6 x 5 ) days,
w i t h a n i n i t i a t i o n a b o u t e v e ry 1 2 (6 X 2 ) y e ars ( t o a t o t a l o f n i n e i n i t i a t i o n s ) .
E ach o f th e s e rites of passage in v o lv e s a le n g th y e d u c a tio n in a n e w lev el of
t r a d i t i o n a l k n o w l e d g e . T h e m o s t i m p o r t a n t is t h e p a s s a g e t o a d u l t h o o d , w h i c h
las ts for six d ay s. I n a d d i t i o n t o t h e s e t i m e m e a s u re s , t h e n u m b e r six a ls o a p p e a rs
in t h e B assa ri c o u n t i n g s y s te m . S t r i n g t a l l i e s , t r a d i t i o n a l l y u s e d f o r r e c o r d i n g
v a r i o u s c o u n t s , o f t e n u sed k n o t s g r o u p e d b y six. T h e B assari e l d e r w h o d e m o n ­
s t r a t e d t h e s e t a l li e s t o m e (fig. 8 . 5 b ) t o l d tne t h a t h e d i d n o t k n o w m u c h a b o u t
t r a d i t i o n a l fo rm s o f c a l c u l a t i o n , b u t h e d id k n o w t h a t in p r e c o i o n i a l t i m e s it was
p e r f o r m e d by sp ecialists w h o w e re tr a i n e d in t h e m e m o r i z a ti o n o f su m s. T h i s p r a c ­
tice may e x p la in th e o rig in s o f th e fam ou s A fric a n A m e r i c a n c a lc u la tin g
p r o d i g y , T h o m a s F u l l e r . I n 1 7 2 4 , a t t h e a g e o f 1 4, h e w a s c a p t u r e d — q u i t e
p o s s ib ly fr o m t h e g e o g r a p h i c a r e a s t h a t i n c l u d e d t h e B a s s a r i^ — a n d s o l d i n t o
sla v e ry in V irg in ia , w h e r e h e a s t o n i s h e d b o t h p o p u l a r a n d p r o f e s s i o n a l a u d i e n c e s
w i t h h is e x t r a o r d i n a r y c a l c u l a t i n g fe a ts ( F a u v e i a n d G erde.s 1 9 9 0 ).
•
F in a lly , t h e r e is t h e Bassata d i v i n a t i o n s y s te m . A l t h o u g h t h e c a s t_sh.eHs are
i n t e r p r e t e d b y im a g e s r a t h e r t h a n a n y n u m e r i c r e a d i n g , t h e y a re c a s t sixL.times.
E a c h c a s t p r o v i d e s t h e a n s w e r t o a sp e c ific q u e s t i o n ( o r v e r i f i c a t i o n o f a p r e v i ­
o us q u e s t i o n ) r e l e v a n t t o t h e c li e n t's p r o b l e m ; t h e final s i x t h c a s t s h o w s t h e p r o b ­
l e m as a w h o l e . If w e c o m p a r e th i s d i v i n a t i o n to t h e i n i t i a t i o n s y s t e m , t h e
n u m b e r six c a n b e s e g n as_a m a r k e r for in fo r m a t i o n c lu s te rs ^ a pu.nctuaL i.on
p o i n t w h i c h , like t h e ta lly s y s te m , a ll o w s t h e d i s t i n c t i o n s t h a t m a i n t a i n a c o m ­
p r e h e n s i v e s t r u c t u r e . A n d l i k e t h e i n i t i a t i o n , e a c h c y c l e o f six p r o v i d e s a n
e x p a n d i n g v ie w o f t h e w h o l e . T h u s it s e e m s lik e ly t h a t t h e s c a l i n g h e x a g o n s o f
th e in itia tio n m ask re p re se n t th is six-stage ite ra tio n o f k n o w le d g e .
N o n l i n e a r s c a l in g i t e r a t i o n s c a n a ls o b e f o u n d in A f r i c a n i n i t i a t i o n m ask s.
F ig u re 8 .6 a s h o w s a B a k w c l e m a s k in w h i c h b o t h size a n d c u r v a t u r e h a v e a n o n ­
lin e a r in c re a s e w ith e a c h stage. M y guess— 1 h a v e n o t fo u n d a n y c u ltu ra l
d e s c r i p t i o n s t h a t c a n c o n f i r m t h i s — is r h a t it s u g g e s ts “t o o p e n y o u r e y e s ” a s a
m e t a p h o r o f k n o w l e d g e , a n d t h u s m a p s t h e s c a l in g i t e r a t i o n s o f t h e m a s k t o i t e r ­
;
Recursion
123
a t i o n s o f k n o w l e d g e g a i n e d in i n i t i a t i o n stages. F ig ure 8 . 6 b s h o w s a B e m b e m a s k
u se d in t h e first o f a t h r e e - s t a g e i n i t i a t i o n f o r a v o l u n t a r y a s s o c i a t i o n , t h e
bioami (B ie b u y ck 1973). Before t h e c e re m o n y , t h e m a s k is h i d d e n b e h i n d a sc reen ,
a n d d u r i n g t h e ritu a l t h e s c r e e n is g ra d u a lly lifte d by a h i g h - r a n k i n g s e n i o r m e m ­
ber. B o th t h e r e l a t i o n b e t w e e n t h e mnnb_er o f eyes in t h e m ask a n d thejaufrTber
o f s t a g e s in i n i t i a t i o n , as w ell as t h i s m e t h o d o f v is u a lly e x p o s i n g t h e p a t t e r n
as a s e q u e n c e , a g a in su gg est i n t e n t i o n a l use o f a s c a l in g g e o m e t r i c d e s i g n t o r e p ­
r e s e n t s c a l in g i t e r a t i o n s o f k n o w l e d g e .
a
'•%
FIGURE 8 . 6
N o n lin e a r s c a lin g in in itia tio n m a s k s
Bcmhe mask, from western Congo, used in the first of a three-stage initiation for a voluntary
Kiaiion, the biwmu. (b) Mask used in initiation by the Bakwele of Congo.
' p/ioiii courtesy Gene Isaacson; b, courtesy Musee d e I’Hornme.)
124
A fr ic a n fra c ta l mathematics
R e p r e s e n t i n g r e c u r s i o n a s a p r o c e s s in t i m e : p a r t I I , k i n s h i p a n d d e s c e n t
If a g e - g r a d e s y s te m s a r e o n e p a r t o f t h e s t a n d a r d a n t h r o p o l o g i c a l e x p l a n a t i o n
for h o w " c la s s le s s ” s o c i e t i e s a r e s t r u c t u r e d , k i n s h i p is t h e o t h e r A K i n s h i p sy s­
t e m s a r e p r i m a r i l y b a s e d o n g e n e t i c tie s ( “b l o o d r e l a t i o n s " ) a n c l j f n a r r i a g e ,
a l t h o u g h m o s t s o c i e t i e s a ls o h a v e - “f i c t i v e ” k i n ( e .g ., a d o p t i o n ) w h i c h a r e ju s t
as re a l— k i n s h i p is a c u l t u r a l p h e n o m e n o n . D e s c e n t j s also c u lt u r a l l y b a se d . M o s t
W e s t e r n E u r o p e a n a n d A m e r i c a n s o c i e t i e s t h i n k o f d e s c e n t as b io l o g i c a l , p u t
t h a t is b e c a u s e m o s t o f t h e m h a v e b i l a t e r a l d e s c e n t , in w h i c h b o t h p a r e n t s
a r e u se d t o e s t a b l i s h k i n s h i p . U n i l i n e al d e s c e n t , w h e r e a k i n g r o u p t r a c e s t h e i r
li n e a g e t h r o u g h o n e se x o n ly , is a c t u a l l y m o r e c o m m o n ( i n a b o u t 6 0 p e r c e n t o f
t h e w o r ld ’s c u lt u re s ) . A "fclan}’ is a u n i l m e a l k i n s h i p g r° u p w h o s e m e m b e r s re p o r t
t h a t t h e y a re d e s c e n d e d fr o m a c o m m o n d i s t a n t a n c e s t o r , o f t e n a m y t h o l o g i c a l
figure.^Clapls o f t e n h a v e i m p o r t a n t re l ig i o u s a n d p o l i t i c a l f u n c t i o n s , a l t h o u g h
t h e y a re ty p i c a ll y s p r e a d o u t a c r o s s m a n Y ^ |] i a g e s i. g n d >ujua]J_y p r o h i b i t m a r r i a g e
b_eXiveen_dan'rnein_bers.
W e h a v e a lr e a d y s e e n h o w t h e B a m a n a use r e c u r s i o n t o g e n e r a t e a b i n a r y
, c o d e in t h e i r d i v i n a t i o n ; h e r e w e w ill l o o k a t t h e i r r e p r e s e n t a t i o n o f d e s c e n t as
; r e c u r s i o n . T h e a n t e l o p e figure in B a m a n a i c o n o g r a p h y is a s s o c i a t e d w i t h b o t h
j h u m a n a n d a g r i c u l t u r a l fe r til it y . In t h e clti w a ra a s s o c i a t i o n , w h i c h is o p e n to
! b o t h m e n a n d w o m e n , t h e a n t e l o p e a p p e a r s in a s t r i k i n g h e a d d r e s s (fig. 8 . 7 a ) ,
!w h ic h represents th e recu rsio n o f re p ro d u c tio n : m o th e r a n d ch ild . W h e n seeing
o n e h e a d d r e s s i n d i v id u a l ly , t h e s c a l i n g s e e m s tr iv i a l, b u t w i t h s e v e r a l e x a m p l e s
: t o g e t h e r t h e e x t r a o r d i n a r y i n s i s t e n c e o n s e lf- s im i la r it y b e c o m e s a p p a r e n t . T h i s
i c o n a c ts as t h e s e e d t r a n s f o r m a t i o n m j m i t e r a t i v e l o o p ^ t h e c h i l d b e c o m e s a
m o t h e r , w h o h a s a c h i l d , w h o ' b e c o m e s a m o t h e ' t r ' a n d so o n . F i g u r e 8 . 7 b sh o w s
t h e d e s c e n t c a r r i e d to a t h i r d i t e r a t i o n .
In c h a p t e r 2 w e sa w s e v e r a l e x a m p l e s in w h i c h d e s c e n t w a s t i e d t o s c a l ­
in g a r c h i t e c t u r e . T h e B a t a m m a l i b a , w h o li v e in t h e n o r t h e r n p a r t s o f G h a n a ,
B e n in , a n d T ogo, h a v e d e v e lo p e d a n e la b o r a te sy ste m for th is re la tio n s h ip
( B li e r i 9 8 7 ) . F ig u re 8 . 8 s h o w s a d i a g r a m o f t h e i r t w o - s t o r y h o u s e , b a s e d o n che
c ir c le o f c irc les f o u n d ir. m u c h o f t h e W e s t A f r i c a n in t e ri o r. I n f r o n t o f t h e h o u se
lies t h e first o f tw o s c a l i n g t r a n s f o r m a t i o n s . It is t h e " s o u l m o u n d , ” a c i r c l e o f
c y l i n d e r s r e p r e s e n t i n g t h e s p i r i t s o f rho.se c u r r e n t l y l i v i n g in t h e h o u s e a n d
p h y s i c a ll y s t r u c t u r e d like a s c a l e d - d o w n v e r s i o n o f t h e h o u s e a r c h i t e c t u r e . As
t h e c u r r e n t fa m ily g iv e s w ay t o a n e w g e n e r a t i o n , t h e s o u l m o u n d u n d e r g o e s a
s e c o n d t r a n s f o r m a t i o n in w h i c h i t is d i v i d e d i n t o a s i n g l e c y l i n d e r a n d is
m o v e d in s id e . A s c a l i n g s e q u e n c e o f t h e s e s i n g l e c y l i n d e r s — o n e fo r e a c h g e n ­
e r a t i o n — c a n be s e e n w r a p p e d a r o u n d r h e c e n t r a ) t o w e r i n s i d e t h e h o u s e .
'■ I I
FIGURE
8.7
R e c u r s io n a n d re p r o d u c tio n in B a m a n a s c u lp tu r e
iM The chi wara figure, used in ritual dances for agricultural fertility, shows a striking self-similarity:
fiAlthough the figures vary widely, each one is similar to itself. This can be attributed to the Bamana
jtvicw of reproduction as cyclic iterations, (b) Here the cycle is carried out to three iterations.
(•: upper le ft , f r o m die de Hevenon Collection, Museum o f A f r i c a n A r t , S m i t h s o n i a n I n s t i t u t i o n ; upper
courtesy Muse'e d e I ' H o m m e ; l o w e r , f r o m C a r n e g i e I n s t i t u t e 1970. b, c o u r t e s y Musee d e I ' H o m m e . )
A frican fra c ta l mathematics
126
F IG U R E
fig
i
>
I'jK ! ((
V
1.
..
8 .8
R e c u r s io n in B a ta m m a lib a a \->v
r c h ite c tu r e
(a) Diagram or the Batammaliba two-story house. In
front of the house lies the “soul mound," representing
the spirits of those currently living in the house.
(b) Inside the house, single mounds representing
ancestors are found in the scaling arrays, with the si2e of
the ancestral mounds increasing from youngest to oldest.
Here only one such array is shown, but typically there
are several in the same household.
( a , fro m B lier 1 9 8 7 .)
ii
r
I ■H ]
S
J
u
r* s
B lier’s d i a g r a m i n d i c a te s t h a t t h e size o f t h e a n c e s t r a l m o u n d s in c re a se s from
y o u n g e s t t o o l d e s t , a n d s h e n o t e s t h a t t h i s reflects t h e B a t a m m a l i b a ’s id e a o f a
s p i r i t u a l p o w e r i n p r o p o r t i o n t o ag e. S o far it w o u l d a p p e a r t h a t t h e r e a r e o n ly
tw o s c a l in g c a s c a d e s — o n e t o s h r i n k h o u s e s to s o u l m o u n d s , a n d a n o t h e r t o
. d i v i d e s o u l m o u n d s i n t o c y l i n d e r ro w s— a n d n o i t e r a t i v e lo o p . B u t if t h e la r g e st
m o u n d re p re s e n ts th e o ld e s t, t h e n re c e n t m o u n d s w o u ld b e in c re a sin g ly
t h r e a t e n e d by v a n i s h i n g sc a le . H o w w o u ld t h e first d e s c e n d a n t h a v e k n o w n h o w
la r g e t o m a k e t h e first m o u n d ? B li e r n o t e s t h a t m a n y o f t h e s y m b o l i c f e a t u r e s
o f t h e a r c h i t e c t u r e a r e r e p l a s t e r e d w i t h a d d i t i o n a l la y e r s o f w e t c la y o n r i t u a l
o c c a s i o n s , a n d w e c a n s u r m i s e t h a t t h i s a p p l i e s t o t h e a n c e s t r a l m o u n d s as
well. T h u s a n i t e r a t i v e lo o p , in w h i c h e a c h n e w . a n c e s t o r a d d s p o w e r to t h e o ld e r
o n e s by i n c r e a s i n g t h e i r m o u n d ’s size, w o u l d b e a t w o r k in t h e s c a l i n g s e q u e n c e
w e se e a c c u m u l a t i n g a r o u n d t h e c e n t r a l tow er.
f t...
'
T h e M i t s o g h o so c ie ty o f G a b o n i n c l u d e s s e v e r a l relig io u s a s s o c ia ti o n s t h a t
are h o u s e d in che s a m e t e m p l e (e b a n d z a }. F ig u r e 8 . 9 a s h o w s che c e n t r a l p o s t o f
a n e b a n d z a fe a tu rin g sc a lin g pairs o f h u m a n figures. A s in. t h e c h i w a ra figure, th e r e
is o n ly o n e i t e r a t io n ; t h e s i g n i f i c a n c e j i e s j p t h i s f i g u r e a s t h e se ed t r a n s f o r m a t i o n
^ ^ a d ^ Lil!§Lv.?.PJ°cess. T h e use o f a cross s h a p e m a y b e d u e t o C i u j s n a n influen c e ,
b u t t h e b ilareral scalin g is q u i t e i n d i g e n o us, as w e see in che classic B a k w e le s c u l p ­
tu r e (fig. 8 . 9 b ) e l s e w h e r e in G a b o n . M o s t i m p o r t a n t , t h e e b a n d z a p o s t p r o v i d e s
a v is u a li z a ti o n for t h e i t e r a t i v e c o n c e p t o f d e s c e n t t h a t is w id ely u s e d in t h i s c u l ­
tu r e area. T h i s is b e a u tifu lly d e s c r ib e d by F e r n a n d e z ( 1 9 8 2 ) in a d e t a i l e d e t h n o g ­
r a p h y o f t h e M i t s o g h o ’s n e i g h b o r s a n d c u l t u r a l r e l a ti v e s , t h e F an g.
A l t h o u g h t h e F a n g a re p a t r i l i n e a l , t h e y b e l i e v e t h a t t h e a c t i v e p r i n c i p l e
o f b i r t h — a ti n y h u m a n ( w h a t w as c a l l e d a “ h o m u n c u l u s ” in early E u r o p e a n m e d ­
ical th e o r y ) — is c o n t a i n e d in t h e fe m a le b lo o d . T h e idea o f t h e n e w existin g w i t h i n
t h e o ld , a n d v ic e versa , is a s t r o n g c u l t u r a l t h e m e . F o r e x a m p l e , in o n e r i t u a l t h e
m o t h e r p l a c e s a n e w b o r n c h i l d o n t h e b a c k o f h e r o l d e s t s i b li n g t o sy m b o liz e
c o n tin u ity o f th e lineage. F e rn a n d e z (1 9 8 2 , 2 5 4 ) n o te s t h a t th e re b irth c o n ­
c e p t is so s t r o n g t h a t “F a n g f a t h e r s o f t e n c a l l e d t h e i r i n f a n t s o n s g t q ...the_
f a m i l i a r f o n n j a / . f a i b e r . ’’ I n m a n y o f t h e F a n g a n d M i t s o g o r e l i g i o u s p r a c t i c e s ,
t h e s p i r i t is e x p l i c i t l y d e s c r i b e d a s t r a v e l i n g a v e r t i c a l c y c l i c p a t h . A n c e s t o r s
rise fr o m t h e e a r t h t o b e c o m e b o r n a g a i n , a n d by p r o p e r l i v i n g t h e y c a n rise
h ig h e r w ith each rebirth.
T h e s e c y c l i c i t e r a t i o n s a r e v i s u a l i z e d in t h e N g a n g a d a n c e o f t h e B w i ti
r e l i g i o n (fig. 8 . 9 c ) . E v e n in C h r i s t i a n - a n i m i s t s y n c r e t i s m , b i b l i c a l c h a r a c t e rs
a r e r e i n t e r p r e t e d as c y c li c _ r e b i r t h s : t h e A f r i c a n g o d s Z a m e a n d N y i n g w a n
b e c o m e A d a m a n d E v e , w h o b e c o m e C a i n a n d A b e l ( u n d e r s t o o d as m a l e a n d
fem ale), w h o b e c o m e C h r is t a n d th e V irg in .M ary . F e rn a n d e z n o te s t h a t th ese
c y c le s a re n o t m e r e r e p e t i t i o n , b u t r a t h e r i t e r a t i v e t r a n s f o r m a t i o n s : " T h e
s p i r i t u a l - f r a t e r n a l r e l a t i o n o f Z a m e a n d h i s s i s t e r is c o n v e r t e d i n t o t h e c a r n a l
relatio n o f A d a m a n d Eve w h ic h d e g e n e ra te s in to th e m aterialistic a n d divisive
r e l a t i o n o f C a i n a n d A b e l w h i c h t h e n is r e g e n e r a t e d a s t h e i m m a c u l a t e a n d
filial r e l a t i o n s h i p o f M a r y a n d J e s u s ” (p. 3 3 9 ) . A c c o r d i n g t o F e r n a n d e z , t h e s e
d e g e n e r a t i o n / r e g e n e r a t i o n d i f f e r e n c e s a r e v i s u a l i z e d as h o r i z o n t a l v e r s u s
v e r t i c a l J w h i c h c o u l d e x p l a i n t h e a l t e r n a t i o n in t h e e b a n d z a p o s t s . I n a p p l y ­
in g t h i s c y c l i c c o n c e p t i o n t o t h e e b a n d z a s t r u c t u r e (fig. 8 . pel), w e c a n s e e t h e
d e s c e n t m o d e l in its full f r a c t a l e x p a n s i o n .
T h e Tabw a, w h o occupy th e ea ste rn s e c tio n of th e D e m o c ra tic R e p u b lic
o f C o n g o ( Z a ir e ) , h a v e also d e v e l o p e d s e v e r a l g e o m e t r i c figures t o se r v e as m o d ­
els for t h e i r c o n c e p t i o n s o f k i n s h i p a n d d e s c e n t . M a u r e r a n d R o b e r t s ( 1 9 8 7 , 2 5 )
e x p l a i n t h a t in t h e T a b w a o r i g i n story, a n a a r d v a r k ’s w i n d i n g t u n n e l re s u lts in
a
b
FIGURE 8.9
R e c u r.s w e k in s lu f) i n Q abon
(a) T h e central post of t h e ebandza temple in western G a b o n suggests an iterative descent
concept. T h is is actually a museum reproduction, (b) Bakwele masks from eastern G a b o n show
similar bilateral scaling.
f a , f r o m F erro is 19 8 6 ; In left, f r o m Pefrois j 9 8 6 ; rig/u, M etro p o lita n M u se u m o f A r t ; f r o m Z aslavsky
197 .}•)
(figure continue*)
Recursion
129
F I G U R E 8 . 9 (continued)
R e c u rs iv e d e s c e n t in Q a b o n
(c) In many of the Fang-and Mitsogo religious practices, the
spirit is explicitly described as traveling a vertical cyclic path.
Ancestors rise from the earth to be born again, and by proper
living they can rise higher with each rebirth. These cyclic
iterations are visualized in the Nganga dance of the Bwiti
religion, (d) We can apply the explicit mapping of cyclic
generations given by the Nganga dance to the iterative posts of
the ebandza temple and see the descent model in its full fractal
expansion. T h e implication of infinite regress is discussed in
chapter 9.
(c./rom Fernandez: 1982.)
a “b o t t o m l e s s s p r i n g ” fr o m w h i c h e m e r g e s t h e first h u m a n , K y o m b a , w h o s e
d e s c e n d a n t s s p r e a d in all d i r e c t i o n s f r o m t h i s c e n t r a l p o i n t . T h i s s p r e a d is v i s u ­
alized by t h e m p an d e, a d is k c u t fr o m t h e e n d o f a c o n e s n a i l, w h i c h is w o r n as
a chest p e n d a n t (fig. 8 . 10a). T h e c e n tr a ! p o i n t is d rilled o u t, r e p r e s e n tin g t h e e m e r ­
ge n c e o f K y o m b a fr o m t h e d e e p s p r in g , a n d t h e l o g a r i t h m i c sp iral o f t h e shell
.end s y m b o lizes t h e e x p a n s i o n o f k in g r o u p s fr o m t h i s origin.®
O n e w ay ro r e p r e s e n t t h e s e e x p a n d i n g i t e r a t i o n s t h r o u g h ti m e is t o ta k e a
series of p o r t r a i t s as t h e s t r u c t u r e c h a n g e s : p r o j e c t i o n s a t d i f f e r e n t p o i n t s a l o n g
th e ti m e axis. F ig ure 8 . 1 0 b s h o w s t h e first s t e p t o w a r d t h i s d e s ig n : a m o r e li n e a r
version o f t h e m p a n d e d isk, in w h i c h a n A r c h i m e d e a n spiral fits b e t w e e n a series
i •
•'i
A fr ic a n fr a c ta l m athem atics
o f t r i a n g l e s ( w h i c h r e p r e s e n t t h e w i v e s o f t h e g u a r d i a n o f t h e a n c e s t o r s ) . In
fi g u r e 8 . i o c w e s e e t h a t t h e l i n e a r s p i r a l h a s b e c o m e c o n c e n t r i c s q u a r e s , b u t
t h e y a r e n o w p o r t r a y e d i n a s c a l i n g s e q u e n c e , s u g g e s tin g a se r ie s o f p o r t r a i t s o f
t h e k i n s h i p s p ira l as it e x p a n d s t h r o u g h t i m e . S i m i l a r s c a l i n g s q u a r e s e q u e n c e s ,
c a r r i e d o u t t o a g r e a t n u m b e r o f i t e r a t i o n s , c a n b e s e e n i n t h e s ta ffs o f t h e i r
n o r t h e r n n e i g h b o r s , t h e B a l u b a (fig. 8 . i o d ) .
b
c
d
FIGURE 8 . I O
T a b w a k i n s h i p r e p r e s e n ta tio n s
(a) T h e m p and e shell worn by C h i e f M and a Kaseke Joseph, (b) A more linear version of the
m pande disk, in w hich an A rc h im e d e a n spiral fits betw een a series of triangles (w hich represent
the wives o f the guardian ancestors), (c) T h e linear spiral has become c o n c e n tric squares, hut they
are now portrayed in a scaling sequence, suggesting a series of portraits o f th e kinsh ip spiral as it
expands th ro ugh time, (d) Sim ilar scaling of square sequences can he seen in d ie sraffs of their
n o rth e rn neighbors, the Baluba.
fa-c,
from
Roberts anti Mai iter 1985; d, Museum f i i r
V o lk e v k tm d e , F rcm hfnrt.)
Recursion
R ecu rsive cosmology
I n a ll che d e s c e n t r e p r e s e n t a t i o n s we h a v e e x a m i n e d , k i n s h i p gro u p s tr a c e t h e m ­
selv es to a m y th o lo g ic a l a n c e s t o r a t t h e b e g i n n i n g o f- th e w o rld , a n d t h u s w e m o v e
fr o m t h e - o r ig i n s o f h u m a n i t y to t h e o r i g i n s ' b f t h e c o s m o s . A f r i c a n c r e a t io n c o n ­
c e p t s a r e o f t e n b a s e d o n a re c u r s iv e n e s t i n g . T h e b e s t - k n o w n e x a m p l e is t h a t o f
t h e D o g o n , as d e sc rib e d by F r e n c h e t h n o g r a p h e r M a r c e l G r i a u l e ( 1 9 6 5 ) . H is w o rk
b e g a n d u r i n g t h e 1 9 3 0 D a k a r - D j i b o u t i e x p e d i t i o n , w h e r e h e first m a d e c o n t a c t
w i t h t h e D o g o n o f S a n g a in w h a t is n o w M a l i . I n 1 9 4 7 h i s s tu d ie s t o o k a d r a ­
m a t i c t u r n o f e v e n t s w h e n o n e o f t h e D o g o n e ld e rs, O g o c e m m e li, a g re e d to i n t r o ­
d u c e G r i a u l e t o t h e i r e l a b o r a t e k n o w l e d g e s y s te m . C l i f f o r d ( 1 9 8 3 ) p r o v i d e s a
d e t a i l e d r e v i e w o f t h e s t r o n g r e a c t i o n s t o G r i a u l e ’s r e s u l t i n g e t h n o g r a p h y .
W h i l e m a n y o f che c r i t i q u e s w e re r e a l ly a b o u t t h e fa ilin g s o f m o d e r n i s t a n t h r o ­
p o l o g y in g e n e r a l — t h e t e n d e n c y t o p r e f e r a s t a t i c p a s t o v e r t h e p r e s e n t , o r a
s i n g u l a r “t r a d i t i o n ” o v e r i n d i v i d u a l i n v e n t i o n — t h e r e w e r e a ls o t h o s e w h o
s i m p l y d id n o t b e l i e v e t h a t s u c h e l a b o r a t e a b s t r a c t i o n s c o u l d b e i n d i g e n o u s .
F o r t h e Dogotir t h e h u m a n s h a p e is n o t o n l y a b io l o g i c a l f o r m , b u t m a p s
m e a n i n g a t a l P l e v e j s: “T h e fa c t t h a t t h e u n i v e r s e is p r o j e c t e d in t h e s a m e
m a n n e r o n a se r ie s o f d i f f e r e n t s c a l e s — t h e c o s m o s , t h e v i l la g e , t h e h o u s e , t h e
i n d i v i d u a l — p ro v i d e s a p r o f o u n d ly u n if y in g e l e m e n t in D o g o n life” (D u ly 1 9 7 9 ).
. T h e ^ Q g o n h o u s e is p h y s i c a ll y s t r u c t u r e d o n a m o d e l o f t h e h u m a n fo r m , w i t h
a la r g e r e c t a n g l e for t h e b o dy , s m a l l e r r e c t a n g l e s o n ^ e a c h , s j d e T o r a rm s , a door,
for t h e m o u t h , a n d so o n . T h e D o g o n v i l l a g e , h o w e v e r , r e p r e s e n t s t h e h u m a n
fo r m w i t h a s y m b o l i c s t r u c t u r e r a t h e r t h a n a g e o m e t r i c s t r u c t u r e : it is n o t p h y s ­
ically a r r a n g e d as a h u m a n s h a p e , b u t v a r i o u s b u i l d i n g s a r e a s s ig n e d m e a n i n g
a c c o r d i n g to t h e i r so cial f u n c t i o n ( t h e s m i t h y s t a n d s for t h e h e a d , t h e m e n s t r u a l
• ••
lo d g e s as h a n d s y a n d ' s T r o v p T ln r v i s e 'o f ' tw o ' d i f f e r e n t s y s t e m s o f repres..e-n t a t io n
p r e v e n t s s e l f - s im i la r it y in t h e p h y s i c a l s t r u c t u r e o f t h e a r c h i te c t u re ^ _ b u t _ s o m e
o f t h e D o g o n ’s re l ig i o u s ic o n s d o s h o w h u m a n fo r m s m a d e o u t o f h u m a n fo r m s
Tfig. 8.1 i a ) .
r
A t h r e e f o l d s c a l i n g a p p e a r s in s e v e r a l a s p e c t s o f t h e D o g o n r e l i g i o n , a n d
1 it is h e r e t h a t we f i n d a n i n d i c a t i o n t h a t t h e D o g o n a re u s i n g m o r e t h a n ju s t
V a c a s c a d e . G r i a u l e ( 1 9 6 5 , 1 3 8 ) s u m m a r i z e s O g o t e m m e l i ’s c r e a t i o n s t o r y :
“G o d . . . h a d t h r e e t i m e s r e o r g a n i z e d t h e w o r l d by m e a n s o f t h r e e su c c e s s iv e
W o r d s , e a c h m o r e e x p l i c i t a n d m o r e w i d e s p r e a d in its r a n g e t h a n t h e o n e
b e f o r e it." B u t t h e s e r e o r g a n i z a t i o n s a r e n o t m e r e l y l a y e r i n g o n e o n t o p o f che
o c h e r ; r a t h e r t h e o u t p u t o f e a c h r e o r g a n i z a t i o n b e c o m e s t h e i n p u t for t h e n e x t .
T h e e a r t h g iv e s b i r t h t o t h e first s p i r it s ; t h e s e “N u m m o ” r e g e n e r a t e a n c e s t r a l
b e in g s i n t o h u m a n l i k e re p tile s; t h e r e p t i l e - a n c e s t o r s a re a g a in r e b o r n as t h e first
tr u e h u m a n s . W i t h i n r e b i r t h , t h e t h r e e f o l d i t e r a t i o n is a g a i n e n a c t e d . In t h e first
( a ) I n t h e D o g o n c o s m o l o g y , t h e s t r u c t u r e o f t h e h u m a n f o r m is
c r e a t e d fro m h u m a n form .
( h ) T h e s y m b o l i s m o f tire s t a c k e d p o t s ,
representing th e b reath o flife, w ith in the
feteus, w ith in th e w o m b . W e c a n use an
iterativ e d raw in g p ro ced u re to b e tte r
u n d e rsta n d ho w this k in d o f scaling can
resu lt from a recu rsiv e lo o p . S u p p o s e we
h a v e a ro u tin e th a t c a n draw th e circle o f
th e p o t given a d iam eter, an d o n e th a t can
d r a w a lid.
W h i l e d i a m e t e r > m i n i m u m do:
D r a w a c i r c l e o f size d i a m e r e r
I f size = m i n i m u m , d r a w a lid
S h r i n k d i a m e t e r h y 2/3
E n d of “ w h ile " loop.
T h i s p r o c e d u r e first c h e c k s t o s e e if w e a r e
p ast t h e sm a lle s t d i a m e t e r p o ssib le. If n o t,
it d raw s a po t, sh rin k s th e d ia m e te r v alu e
b y 2/ 3 S, a n d t h e n g o e s b a c k t o t h e s t a r t o f
t h e w h ile lo o p . In o t h e r w o rd s, t h e o u t p u t
o f o n e ite ra tio n — a given d ia m e te r—
b e c o m e s t h e i n p u t for t h e n e x t ite r a t io n .
(c )
D o g o n recursive im age o f m o th e r a n d
child.
FIGURE 8 . 1 I
S c a lin g in D o g o n re lig io u s ic o n s
(n,
from Lau de 197
J a y C . L e ff.)
3;
c o u r t e s y Leste r W io id e r m a n ;
c,
f r o m C a r n e g i e Institute
19 7 0 ;
co u rtesy
0/
Recursion
r e g e n e r a t i o n , fo r e x a m p l e , e a c h a n c e s t r a l b e i n g e n t e r s t h e e a r t h 's w o m b , w h i c h
t u r n s e a c h o f t h e m i n t o a fetus, w h i c h a ll o w s t h e b r e a t h o f life (n u m m o ) t o e n te r.
T h e c o s m o lo g ic a l n a r r a tiv e su g g ests t h a t in th e D o g o n v iew t h e b ir th i n g
p ro c e s s e s a t a ll s c a l e s a r e , in s o m e s e n s e , i t e r a t i o n s t h r o u g h t h e s a m e t r a n s f o r ­
m a t i o n , a n d t h a t t h e s e i t e r a t i o n s a r e a c t u a l l y n e s t e d lo o ps.
W h y s h o u ld th e D o g o n req u ire s u c h d e e p iterativ e n esting? 1 su sp ect th a t
t h e r e a r e tw o m o t i v a t i o n s / F ^ r s t ^ t h e r e is a n i n s i g h t i n t o m o d e l i n g t h e w o rld :
r e c u r s i o n is a n i m p o r t a n t f e a t u r e i n b i o l o g i c a l m o r p h o g e n e s i s , as w e l l as in
e n v i r o n m e n t a l a n d s o c j a l _ c h a n g e . T h e ;s e c o n d .is t h e c u l t u r a l c o n t e x t o f t h i s
T cn ow led ge: e l d e r s n e e d t o e n s u r e t h a t t h e y o u n g e r g e n e r a t i o n r e s p e c t s t h e i r
au th o rity , w h ic h c a n o n ly b e d o n e by g iv jn g th e m g radual access to t h e source
o f t h i s p o w e r , w h i c h is k n o w l e d g e . A c k n o w l e d g e s y s te m in w h i c h e n d l e s s e x e ­
gesis is p o s s i b l e m a k e s t h e i n i t i a t i o n p r o c e s s a l i f e t i m e a c t i v i t y . B u t h a v i n g so
m u c h e x p la n a to ry elb o w ro o m also p re s e n ts a p ro b le m w ith tr a n s la tin g su c h
n a r r a t i v e s i n t o m a t h e m a t i c s . ^ W e h a d t o b e c a r e f u l w i t h t r a n s l a t i o n s fo r m o r e
fo rm a l p r a c t i c e s , s u c h as i n t e r p r e t i n g t h e B a m a n a d i v i n a t i o n s y s te m as a b in a r y
c o d e , o r a d ire c l o t h as a g e o m e t r i c a l g o r i t h m . A n a r r a t i v e is n o t a q u a n t i t a t i v e
o r g e o m e t r i c p a t t e r n , a n d its a m b i g u i t y r e q u i r e s ail t h e m o r e ; c a r e i n p r o d u c ­
in g a m a t h e m a t i c a l t r a n s l a d o n ^ t h a t . c l o e s n o t e m b e d i $ h J n d j g e j } O u s c o n c e p t s ,
firs t, we h a v e to d is tin g u is h b e tw e e n m o d e lin g th e n a r r a tiv e — s o m e th in g a
s tru c tu r a l a n th r o p o lo g is t lik e C la u d e L e v i-S tra u ss w o u ld d o — a n d t h e n a r r a ­
t i v e as a n i n d i g e n o u s m o d e l , s u c h as t h e D o g o n ’s s y s te m for r e p r e s e n t i n g t h e i r
o w n a b s t r a c t ideas. T h e b e s t way t o li m i t o u r t r a n s l a t i o n to ideas t h a t t h e D o g o n
t h e m s e l v e s are t r y i n g t o c o n v e y is t o c o m p a r e t h e s e a b s t r a c t i o n s o f t h e n a r r a ­
ti v e w i t h o t h e r , m o r e f o r m a l D o g o n s y s te m s . T h i s m e a n s m i s s i n g s o m e id e a s
t h a t d o n o t h a v e s u c h f o r m a l c o u n t e r p a r t s , b u t it is b e t t e p t o e r r on. t h e safe. s|de.
- in t h i s c o n t e x t .
. T h e m a t e r i a l d e s ig n s o f t h e D o g o n a re m o r e r e s tr i c te d t h a n ^ t h e . n a r r a ti v e
in te r m s o f t h e i r i t e r a t i v e d e p t h . T h e b e s t c a s e is p r o b a b l y in t h e i c o n o g r a p h y
of t h e g ra n a r y , w h e r e O g o t e m m e l i e x p l a i n s a s t a c k o f t h r e e p o ts : che la r g e s t rep. re s e n ts t h e w o m b ; t h e o n e o n t o p o f it, c r e a t i n g its lid, r e p r e s e n t s t h e fetus; a n d
che lid o f t h a t p o t is t h e s m a l l e s t p o t , c o n t a i n i n g a p e r f u m e t h a t r e p r e s e n t s t h e
b r e a t h o f life ( G r i a u l e 1 9 6 5 , 3 9 ) . T h e s m a l l e s t p o t is c a p p e d by a n o r m a l lid; at
chis p o i n t t h e r e c u r s io n “b o t t o m s o u t . ” T h i s is n o t m erely a s t a c k o f d i f f e r e n t sizes;
in th e D o g o n vie w t h e w o m b c r e a t e s t h e p r e c o n d i t i o n s t h a t give rise t o t h e fetus,
w hich is th e p re c o n d i ti o n for t h e e n tr y o f th e b re a th o f life. T h e recu rsion is e m p h a ­
sized in t h e way t h a t e a c h n e w poc b e g in s before che p rev io u s p o t e n d s (fig. 8.1 i b ) ,
chat is, o n e p o t ’s lid is t h e n e x t p o t ’s b o d y ( G r i a u l e 1 9 65 , 19 9 ). In t h e s c u l p t u r e
in figure 8.1 i c t h e m o t h e r ’s b re a s ts b e c o m e t h e c h i l d ’s h e a d — a g a in , a n e w o n e
133
134
A fr ic a n fractal m athem atics
b e g in s b e f o r e t h e p r e v i o u s o n e e n d s . A s wc saw in t h e c h i w a r a s c u l p t u r e o f t h e
D o g o n ’s B a m a n a n e i g h b o r s , r e p r o d u c t i o n is m o d e l e d as r e c u r s i o n .
T h e D o g o n v i e w o f a c o s m o s s t r u c t u r e d as n e s t e d h u m a n - f o r m is q u i t e
s im ila r to c e r t a in a n c i e n t E g y p tia n re p r e s e n ta tio n s . F igufe 8 .1 2 sh o w s a re lie f
f r o m a t o m b in w h i c h t h e c o s m o s e n c l o s e s t h e sky, w h i c h e n c l o s e s t h e e a r t h .
I t is i n t e r e s t i n g t o n o t e t h a t t h e r e a r e a g a i n t h r e e i t e r a t i o n s o f s c a l e . A t h r e e i t e r a t i o n n u m e r i c l o o p is i n d i c a t e d f o r t h e E g y p t i a n g o d o f w i s d o m , T h o t h . H e
is re f e r r e d t o as H e r m e s T r i s m e g e s t u s , w h i c h m e a n s “ t h r i c e g r e a t H e r m e s , ’' b u t
h e is a ls o re f e r r e d t o a s “ e i g h t t i m e s g r e a t H e r m e s . " W h y b o t h t h r e e a n d e i g h t ?
It m a k e s sen se if we t h i n k in te r m s o f t h o s e c o m m o n e l e m e n t s o f A f r i c a n n u m e r ic
s y s te m s , r e c u r s i o n a n d b a s e - t w o a r i t h m e t i c . T h r i c e g r e a t b e c a u s e w h i l e a n
o r d i n a r y h u m a n m a y rise as h i g h as t h e m a s t e r o f m a s t e r s , H e r m e s T r is m e g e s t u s
is t h e m a s t e r o f m a s t e r s o f m a s t e r s ( t h r e e i t e r a t i o n s ) ; t h u s w e c a n s u r m i s e “e i g h t
t i m e s g r e a t ” re f e rs t o 2^ = 8.
FIG U RE 8 . 1 2
R e c u r s io n in th e c osm olog y o f a n c i e n t Egyf)t
Gelt, the Earth, enclosed hy Shu, space, enclosed hy Nut, the stellar canopy.
(From Fourier 182 1.)
Recursion
M a n y o f t h e p r o c e s s i o n a l c ro s se s o f E t h o p i a also i n d i c a t e a t h r e e f o l d i t e r ­
a t i o n (fig. 8 .1 3 ) . A l t h o u g h t h e cro sse s a r e n o w u sed in C h r i s t i a n c h u r c h p r o ­
c e e d in g s , P e rc z e l ( 1 9 8 1 ) r e p o r t s t h a t r e l a t e d d e sig n s c a n b e f o u n d o n o r n a m e n t s
e x c a v a t e d f r o m t h e c it y o f A x u m in n o r t h e r n E t h o p i a in t h e s e c o n d h a l f o f th e
first m i l l e n n i u m b .c . e ., so w e s h o u l d n o t a s s u m e t h a t t h e t h r e e f o l d i t e r a t i o n was
o riginally re la te d to th e C h r is ti a n trinity, a lth o u g h a c o n n e c t i o n m ay h a v e
oc curred later (fig. 8 .1 3 b ). C o u l d th e r e be a c o m m o n history b e h in d all th e s e o c c u r­
r e n c e s o f tr i p l e i t e r a t i o n s in t h e re lig io u s i c o n s o f t h e S u d a n a n d N o r t h A fric a ?
I t h i n k t h e c o m m o n u se o f r e c u r s i o n it s e l f is d u e to a m u t u a l in f lu e n c e , b u t t h e
o c c u r r e n c e o f tr i p l e i t e r a t i o n m a y b e o n l y d u e to t h e s i m i l a r i t y o f c i r c u m s t a n c e s
r a t h e r t h a n d if fu s i o n . F o r o n e t h i n g , g i v e n t h e m a t e r i a ls t h e a r t is a n s a r e w o r k ­
ing w i t h , m i n u t e s c a le s a re d if fic u lt, so t h a t t h e t e n d e n c y t o be l i m i t e d t o t h r e e
ite ra tio n s m a y sim p ly b e a p ra c tic a l c o n s e q u e n c e o f t h e c ra f t m e t h o d s . It m a y also
be t h a t if o n e w is h e s t o g e t t h e c o n c e p t o f i t e r a t i o n across, tw o is t o o f e w ^ w h ile
m o re t h a n t h r e e is u n n e c e s s a r y ( w h i c h is w h y m o d e r n m a t h e m a t i c i a n s o f t e n r e p ­
r e s e n t a n in f in i te series by t h e first t h r e e e l e m e n t s , e.g., " 1 ,2 ,3 . . .”). O n t h e o t h e r
h a n d , t h e r e a re ca ses w h e r e m a n y s u c h “ u n n e c e s s a r y ” i t e r a t i o n s are m a d e in t h e
m o s t d iffic u lt o f c r a f t m a t e r i a ls . F ig u r e 8 .1 4 s h o w s a n a n c i e n t E g y p ti a n d e s ig n ,
c a rv e d in s t o n e , r e p r e s e n t i n g t h e o rig in m y t h in w h i c h t h e lo tu s flow er (its petalsw ith i n - p e t a ls illu s tra te d by a m u l t i t u d e o f sc a lin g lines) b eg ins che se lf-g e n e r a tin g
c r e a t i o n o f t h e m a t e r i a l w o rld .
-reference
S e l f - r e f e r e n c e is t h e m o s t p o w e r f u l ty p e o f r e c u r s i o n . T h e a b il it y o f a s y s t e m to
re fle c t o n it s e l f is a t t h e h e a r t o f b o t h che li m i ts o f . m a t h e m a t i c a l c o m p u t a t i o n
as w ell as o u r s u b j e c t i v e e x p e r i e n c e o f c o n s c i o u s n e s s . B u t t h e r e a re r e l a t i v e l y
triv ial a p p l i c a t i o n s o f s e l f - r e f e r e n c e as w ell ( o n e c a n a lw a y s use a b l o w t o r c h to
lig h t a c a n d l e ) . S e l f - r e f e r e n c e first c a m e to t h e a t t e n t i o n o f m a t h e m a t i c i a n s in
s im p le e x a m p l e s o f lo g ic a l p a r a d o x ; for e x a m p l e , t h e “ li a r ’s p a r a d o x ” w e e x a m ­
in ed e a rlie r . T o s e e h o w s e l f - r e f e r e n c e c a n b e m o r e t h a n j u s t a l o g i c i a n ’s j o k e ,
le t’s e x a m i n e h o w it w o r k s in p r o g r a m m i n g . R e c all t h a t j i s i in p j e c a s c a d e c o u ld
n o t b g . u s e d ■if we.,did- n ot.-k no w . h o w ..m a n y j_ t ^ n 5 fp r m a ti o n s _ w e re . n e e d e d .a h e a d
of tim e. T h e s a m e p r o b l e m o c c u r r e d for t h e B a t a m m a l i b a a n c e s t r a l m o u n d s ; sin c e
t h e first d e s c e n d a n t d i d n o t k n o w h o w m a n y w o u ld b e n e e d e d , t h e s y s te m h a s
to allow for i t e r a t i v e resizing. W e a lso saw t h e possibility o f n e s t e d it e ra t iv e loops,
illustrated by t h e t w o - l o o p d r a w i n g p r o g r a m fo r M o k o u l e k a r c h i t e c t u r e . B u t su p ­
pose we d i d n ’t k n o w h o w m a n y n e s t e d lo o p s we were g o in g t o n e e d ? I n t h e s a m e
way t h a t t h e r e c u r s i v e c a s c a d e c o u l d n o t d e a l w i t h a n u n k n o w n n u m b e r o f iter-
135
S e e d sh a p e
( a ll tin e s are
a c tiv e lin es)
S e c o n d iteration
FIGURE 8 . 1 3
F r a c t a l s in E t h i o p i a n
p ro c e s s io n a l c ro sse s
(a) Fractal simulations for Ethiopian
processional crosses through three iterations(b) Ethiopia converted to Christianity in
333 c.ti., and in the thirteenth century King
Lalibela directed the construction of churches
to be cut from massive rocks in one of the
mountain regions. T he church of St. George
{at right) shows a triple iteration of nested
crosses:
(a, a l l Ethiopian processional crosses f r o m P o r t l a n d
Museum in Oregon; pliotos c o u r t e s y o f Cs ilia
P e r c z e l , b, photo hy Georg Gerstcr.)
Recursion
FIGURE
137
8.14
T h e l o t t t s i c o n in a n c i e n t E g y p t i a n cosm ology
In the origin story of ancient Egypt the lotus flower was often used as an image of the unfolding of
the universe, its petals-within-petals signifying the expansion of scales. This is a very stylized
representation used in the capitals of columns in temples.
(From F ourier 1 8 2 1 .)
a t i o n s , n e s t e d i t e r a t i o n h a s t r o u b l e w i t h a n u n k n o w n n u m b e r o f l o o p s . 10 H e r e
is w h e r e s e l f - r e f e r e n c e c a n h e l p o u t . A n e x a m p l e o f s e l f - r e f e r e n c e in p r o g r a m ­
m i n g is il l u s t r a t e d for t h e D o g o n p o t s t a c k in fig u re 8 .1 5 .
W e k n o w t h a t t h e D o g o n p o t s t a c k c a n b e d r a w n w i t h a s i n g le i t e r a t i v e
l o o p — it d o e s n o t re q u ire s e l f - r e f e r e n c e . B u t t h e ta s k c a n be a c c o m p l i s h e d by
s e l f - r e f e r e n c e , a n d w e m i g h t s i m d a r l y ask if t h e r e a re 'c a s e s o f s c a l in g in A f r i c a n
d e s i g n s in w h i c h s e l f - r e f e r e n c e p la y s a ro le , re g a r d le ss o f w h e t h e r it is re q u i r e d .
In. E u r o p e a n h i s t o r y , s e l f - r e f e r e n c e b e g i n s w i t h t h e s t o r y o f E p i m e n i d e s o f
C r e t e , t h e “ li a r ’s p a r a d o x . ” S i m i l a r u t i l i z a t i o n s o f n a r r a t i v e se lf-r e fe re n c e t o c r e ­
a t e u n c e r t a i n t y c a n be f o u n d in c e r t a i n A f r i c a n t r i c k s t e r s to rie s . F o r e x a m p l e , '
iii a n A s h a n t i s t o r y o f A n a n s e ( w h o b e c a m e “A u n t N a n c y ” in A f r i c a n A m e r i ­
c a n f o l k l o r e ) , a m a n n a m e d “ H a t e s - t o - b e - c o n t r a d i c t e d ” is t r i c k e d i n t o c o n ­
tr a d ic tin g h im self. P e ito n (1 9 8 0 , 5 1 ) n o te s t h a t th e a p p lic a tio n o f such
s e l f - r e f e r e n t i a l p a r a d o x is a t h e m e in m a n y A n a n s e st o r ie s : “T h u s A n a n s e
r e j e c ts t r u t h in f a v o r o f ly in g , b u t o n l y for t h e s a k e o f s p e e c h ; t e m p e r a n c e in
fav o r o f g l u t t o n y for t h e s a k e o f e a t i n g ; c h a s t i t y in f a v o r o f la s c i v i o u s n e s s for
t h e s a k e o f s e x ." T h e f o l l o w i n g t a l e is n o t n e a r l y a s sp a r s e b u t c a r r ie s t h e f l a ­
v or o f s e l f - r e f e r e n t i a l p a r a d o x q u i t e w ell:
O n e of the most com m on of all stories in Africa describes the e n c o u n te r of a
man and a hum an skull in the bush. A m ong the N u pe of Nigeria, for instance,
they tell of the h u n te r who trips over a skull while in pursuit of game and
exclaim s in w o n d e rm e n t, “W h a t is this? How did it get here?" “Talking
A fric a n fractal mathematics
FIGURE 8 . 1 5
D r a w in g th e D o g o n p o t s ta c k by s e lf-re fe re n c e
T h e symbolism of the stacked pots represents th e b reath of life,
w ith in th e fetus, w ithin th e wotnB.' W e have aready seen h o w this
can be drawn using an iterative loop; now let’s see h ow it ca n be
drawn using self-reference.
Suppose we h av e a ro u tin e th a t can draw the semicircle of th e
p o t given a diameter.
Procedure D R A W -P O T
If size = m inim um , draw a lid.
Else
Draw a circle of size diam eter
S h rin k diam eter by V3
D R A W -P O T
End of "else" clause
End o f procedure
N otice th a t this procedure first checks to see if we are at the
smallest dia m e ter possible. If not, it draws a pot, shrinks the
diam eter value it by 2/ss, and th e n calls itself— an application of
self-reference. N ow th e program has to execute n D R A W -P O T
procedure again. T h e recursion will “bo tto m -o u t" w hen it finally
draws a lid. T h e program th e n skips to the “End of procedure" line
an d can finally pop hack up to th e place it left off after ex ecutin g
the previous D R A W -P O T call.
b r o u g h t m e h e r e , ” th e sk u ll re p lie s. N a t u r a l l y t h e h u n t e r is a m az ed a n d
q u ic k ly ru n s b ac k to his village, e x c la im i n g a b o u t w h a t h e h a s fou nd . E v e n ­
tu ally t h e kin g h ea rs ab o u t th is w o n d e r an d d e m a n d s t h a t th e h u n t e r ta k e h im
/
to see it. T h e y re t u r n to th e p la c e in t h e b u s h w h e r e t h e sk u ll is s ittin g , a n d
t h e h u n t e r p o in t s ir o u t to h is k in g , w h o n a t u r a l l y w a n t s to h e a r t h e s k u ll’s
message. T h e - h u n t e r r e p e a t s .t h e question.:..“H y w d i d you g e t h e re ? ” h u t th e
skull says n o t h i n g . T h e k ing, angry now, accu se s t h e h u n t e r o f d e c e p t io n , an d
o rders his h e a d c u t off o n th e sp ot. W h e n t h e royal p a rty d e p a rt s , t h e skull
speaks o u t, asking th e h u n t e r “W h a t is this? H o w d id you get h e re ? ” T h e he ad
replies, “T a lk i n g b ro u g h t m e h e r e ! ”
( A b r a h a m s 1983, 1)
S e lf -re f e re n c e is -also visually p o rtra y e d in s o m e A f r i c a n de sign s. Figure 8. i6n
s h o w s a n o t h e r a b b i a c a r v i n g fr o m C a m e r o o n , s e e n a ls o in t h e n e s t e d fish e a rl ie r
in th i s c h a p t e r . B u t th is a b b ia c a r v i n g is a n ic o n for itse lf— it is a n a b b ia o f a b b ia .
A c c o r d i n g to t h e C a m e r o o n C u l t u r a l Review ( i n s i d e c o v e r , J u n e 1 9 7 9 ), its m e a n ­
i n g is “r e p r o d u c t i o n . " A n o t h e r e x a m p l e o f s e l f - r e f e r e n c e f r o m C a m e r o o n is
s h o w n in figure 8 . 1 6 b , a life-size b r o n z e s t a t u e o f t h e k i n g o f F o u m b a n . H e r e we
s e e t h e k i n g s m o k i n g h is p ip e , t h e b o w l o f w h i c h is a fig u re o f t h e k i n g s m o k ­
in g h is p ip e , t h e h o w l o f w h i c h is a figure o f t h e k i n g s m o k i n g h i s p ip e . L ik e t h e
K e l l o g g ’s c o r n f l a k e s b o x d e s c r i b e d e a rl ie r , t h e v is u a l s e l f - r e f e r e n c e i n s t a n t l y
leads to in f in ite regress. B u t it c o u ld be m o r e t h a n ju st h u m o r in t h e b ro n z e sculp-
Recursion
139
tu r e . S i n c e t h e p ip e is a w e l l - k n o w n s y m b o l o f ro y a l p re s tig e in F o u m b a n , it m ay
be t h a t t h e a rtisa n s w ere m a k i n g p u rp o s e fu l u se o f t h e in fin ite regress: “T h e k i n g ’s
p o w e r is n e v e r - e n d i n g . ”
Figure 8 .1 6 c s h o w s a B a m a n a h e a d d r e s s , t h a t is, a s c u l p t u r e w o r n o n t h e
h e a d d u r i n g c e r e m o n i e s . Fag g ( 1 9 6 7 ) s u g g e s ts t h a t t h i s e n a c t s s e l f - r e f e r e n c e :
a h e a d d re s s of a p erson w e a rin g a h e a d d re s s of a p erson w earin g a h e ad d ress.
O t h e r s (cf. A r n o l d i 1 9 7 7 ) h a v e d e s c r i b e d t h i s as a s y m b o l o f f e r til it y sp irits, b u t
th e tw o in te r p r e ta tio n s m ay n o t be m u tu a lly ex c lu s iv e . R e tu r n in g to th e
FIGURE 8 . l 6
S e l f 't e f e r e n c e in A f r i c a n i c o n s
1) The abbia carvings from C am e ro o n show a wide variety o f images, but this abbia carving is
icon lor itself— it is an abbia of abbia. (b) A life-size bronze statue of the king of Foumban.
ere wc sec the king smoking his pipe, .die bowl of w hich is a figure of the king sm oking his pipe.
Bamana headdress.
Jwu'ing Iwsed on a b b i a p i c t u r e d on
iu University Museum o f A f r i c a n
th e c o v e r o f
A r t.)
Cameroon Cultural Review, 1979; c, photo
courtesy
140
A fr ic a n fra c ta l ?nacliema£ic5
B a m a n a ’s c lo s e c u l t u r a l r e l a t i v e s t h e D o g o n , w e se e s e l f - r e f e r e n c e s u g g e s te d by
O g o t e m m e l l i ’s d e s c r i p t i o n o f h o w t h e e i g h t h a n c e s to r, “w h o w as W o r d itself,” was
a b le to use W o r d ( t h a t is, t h e b r e a t h o f life) t o s e l f - g e n e r a t e i n t o ..the n e x t i t e r ­
a t i o n o f h u m a n i t y . I n e x a m i n i n g t h e s e lf-s im ila r it e r a t io n s * o f t h e D o g o n m o t h e r
a n d c h i l d in fi g u r e 8 . 1 1 c , w e n o t e d a s t r u c t u r a l c h a r a c t e r i s t i c t h a t c a n b e
e x p re s s e d in t h e p h r a s e “ a n e w o n e b e g in s b e f o r e t h e o l d o n e e n d s . " T h i s w o u ld
also d e scrib e t h e s t r u c t u r e o f t h e pip e in t h e s t a t u e o f t h e k i n g o f F o u m b a n , w h i c h
w e k n o w t o b e e x p l i c i t l y s e l f - r e f e r e n t ia l . P e r h a p s t h e s e l f - r e f e r e n t i a l v e r s i o n o f
t h e D o g o n p o t s t a c k w a s t h e c o r r e c t o n e a fte r all.
I c o n i c r e p r e s e n t a tio n s o f r e c u r s io n
T h e a b b i a o f a b b i a , as a s y m b o l o f ( ^ e p r o d u c t i o n , ’.”>is m o r e t h a n j u s t a n a p p l i ­
c a t i o n o f s e l f - r e f e r e n c e ; it r e p r e s e n t s ~ t h e c o n c e p £ , i t s e l f . I f r e c u r s i o n is r e a l l y a
c o n s c i o u s ( t h a t is, s e l f - c o n s c i o u s ! ) a s p e c t o f A f r i c a n k n o w l e d g e s y s t e m s , t h e n
w e s h o u l d e x p e c t s u c h r e p r e s e n t a t i o n s , r a t h e r t h a n j u s t i n s t a n c e s in w h i c h t h e
c o n c e p t is a p p l i e d . F i g u r e 8 . 1 7 a s h o w s t h e a p p l i c a t i o n o f r e c u r s i o n in t h e t r a -
fig u re
8 .1 7
R e /lu x
(a) This sketch from the notebook of a nineteenth-century ethnographer in southern Senegal
shows an indigenous apparatus for the distillation of liquor from palm wine using a scaling cascade.
(b) A ncient Egyptian alchemists drew this snake symbol to represent their reflux technique.
A tube comes out of a heated pot and reenters after cooling. This cyclic refinement was used in
the creation of dyes and perfumes, bur it also symbolized the alchemists’ goal of refinement of the
human soul.
(a, pfioto c o u r t e s y /FAN, D a k a r - , b, drawing b a s e d on T o y lor 1 930.)
Recursion
141
d i t i o n a l d i s t i l l a t i o n o f p a l m w i n e i n t o l i q u o r in t h e C a s a m a n c e r e g i o n o f
S e n e g a ! . S u c h d i s t i l l a t i o n t e c h n i q u e s w e r e d e v e l o p e d t o s o p h i s t i c a t e d le v e ls
in a n c i e n t E g y p t, w h e r e t h e p r o c e s s b e c a m e a n i t e r a t i v e l o o p w h i c h m o d e r n
c h e m i s t s c a ll a “re f lu x ” a p p a r a t u s . F ig u r e 8 . 1 7 b shovirs-the i c o n i c r e p r e s e n t a t i o n
o f t h e reflu x sy s te m in t h e o ld e s t k n o w n a l c h e m i c a l w r i t i n g s (first c e n t u r y
c .e
.),
w h i c h a r e a t t r i b u t e d t o M a r i a ( w h o w r o t e u n d e r t h e n a m e o f M i r i a m , s is te r o f
M o s e s ) , C l e o p a t r a ( n o t t h e f a m o u s q u e e n ) , C o m a r i u s , a n d t h e m y t h i c figure
o f H e rm e s T rism eg estu s ( T h o t h ) . T a y lo r (1 9 3 0 ) n o te s t h a t a lth o u g h th ese
w e r e w r i t t e n in G r e e k , “t h e re l ig i o u s e l e m e n t . . . l i n k s t h e m t o E g y p t r a t h e r
t h a n t o G r e e c e , ” a n d h e s u g g e s ts t h a t t h e m o s t li k e ly o r i g i n is f r o m t h e t r a d i ­
t i o n s o f t h e a n c i e n t E g y p ti a n p r i e s t h o o d . 11 I n t h e s e w r i t i n g s w e f i n d t h e re flu x
i c o n a s s o c i a t e d w i t h t h e a p h o r i s m “ as a b o v e , so b e l o w , ” r e c a l l i n g t h e selfs i m i l a r s c a l in g c o s m o lo g y w e h a v e s e e n in s u b - S a h a r a n A fric a , as well as its lin k s
t o t h e r e c u r s i o n o f s e l f - f e r t i l i z a t i o n . 1^
O f c o u r s e , o n e c a n go t o o far in a t t r i b u t i n g li n k s b e t w e e n a n c i e n t E gy pt
a n d s u b - S a h a r a n A f r i c a (see O r i t z d e M o n t e l l a n o 1 9 9 3 ; M a r t e l 19 94; L efk o w itz
1 9 9 6 ). T h e r e is goocf_e_yidence for th e o r i g in s o f t h e E g y p ti a n b a s e - tw o a r i t h m e t i c
sy s te m fro m s u b - S a h a r a n A fric a , a n d for t h e p e r s i s t e n t use o f r e c u r s io n in k n o w l ­
e d g e sy stem s acro ss t h e A f r i c a n c o n t i n e n t . B u t it w o u ld b e u n w is e to a ss u m e t h a t
o n e c a n a t t r i b u t e m o r e sp ecific f e a t u r e s t o d if fu s i o n . I n p a r t i c u l a r , it is h ig h l y
u n l i k e l y t h a t t h e s a m e fig u re o f a s e r p e n t b i t i n g .its ta i l, a p p e a r i n g as a n ic o n
for t h e g od D a n in t h e v o d u n r e l i g i o n o f B e n i n (fig. 8 . 1 8 a ) c o u ld h a v e d e r i v e d
f r o m tlie E g y p t i a n i m a g e , o r v i c e v e r s a . A s w e s h a l l s e e , t h e m e a n i n g o f t h e
v o d u n ic o n h a s n o t h i n g to d o with, t h e E g y p t i a n reflux c o n c e p t .
In A u g u s t 1994, t h a n k s to t h e a id o f M a r t i n e d e S o u s a ( o n e o f t h e A f r i c a n
d e s c e n d a n t s o f t h e fa m e d F r a n c i s c o d e S o u z a ) , 1 was g r a n t e d a n i n t e r v i e w w ith
t h e c h i e f o f t h e D a n t e m p l e in O u i d a h , B e n i n . B o t h t h e c h i e f a n d his wife w e re
q u i t e r e s p o n s iv e to my i n t e r e s t in t h e g e o m e t r i c fe a tu re s o f D a n ’s r e p r e s e n t a t i o n s
a n d identified th e sinusoidal ico n in ir o n (fig. 8. j 8 b ) as " D a n a t w ork in t h e w o rld ,”
p o i n t i n g o u t t h a t h e c r e a t e s o r d e r in w i n d a n d w a te r. T h e c y c li c D a n w as m o r e
a b s t r a c t , e x i s t i n g in a d o m a i n w h e r e h e w as in c o m m u n i c a t i o n w i t h o t h e r gods
o f v o d u n . M a u p o i l ( 1 9 8 1 , 7 9 ) a is o f o u n d t h a t D a n ( D a n g b e ) w as t h e r e “to
ass u re t h e r e g u l a r i z a t i o n o f t h e fo r c e s ,” a n d B li e r ( 1 9 9 5 ) s u m m a r i z e s his ro le as
“p o w e r s o f m o v e m e n t t h r o u g h life, a n d n a t u r e ’s b le s s in g s .” R e g u l a r p h e n o m e n a
in n a t u r e — t h e p e r i o d i c a s p e c t s o f w e a t h e r , w a t e r w a v e s , b i o l o g i c a l c y c le s,
e t c . — a re a t t r i b u t e d t o t h e a c t i o n o f D a n .
T h e r e l a t i o n b e t w e e n t h e u n d u l a t o r y D a n “a t w o r k in t h e w o r l d ” a n d _ th e
c i r c u l a r fo rm o f D a n as a m o r e a b s t r a c t s p i r i t u a l f o r c e m a p s n e a t l y o n to t h e d i f ­
f e r e n c e b e t w e e n t h e s i n u s o i d a l w a v e s w e s e e in s p a c e a n d t i m e — w a v e s in
noise {external temperature changes)
input (desired
temperature)
T h e t h e r m o s t a t t h a t re g u l a t e s t e m p e r a t u r e in a h o u s e is a n e g a t i v e f e e d b a c k l o op . T h e w o r d “ n e g a t i v e "
is used b e c a u s e we s u b t r a c t t h e c u r r e n t r o o m t e m p e r a t u r e fro m t h e d e s i r e d t e m p e r a t u r e s e t by t h e
t h e r m o s t a t c o n t r o l . O v e r t i m e t h i s wi ll t e n d t o p r o d u c e cy cles o f h e a t a n d co ld .
noise {road bumps)
D r i v i n g a c a r c a n als o be m o d e l e d by a n e g a t i v e fe e d b a c k lo o p . T h e d r i v e r a t t e m p t s t o stay in t h e c e n t e r
o f t h e la n e , a n d will c o r r e c t t o a d j u s t for b u m p s . A g a i n , g i v e n e n o u g h b u m p s ,' w e will t e n d to see cy cles
o f s w e rv i n g t o g e t b a c k to t h e c e n t e r .
c
FIGURE 8 . l 8
T h e v o d u n god D a n
I n t h e v o d u n r e l i g i o n o f B e n i n , t h e s n a k e g o d D a n r e p r e s e n t s t h e c y c l i c o r d e r o f n a t u r e . D a n 's
s h a p e r e fle c ts t h i s i d e a in t w o w ay s . A s a n a b s t r a c t f o r c e , h e is r e p r e s e n t e d as a f e e d b a c k l o o p ( a ) .
A s a c o n c r e t e m a n i f e s t a t i o n , h i s b o d y is a l w a y s o s c i l l a t i n g in a p e r i o d i c w a v e ( b ) . T h i s s a m e id ea
o f a p e r i o d i c t i m e s e r i e s f r o m c y c l i c f e e d b a c k is a l s o u s e d in W e s t e r n m o d e l s o f n a t u r e (c ).
( a , p/ioto courtesy J F A N , D akar.)
R ecursion
w a t e r a n ti c ir r u s c lo u d s , d a il y f l u c t u a t i o n s in h e a t a n d li g h t, t h e b i a n n u a l ra i n y
seaso ns, e tc .— a n d th e a b s t r a c t id e a o f a n i t e r a t iv e l o o p t h a t g e n e ra te s th e s e w a v e ­
fo rm s. T h e a s s o c i a t i o n c a n b e d e r i v e d f r o m i:he k i n d o f e m p i r i c a l o b s e r v a t i o n
o n e . g e ts in e v e r y d a y o c c u r r e n c e s . A d o p s i d e d w h e e l w ill p r o d u c e u n d u l a t o r y
t r a c k s in s a n d ; f r i e n d s w h o p e r i o d i c a l l y g i v e gifts a r e in a “c y c le o f e x c h a n g e , ”
a n d so f o r t h . W h a t d id t a k e g r e a t i n s i g h t a n d i n t e l l e c t u a l lab o r, h o w e v e r , was
t h e re l ig i o u s p r a c t i t i o n e r s ’ g e n e r a l i z a t i o n o f s u c h o b s e r v a t i o n s i n t o s p e c ific ,
a b s t r a c t , u n iv e r s a ll y a p p l i c a b l e c a t e g o r i e s , r e p r e s e n t e d by i c o n s w i t h t h e a p p r o ­
priate g eo m etric stru ctu re. ^
T h e m a t h e m a t i c a l e q u i v a l e n t s i n n o n l i n e a r d y n a m i c s a r e li m i t c y c le s a n d
p o i n t a t t r a c t o r s — t h e re s u lts o f w h a t e n g i n e e r s c a ll a “n e g a t i v e f e e d b a c k l o o p . ”
W e h a v e a l r e a d y s e e n s u c h c h a r a c t e r i z a t i o n s in c e l l u l a r a u t o m a t a a n d o w a r i,
w h e r e s p a t i a l p a t t e r n s r e m a i n b o u n d e d w i t h i n a c y c le o r fr o z e n in a s t a t i c p a t ­
te rn . Figure 8 .1 8 c sh ow s s o m e c o m m o n p l a c e e x a m p l e s o f n e g a ti v e fee d b a c k loops,
a n d h o w th e y a c t to k e e p th e b e h a v io r o f system s b o u n d e d o r stabilized, e v e n
in che p r e s e n c e o f n oise . B u t che v o d u n s y s te m w o u ld n o t b e c o m p l e t e if it c o u ld
o n l y a c c o u n t fo r r e g u l a r i t y — w h a t c a u s e s d e v i a t i o n in t h e first p l a c e ? H e n c e
t h e ro le o f L g g b a, g o d o f c h a o s . F igu re 8 . 1 9 a s h o w s a n o t h e r i r o n i c o n , t h e fo r k e d
p a t h o f L e g b a , “g o d o f t h e c r o s s r o a d s . ” A s e x p l a i n e d to m e by K a k e S . A l f r e d ,
1
- ; I------ ------- -..w
a d i v i n a t i o n p r i e s t o f v o d u n i n C o t o n o u , B e n i n , L e g b a is r e p r e s e n t e d by t h e
fork b e c a u s e " t h e a n s w e r c o u l d b e yes o r n o ; y o u d o n ’t k n o w w h i c h p a t h h e wilt
t a k e . ” F or d i v i n a t i o n , in w h i c h a “p a t h ” ( q u e s t i o n ) is o f t e n p u r s u e d fo r f u r c h e r
q u e s t i o n s , t h e im a g e b e c o m e s o n e o f e n d l e s s b i f u r c a t i o n s . A t t h e P a l a i s R o y a l
in P o r t o N o v o , B e n i n , I w a s t o l d t h a t t h e s h r i n e t o L e g b a w as p l a c e d a t t h e
t h r e s h o l d b e c a u s e h is f o r c e w a s so d i s r u p t i v e t h a t it w o u ld u n d o b o c h g o o d a n d
e v il, c r e a t i n g a p u r i f i c a t i o n a t the-er.trav.-ce-; K-akc a ls o e x p l a i n e d t h a t w h i l e t h e
m u sic o f D a n was slow a n d re g u la r, t h e m u s ic o f L e g b a w as b o t h fast a n d s lo w —
sig n ify in g h i s u n p r e d i c t a b l e n a t u r e — a n o b s e r v a t i o n 1 was a b l e to c o n f i r m by
r e c o r d i n g t h e d r u m m i n g t h a t w a s u se d t o c a l l e a c h g o d a t t h e t e m p l e o f D a n in
O u i d a h . 1-5 A s t h e c o n v e r s e to D a n , t h e b i f u r c a t i n g u n c e r t a i n t i e s o f L e g b a are
like a p o s i t i v e fe e d b a c k l o o p , a m p l i f y i n g d e v i a t i o n a n d n o is e (fig. 8 . 1 9 b ) .
C o n t r a s t s b e t w e e n a n e g a t i v e fe e d b a c k lo o p , c r e a t i n g stability, a n d t h e p o s­
itive f e e d b a c k o f u n c o n t r o l l e d d i s o r d e r a r e a l s o f e a t u r e d in t h e .ico n ic c a r v i n g s
o f t h e Battle. V o g el ( 1 9 7 7 , 5 3 ) n o t e s t h a t t h e B a u le c h i e f is c h o s e n by c o n s e n T iisT a h d t h a t in all i m p o r t a n t d e c i s i o n s h e s e r v e s as m e d i a t o r in p u b li c m e e t i n g s
r a t h e r t h a n as a n a u t o c r a t . T h e B a u le c a r v i n g in figure 8 .2 0 a s h o w s t w o c a i m a n s
(relativ es o f th e a llig a to r) b i t i n g e a c h o t h e r ’s tails. It is said to r e p r e s e n t t h e c h i e f
a n d t h e p e o p l e in b a l a n c e — if o n e b i t e s , t h e o t h e r will b i t e b a c k , k n i c e l y
recalls th e k in d s o f n e g a tiv e fe e d b a c k lo op m o d e l s th a t a re o fte n p ro p o s ed in W est-
143
FIGURE
8.19
L egba
(a) T h e v o du n god Legba represents th e forces o f disorder.
Vodun d iv in a tio n priests explain this icon as'the p a th to the
future: w ith Legba there is no way to’h n o w w h ic h p a th will be
caken. S in c e o n e crossroad leads to another, th e resulting image
is o n e o f bifurcating u n know ns, th e u n certain ty multiplying with
ea c h crossroad.
n oise (road bum ps)
output (n ew road position
In contrast to negative feedback, which wili help stabilize a system, positive feedback will destabilize it.
A drunken driver, for example, can overshoot the center line and create increasingly large oscillations,
eventually running off the road.
Here we see positive feedback in an arms race.
R ecursion
FIGURE 8 .2 0
F e e d b a c k loops in B a u l e ic o n o g r a p h y
(a) This Baule carving shows two crocodiles biting
e ac h other’s tails. It is a symbol showing the chief and
the people in equal power, the idea of social forces in a
cycle of balance, (b) Baule door. Holas (1952, 49-50)
describes this as a c i r c u i t f e r i n e of f e c o n d i t e (closed
tircuir of fecundity); Soppeiasa (1974) and Odica
(ii;71) identify these animals as symbols of ‘‘increase.’’
(a,md b, [iliots) courtesy of 1FAN, Dakar.)
orn p o li ti c a l th e o r y , h u t t h i s f l o w c h a r t is a p u r e l y in d i g e n o u s i n v e n t i o n . S o , coo,
is th e B au le p o s i t i v e f e e d b a c k l o o p o f figure 8 . 2 0 b , s h o w i n g t h a t “ p o w e r c r e a t e s
the a p p e t i t e for m o r e p o w e r ”— l i t t l e fish a r e e a t e n by b ig g e r fish, w h o t h e n
b e c o m e e v e n b ig g e r fish, T h e f i s h - w i t h i n - f i s h a b b i a f r o m C a m e r o o n we saw
earlier m ay h a v e h a d s i m i l a r c o n n o t a t i o n s .
[C o n c lu sio n
R ecu rsio n c_giibe--found. in a l m o s t e v e r y c o r n e r o f A f r i c a n m a t e r i a l c u l t u r e a n d
design, fr o m c o n s t r u c t i o n t e c h n i q u e s t o e s t h e t i c d e s i g n , a n d in c u l t u r a l r e p r e ­
s e n t a t i o n s fr o m k i n s h i p t o c o s m o l o g y . M o s t o f t h e s e a r e s p e c i f i c e n o u g h to
allow us to d i s t i n g u i s h b e t w e e n t h e first t w o ty p es o f r e c u r s i o n — c a s c a d e v ersu s
I 45
A fr ic a n fra c ta l mat/icniatics
i t e r a t i o n — a n d in s o m e c a s e s t h e t h i r d t y p e , s e l f - r e f e r e n c e , is a ls o m a d e e x p l i c i t
by t h e i n d i g e n o u s k n o w l e d g e s y s te m . W e h a v e s e e n s e v e r a l c a s e s i n w h i c h t h e
i t e r a t i v e lo o p s a r e n e s t e d , b u t t h e s e a r e ra r e ly m o r e t h a n t w o lo o p s d e e p , so it
w o u l d n o t a p p e a r t h a t t h e a p p l i c a t i o n o f s e l f - r e f e r e n c e is t n o t i v a t e a by t h e c o m ­
p le x ity o f th e c o m p u t a t i o n . T h e o n l y p o t e n t i a l e x c e p t i o n is t h e c o s m o lo g i c a l n a r ­
r a t i v e o f t h e D o g o n , a n d t h i s n a r r a t i v e is t o o v a g u e t o s e r v e as a m a t h e m a t i c a l
f o u n d a t i o n . T h e r e is, h o w e v e r , a n o t h e r r o u t e t o t h e l i m i t s o f c o m p u t a t i o n . A s
w e will find in c h a p t e r 10, t h e c o m b i n a t i o n o f n e g a t i v e a n d p o s i t i v e f e e d b a c k
in d ic a te d , by c e r t a i n r e c u r s i o n icons, p r o v i d e s / a n o t h e r p a t h t o t l ^ . h e l g h t s . o f c o m ­
p u t a t i o n a l c o m p l e x i t y , o n e we w ill e x p l o r e in d e t a i l . B u t first, w e n e e d t o t a k e
a 's K o rt’d e t o u r t h r o u g h ~ m f i n ky.
CHAPTER
-Infinity-
•T h e first t i m e 1 s u b m it te d a j o u r n a l a rt ic l e o n A f r i c a n fractals, o n e re v ie w e r replied
t h a t A f r i c a n s c o u l d n o t h a v e “ t r u e ” f r a c ta l g e o m e t r y b e c a u s e t h e y la c k e d t h e
a d v a n c e d m a t h e m a t i c a l c o n c e p t o f in finity. O n t h e o n e h a n d , t h a t r e v i e w e r was
w ro n g a b o u t fra c ta ls a t a p r a g m a t i c le v e l. If h e o r s h e saw a fra ctal o n a c o m p u t e r
s c r e e n it w o u ld b e t a k e n as a “ t r u e ” e x a m p l e , a n d in f a c t n o p h y s ic a lly e x i s t i n g
fractal is i n f i n i t e in its scales; a t b e s t i t will h a v e t o b o t t o m o u t i n t o s u b a t o m i c
p articles. O n t h e o t h e r h a n d , it raises a n i n t e r e s t i n g q u e s t i o n . I n f in ity h a s b e e n
an i m p o r t a n t p a r t o f f r a c ta l m a t h e m a t i c s in E u r o p e ; so h o w d o e s t h a t c o m p a r e
to th e use o f in f in ity in A fri c a ?
T o t h e a n c i e n t G r e e k s , in f in i ty w as a s s o c i a t e d w i t h w h a t t h e y t h o u g h t (j>f
as t h e h o r r o r s o f j n f i n i t e r e g r e s s . A r i s r o t l e t a m e d t h i s p r o b l e m b y r e d e f i n i n g
infinity: it w as a li m i t t h a t o n e c o u l d t e n d t o w a r d , b u t it w as n o t c o n s i d e r e d to
he a l e g i t i m a t e o b j e c t o f m a t h e m a t i c a l i n q u i r y i n itself. M o s t E u r o p e a n m a t h e ­
m a t ic i a n s k e p t to t h i s d e f i n i t i o n u n t i l t h e C a n t o r s e t, E u r o p e ’s first fr a c ta l , c r e ­
ate d t h e p r o p e r d e f i n i t i o n o f a n i n f i n i t e s e t , t h u s a l l o w i n g in fin ity its e lf t o b e
c o n s i d e re d . W e w ill d is c u ss t h i s in m o r e d e t a i l in c h a p t e r 13, b u t fo r n o w it is
sufficient to n o t e t h a t t h i s d i s t i n c t i o n d o e s n o t s h a p e A f r i c a n c o n c e p t s o f i n f i n ­
ity. M a n y A f r i c a n k n o w l e d g e s y s t e m s using^ip i m i t y in t h e s e n s e o f a p ro g r e s s i o n
w i t h o u t li m i t d o n o t h e s i t a t e to r e p r e s e n t it w i t h i c o n i c s y m b o ls s u g g e s tin g
147
148
A fr ic a n fra c ta l tnmhematics
“t h e in f in i te " in its C a n t o r u m s e n s e as a c o m p l e t e d w h o l e . T h i s is h y n o m e a n s
a m ore so p h istic a te d o r ela b o ra te d d e fin itio n th a n th a t o f p r e - C a n to r ia n E u ro ­
p e a n m a t h e m a t i c s ; i t is r a r e ly l i n k e d to m u c h m o r e t h a n e i t h e r a . n a r r a t i v e o r a
g e o m e t r i c v is u a li z a ti o n . B u t far f r o m b e i n g n o n e x i s t e n t , t h e s e c u l t u r a l l y sp ecific
r e p r e s e n ta tio n s sh o w a s tro n g e n g a g e m e n t w ith th e sam e c o n c e p ts t h a t c o u p led
in f in i ty a n d fr a c ta ls in c o n t e m p o r a r y W e s t e r n m a t h e m a t i c s .
T h e m o s t c o m m o n A f r i c a n v i s u a li z a ti o n s for in f in i ty a r e s n a i l s h e l ls . T h e
( B n lu b a ) fo r e x a m p l e , u s e s p ira l la n d s n a i ls (fig. 9 . 1 ) , a n d t h e r e i n itse t h e spiral
e n d o f a sea s n a i l, w h i c h fo r m s a d r i n k i n g c u p t h a t c a n o n l y b e u se d by th e ch ief.
U n l i k e t h e a n c i e n t G r e e k a s s o c ia tio n s w i t h t r o u b l i n g p a r a d o x a n d p a th o l o g y , th e
Africa_n in f i n ite is typi c a l l y a p o s i ti v e a s s o c ia tio n , in th is c a s e t o in v o k e p ro s p e rity
w i t h o u t e n d . I f j h e s e in f in i ty i c o n s w e r e o n ly m e a n t t o c o m m u n i c a t e t h i s d e s ire
t h e y w o u l d j i t ' A r i s t o t l e ’s d e f i n i t i o n : a p ro c e s s w i t h o u t e n d . B u t t h e s p i r it u a l e l e ­
m e n t qf_these i c o n s .a d d s a n o t h e r r e q u i r e m e n t : t h e ic o n s n e e d t o c o n v e y t h e se nse
t h a t th ex ar.e.d raw .in g o n t h e p o w e r o f in f in ity jt_self{ S n a i l s h e l ls a re used b e c a u s e
o f t h e sc a lin g p ro p e r tie s o f t h e i r l o g a r it h m i c spirals; o n e c a n c le a rly see th e p o t e n ­
tial for th e spiral to c o n t i n u e w i t h o u t e n d d espite its c o n t a i n m e n t in a finite space—
i n d e e d , it is o n l y b e c a u s e o f its c o n t a i n m e n t in a fi n it e s p a c e t h a t t h e r e is a sen se
o f h a v i n g g a i n e d a c c e s s to o r g r a s p e d a t t h e in f in ite )
W e h a v e a lr e a d y s e e n a n o t h e r e x a m p l e o f a n in f i n i t y i c o n in t h e N a n k a n i
a r c h i t e c t u r e d is c u s s e d i n c h a p t e r 2. T h e r e t h e c o i l s o f a s e r p e n t o f i n f i n i t e
FIGURE 9 . J
B a l u b a u s e o f s n a il s h e lls
to sy m b o lize in /im ty
Davidson (197 1, 120) describes this as a fertilit
figure, and notes that the snail shells represent
endless growth.
( CoUecfifW Tm;m: T zara, Pnrii; }>lv>ii> hy Elio!
Eliso/on.)
In fin ity
l e n g t h , s c u l p t e d i n t o t h e h o u s e w alls, m a d e u s e o f che s a m e a s s o c i a t i o n b e t w e e n
p r o s p e r it y w i t h o u t e n d , a n d a g e o m e t r i c l e n g t h w i t h o u t e n d . ^ h e c o n s c i o u s
c r e a t i o n o f t h i s in f in ity c o n c e p t is m o r e c le a r t h a n in t h e c ase o f t h e sn a il sh ells,
b e c a u s e o n e c a n n o t a c t u a l l y se e t h e i n f i n i t e c o ils 'tr f-th e s n a k e . A n d u n l i k e t h e
n a t u r a l l y o c c u r r i n g sh e lls , t h e p a c k i n g o f t h i s i n f in i te l e n g t h i n t o a fin ite s p a c e
( t h e N a n k a n i d e s c r i b e it as “c o i l i n g b a c k o n i t s e l f i n d e f i n i t e l y ” ) c a n n o t be m i s ­
t a k e n for m e r e m i m i c r y o f n a t u r e ; it is r a t h e r t h e artifice o f f r a c t a l s ^ T h i s s n a k e
ic o n d o e s n o t e x is t in is o la t io n ; we saw t h a t - t h e N a n k a n i m a p o u t a s c a l in g p r o ­
g r e s s io n t h a t p a s s e s ' t h r o u g h t h e i r a r c h i t e c t u r e , t h e za h in g a a n d t h e k itm p io ,
w h i c h p r o v i d e s a re c u r s iv e p a t h w a y to t h i s c o n c e p t o f in finity.
In c h a p te r 8 we d iscu ssed th e M i ts o g h o a n d F an g ite r a tiv e m o d e l o f
d e s c e n t . F e r n a n d e z ( 1 9 8 2 , 3 3 8 ) n o t e s t h e c o n t r a s t to C h r i s t i a n th e o l o g y : “T h e
q u e s t i o n as t o w h e t h e r G o d w a s o n e o r m a n y m a y h a v e b o t h e r e d t h e m i s s i o n ­
aries i n t h e i r c o n t a c t s w i t h F a n g m o r e t h a n t h e F a n g t h e m s e l v e s . H o l d i n g C h r i s ­
ti a n b eliefs in t h e ‘U n c r e a t e d C r e a t o r ’ a n d ‘U n m o v e d M o v e r , ’ m i s s i o n a r i e s w ere
c h a l l e n g e d by t h e ‘i n f in i te re g r e ss’ o f t h e g e n e a l o g i c a l m o d e l e m p l o y e d by t h e
F a n g — t h e i r b e l i e f t h a t che G o d o f t h i s w o r ld is o n e o f a l o n g li n e o f g o d s a n d
like m a n h a s h is o w n g e n e a l o g y .”
f
T h e F a n g th e o r y o f in f in ite regress is a c o m p l e t e , c o h e r e n t view ; it d o e s n o t
n e e d f u r t h e r a m e n d m e n t , for t h e C h r i s t i a n th e o r y o f u n c r e a t e d c r e a t o r is n o m o r e
free o f c o n t r a d i c t i o n — a n d p e r h a p s less s o A O f c o u r s e , as F e r n a n d e z h i m s e l f
warns, o n e c a n n o t simply p ro c laim t h a t a p a rtic u la r A fric a n n a rr a tiv e is just a n o t h e r
w o rk o f t h e o l o g y o r p h i l o s o p h y — or, f o r t h a t m a t t e r , m a t h e m a t i c s . R e c e n t w orks
s u c h as M u d i m b e ’s In v e n tio n o f A fr ic a ( 1 9 8 8 ) h a v e s h o w n t h a t s u c h t r a n s l a t i o n s
to specific E u r o p e a n d is c ip lin e s are alw ays p a rt ia l , hig h ly in t e r p r e tiv e , a n d in d a n ­
ger o f m i s r e p r e s e n t i n g t h e in d i g e n o u s view. Yet M u d i m b e is also res p e ctfu l o f th e
w ork t h a t h a s b e e n d o n e . O f p a r t i c u l a r r e l e v a n c e h e r e a re h is c i t a t i o n s o f A f r i c a n
th eo lo g ian E n g elb ert M v en g .
Y M v e p g i n c l u d e d s e v e r a l n o t e s o n i n f i n i t y in his s t u d i e s o f t h e r e l a t i o n
b e t w e e n t h e A f r i c a n a n d C h r i s t i a n view s. H is b e a u ti f u l te x t, L 'A rt d 'A /r(que N oire
( 1 9 6 4 ) , c o n t a ins d i a g r a m s (p p . 1 0 0 - - 1 0 3 ) s h o w i n g w h a t h e t e r m e d “r a d i a t i o n
a m p l i f i c a t i v e ,” s c a l i n g p a t t e r n s in_Afncan_art_a_hd mus_i_c_that he_ i n t e r p r e t e d as
r e p r e s e n t a t i o n s o f a t r a n s c e n d e n t a l p a t h to in fin ity . “U n e fois d e plus, n o u s
d e c o u v r o n s q u e le m o u v e m e n t r y t h m i q u e , d a n s n o t r e a r t , n ’esc a u t r e c h o s e
q u ’u n e c o u rs e v e rs 1'i n f i n i ” ( O n c e a g a i n , w e d i s c o v e r t h a t t h e r h y t h m i c m o v e ­
m e n t in o u r a r t is n o n e o t h e r t h a n t h e p a t h t o w a r d i n f i n i t y ) (p. 1 0 2 ). F a t h e r
M v e n g w as a w o n d e r f u l i n s p i r a t i o n d u r i n g m y r e s e a r c h in C a m e r o o n , b o t h fo r
his d e e p c u l t u r a l k n o w l e d g e as w ell as fo r his c o u r a g e o u s w o r k as a c ro s s - c u l tu r a l
m e d ia to r. D u r i n g o u r last m e e t i n g we d is c u s s e d M u d i m b e ’s b o o k , a n d 1 p ro m is e d
149
A fr ic a n fra c ta l mathematics
t o s e n d h i m a cop y- S h o r t l y a f t e r d o i n g s o a r e p l y c a m e f r o m t h e A m e r i c a n
C u l t u r a l C e n t e r in Y a o u n d e M v e n g h a d b e e n m u r d e r e d “u n d e r s u s p i c i o u s
c i r c u m s t a n c e s ”— a p p a r e n t l y t h e r e s u l t o f o p p o s i t i o n t o h i s c r o s s - c u l t u r a l
a c t i v i s m . H e h a s f i n a l l y t a k e n t h e course vers V in f ini.
^
CHAPTER
Com plexity
IO
In o r d i n a ry s p e e c h C ^ c o m p I g ^ ” j u s t m e a n s t h a t t h e r e is a lo t g o i n g o n . B u t for
m a t h e m a t i c i a n s t h e t e r m is p r e c i s e l y d e f i n e d , a n d it g iv es us a n e w w ay to
a p p r o a c h m a t h e m a t i c s in A f r i c a n m a t e r i a l c u l t u r e . In c h a p t e r 7 we saw h o w c e r ­
ta i n A f r i c a n s y m b o l i c s y s t e m s , l i k e t h e B a m a n a d i v i n a t i o n c o d e , c o u l d be
g e n e r a t e d b y a r e c u r s i v e l o o p . S u c h n u m e r i c s y s te m s c l e a r l y t r a n s l a t e i n t o th e
W e s t e r n d e f i n i t i o n s o f w h a t it m e a n s t o “c o m p u t e . ” B u t t h e t r a n s l a t i o n w as less
c le a r for s o m e of t h e p h y s i c a ll y re c u r s iv e s t r u c t u r e s in A f r i c a n m a t e r i a l c u l t u r e .
C a n a s y s t e m o f p h y s ical d y n a m i c s b e s a i d t o “c o m p u t e ” ? M a t h e m a t i c a l c o m ­
p le x i ty t h e o r y , w h i c h is b a s e d o n f r a c t a l g e o m e t r y , p r o v i d e s a w ay to m e a s u r e
th e c o m p u t a t i o n e m b e d d e d in p h y s i c a l s t r u c t u r e s , r a t h e r t h a n j u s t s y m b o l sy s­
tem s. By l o o k i n g a t A f r i c a n m a t e r i a l c u l t u r e in t h e f r a m e w o r k o f c o m p le x ity ^
th eory, we c a n b e t t e r im d e rs ta n .d j.h e . p r e s e n c e o f f r a c ta l g e o m e t r y as a n A f r i c a n ) j
k n o w ledg e sy stem ..
A n alo g c o m p u tin g
By th e mid.-,.i_p6ps it w as c l e a r t o m a n y r e s e a r c h e r s c h a t d ig ita l c o m p u t e r s w o u ld
be th e w a v e o f t h e fu tu re . B u t be fo re t h e n , a n a l o g c o m p u t e r s h e l d t h e i r o w n , a n d
th ey m a y y e t m a k e a c o m e b a c k . I n d i g i ta l sy s tem s , i n f o r m a t i o n is r e p r e s e n t e d by
A fr ic a n fra cta l m athematics
p h y s i c a ll y a r b i tr a r y s y m b o ls. A s B a t e s o h ( 1 9 7 2 ) said , “T h e r e is n o t h i n g s9.yeni.sh
a b o u t t h e n u m e r a l.$ e v .e n .” T h e g e o m e t r i c s t r u c t u r e o f a d i g i t a l s y m b o l h a s li t t l e
o r n o t h i n g t o do. w i t h it s m e a n i n g , w h i c h is s i m p l y a s s ig n e d t o it-.-hy s o c i a l c o n ­
v e n t i o n . I n a n a l o g sy s te m s , t h e p h y s i c a l s t r u c t u r e o f t h e 'S i g n a l c h a n g e s in p ro -\
p o r t i o n t o c h a n g e s in t h e i n f o r m a t i o n it r e p r e s e n t s . 1 R a t h e r t h a n b e i n g arb itra ry ,
t h e p h y s i c a l s m ic .tu re is a d i r e c t r e f le c t io n o f its i n f o r m a t i o n . L o u d n e s s in h u m a n
s p e e c h is a g o o d e x a m p l e o f a n a l o g r e p r e s e n t a t i o n . A s I g e t m o r e e x c i t e d , I s p e a k
l o u d e r : t h e p h y s i c a l p a r a m e t e r c h a n g e s in p r o p o r t i o n t o t h e s e m a n t i c p a r a m e ter. T h i s is n o t tr u e for t h e d ig ital p arts o f sp e e c h , s u c h as t h e av erag e p i t c h (“ fo m a t
f r e q u e n c y ”) o f e a c h w o rd . I n E n g l i s h t h e w o r d “c a t ” h a s a h i g h e r p i t c h t h a n t h e
w o r d “d o g ,” b u t t h a t d o e s n o t in fe r a r e l a t i o n in m e a n i n g — in fa c t, t h e d if f e r e n c e
is r e v e r s e d in S p a n i s h , s i n c e “g a t o ” h a s a lo w e r a v e r a g e p i t c h t h a n “p e r r o ." T h j s
s a m e a n a l o g / d i g i t a l d i s t i n c t i o n o c c u r s in n e u r a j , s i g n a l s . . I n t h e frog r e t i n a , fo r
e x a m p l e , s o m e n e u r o n s h a v e a firing r a t e in p r o p o r t i o n to t h e sp e e d o f sm all m o v - ^
i n g im ag es ( G r u s s e r a n d G r u s s e r - C o r n e h l s 1 9 7 6 ) . T h a t is, t h e fa s te r a fly m o v e s C
acro s s t h e ey e, t h e f a s te r t h e p u ls e s o f t h e n e u r o n : a n a n a lo g s y s te m . A d i g i t a l
e x a m p l e c a n b e fo u n d in th e_rnq tor n e u r o n s t h a t fling o p e n t h e crayfish claw. H e r e p
a sp ecific firing p a t t e r n ( o f f - o t v o n - o f f ) s w i t c h e s t h e c l a w t o t h i s d e f e n s e re fle x (
( W i l s o n a n d D a v is 1 9 6 5 ) .
S o far w e h a v e o n l y e x a m i n e d h o w a n a l o g s y s t e m s c a n r e p r e s e n t i n f o r ­
m a t i o n ; fig u re
to
.1
s h o w s a s i m p l e e x a m p l e o f h o w a n a l o g c o m p u t i n g w o rk s .
A l t h o u g h m o s t c o m p u t e r s c i e n t i s t s e v e n t u a l l y s e t t l e d o n d i g i t a l s y s te m s , a n a ­
lo g c o m p u t e r s w e r e q u i t e p o p u l a r u p u n t i l t h e 1 9 6 0 s. E v e n w h e n t h e y b e g a n t o
d i e o u t as p r a c t i c a l m a c h i n e s , t h e r e w as a n i n c r e a s i n g j r w a r e n e s s t h a t m u c h o f
o u r o w n b r a i n . p p e r a . t e s . b y a n a l o g c o m p u t i n g , a n d t h i s J e d some... s c i e n t i s t s
t o w a r d t h e d e v e l o p m e n t o f w h a t .are. n o w c a l l e d .“ n e u r a l , n e t s ”— c o m p u t i n g
d e v i c e s t h a t m i m i c t h e a n a l o g o p e r a t i o n s o f n a t u r a l n e u r o n s (fig. 1 0 .2 ) . By t h e
m i d - 10 8 0s n e u r a l n e .ts .a n d r e l a t e d a n a l o g d e v i c e s h a d a c h i e v e d e n o u g h s u c c e s s
( a n d dig ital c o m p u te r s h a d ru n in t o e n o u g h b a rrie rs ) to b e g in to c o m p a r e th e
t w o . T h e r e w a s a n o d d m o m e n t o f a n a l o g o p t i m i s m , w h e n a few b r a s h c l a i m s
w ere m ad e a b o u t th e p o te n tia l su p erio rity o f a n a lo g c o m p u tin g (see D e w d n e y
1 9 8 5 ; V e rg is e t a l. 1 9 8 5 ) , b u t t h e s e a s s e r t i o n s w e r e e v e n t u a l l y p r o v e d i n c o r ­
re c t (B lu m , S h u b , a n d S m a le 1989; R u b e l 1 9 8 9 ). A s it tu r n s o u t, a n a lo g sys­
t e m s h a v e t h e s a m e t h e o r e t i c a l lim its to c o m p u t i n g as d ig i ta l s y s te m s .2
A l t h o u g h t h e s t u d i e s d i d n o t r e s u l t in r e l e a s i n g t h e k n o w n l i m i t a t i o n s , t h e y
d id p r o d u c e a n e w f r a m e w o r k for t h i n k i n g a b o u t c o m p u t i n g in p h y s i c a l d y n a m ics: c o m p l e x i t y t h e o r y .
Before th is tim e , m a t h e m a t ic i a n s h a d d e fin e d c o m p l e x i t y j n te rm s of
ra n d o m n e ss, p rim arily based o n th e w o rk o f S o v ie t m a th e m a tic ia n A . N .
C o m p le x ity
*53
A n a l o g c o m p x ita tio n
Dewdney (1985) shows a great variety of simple physical devices that demonstrate analog
computing. This device, created by J. H. Luerh of the U.S. Metals Refining Company, solves the
following optimization problem; a refinery must be located to minimize its costs. If transportation
in dollars per mile of ore, coal, and limestone are values of O, C, and L, and disrances of these
sources are o, c, and !, then the refinery should be located at the point where oO + cC + 1L is at
;t minimum. T he holes through which the strings pass are at the source locations, and the weights
on the ends of the strings are proportionate to O, C, and L. T h e brass ring attached tc the strings
ijitickly moves to the optimal location on the map.
iC o u n e s y A . K. D e w d n e y )
K olm ogorov a n d A m e ric a n s G re g o ry C h a i t i n a n d R ay S o lo m o n o ff. In this d ef­
in i t i o n , t h e c o m R l e x jt jf.o f a s ig n a l ( e i t h e r a n a l o g o r d i g i t a l ) is m e a s u r e d by t h e
le n g t h o f t h e s h o r t e s t a l g o r i t h m r e q u i r e d t o p r o d u c e it (fig. 1 0 .3 ) . T h i s m e a n s
t h a t p e r i o d i c n u m b e r s ( s u c h as . 2 7 2 7 2 7 2 . . . ) will h a v e a lo w a l g o r i t h m i c c o m 1
plexity. E v e n if t h e n u m b e r is i n f i n i t e l y lo n g , t h e a l g o r i t h m c a n s i m p l y say,
“W r i t e a d e c i m a l p o i n t fo l lo w e d by e n d l e ss r e p e t i t i o j i s o f ‘2 7 , ” ' o r e v e n j h o r t e r :
“3 / 1 1 . " T r u l y r a n d o m n u m b e rs ( e .g ., a s t r i n g o f n u m b e r s p r o d u c e d by r o l l i n g
d i c e ) w ill h a y e j _ h e J x i g h e s t a l g o r i t h m i c c o m p l e x i t y p o s s i b l e , s i n c e t h e i r o n l y
a l g o r i t h m is_che n u m b . y . . i t s e l f — f o r . a n j n f i n i t e l e n g t h , y o u g e t i n f i n i t e c o m p l e x i t y . J n a n a l o g s y s t e m s a p e r i o d i c s i g n a l s u c h as t h e v i b r a t i o n fr o m a s i n g l e
g u ita r s t r in g o r t h e r e p e t i t i v e s w in g s o f a p e n d u l u m w o u ld h a v e t h e lo w e st a l g o ­
r i t h m i c c o m p l e x i t y , a n d r a n d o m n o i s e s u c h as s t a t i c f r o m a r a d i o t h a t h a s lo s t
A fr ic a n fra c ta l m a th em a tics
154
its s t a t i o n ( w h a t is o f t e n c a l l e d “w h i t e n o i s e " ) w o u l d h a v e t h e h i g h e s t a l g o rith m ic co m p lex ity.
O n e p r o b l e m w i t h d e fi n in g c o m p l e x it y in te r m s o f r a n d o m n e s s is t h a t it does
n o t m a t c h o u r i n t u i t i o n . W h i l e i t ’s t r u e t h a t t h e p e r i o d i c s i g n a l o f a t i c k i n g
m e t r o n o m e is so s i m p l e t h a t it b e c o m e s h y p n o t i c a l l y b o r i n g , t h e s a m e c o u l d be
said for w h i t e n o i s e — in f a c t , 1 s o m e t i m e s t u n e m y r a d i o b e t w e e n s t a t i o n s if 1
w a n t t o fall a s l e e p . B u t if I w a n t t o s t a y a w a k e 1 l i s t e n t o m u s ic . M u s i c s o m eh o w sa tisfie s o u r j n t u i t i v e c o n c e p t o f c o m p l e x i t y : it is p r e d i c t a b l e e n o u g h t o fo l­
low a lo n g , b u t s u rp ris in g e n o u g h t o k e e p us p l e a s a n t l y a t t e n t i v e . M a t h e m a t i c i a n s
e v e n t u a l l y c a u g h t u p w i t h t h e i r i n t u i t i o n a n d d e v e l o p e d a n e w m e a s u r e in
w h i c h t h e m o s t c o m p l e x s i g n a ls a re n e i t h e r c o m p l e t e l y o r d e r e d n o r c o m p l e t e l y
d i s o r d e r e d , b u t r a t h e r a r e h a l f w a y in b e t w e e n . T h e s e p a t t e r n s ( w h i c h i n c l u d e
a l m o s t e v e r y ty p e o f i n s t r u m e n t a l m u s i c ) a ls o h a p p e n to b e f r a c t a l s — in fa c t, as
w e ^ l T s e e , th e n ew c o m p le x ity m easu re e x a c tly c o in c j d e s j v it h th e m easure-of
fractal d im e n sio n .
T h e first ste p in this d i r e c ti o n was t h r o u g h stu dies o f c e llu la r a u to m a ta . R eca ll
fr o m c h a p t e r 7 t h a t c o m p u t e r s c i e n t i s t s in t h e early( i g 8 g s-t\ad s t a r t e d t o t h i n k
N e u r a l n ets
(n) Suppose we balance a ball on a teeter-totter. Unless the ball is at the precise center, the
teeter-totter wtl! start to slope toward one side, which will cause the ball t o roll even farther
toward that side. In other words, there are two stable states, and anything in between (except ft
tiny neutral point) will get caught up in the positive-feedhack loop leading rapidly to a stable st
(b) This is an electrical circuit that works much like the teeter-totter. Each triangle is an ampli
with two outputs, one normal and the other (black circle) an inverted output. Since the inverte
output is connected to the input of the other amplifier in each, they will balance out like the b;
at the exact center of the teeter-totter, but rapidly flip to one of the two stable states in which t
amplifier is at its maximum (“saturated”). T h a t means that this circuit can solve a simple task:
which of two numhers is larger? By putting an initial charge proportionate to one of the two
numbers at each inpur, the system rapidly (lips to the saturated stable state favored hy the larget
number. Linking thousands of these simple amplifiers Together allows computer scientists to m;i
sophisticated machines for pattern recognition and other artificial intelligence tasks.
C o m p le x ity
155
a b o u t c e llu la r a u t o m a t a as t h e s i m u l a t i o n o f c o m p l i c a t e d p h y s ic a l d y n a m ic s , s u c h
as t h a t j e e n in l i v in g o rg a n i s m s . P h y s i c i s t S t e p h e n W o l f r a m b e g a n to w o n d e r :
ju s t h o w c o m p l i c a t e d is it? C le a r l y , li v i n g s y s te m s a r e m o r e c o m p l e x t h a n r a n ­
d o m n o is e , so h e k n e w t h a t t h e o ld c o m p l e x i t y m e a s u r e o f K o l m o g o r o v w o u ld
n o t do. B ut W o lfram h a d studied a good d e a l o f c o m p u te r science, a n d h e re a l­
ized t h a t ch e way in w h i c h d i f f e r e n t ty p es o f r e c u r s i o n s a re used t o m e a s u r e c o m p u t in g p o w e r c o u ld a ls o be a p p l i e d t o p h y s i c a l d y n a m i c s . R e c a l l fro m c h a p t e r 8
t h a t we d iv i d e d re c u r s io n in t o t h r e e types: c a sc a d e s , it e ra t io n s , a n d self-reference.
FIGURE
IO.3
K o n n o g o r o a t - C h u / t m c o m p le x ity m e a s u r e
( j) W h e t h e r it is in d i g i t a l o r a n a l o g s i g n a l s , c o m p l e x i t y c a n b e
measured in t e r m s o f t h e i n f o r m a t i o n c o n t e n t . T h e first s u c h
measure w as t h a t o f K o l m o g o r o v a n d C h a i t i n , w h o t h o u g h t o f
£
co m p lexity in t e r m s o f r a n d o m n e s s . T h e s i n e w a v e is a b o u t as
n o n r a n d o m as w e c a n g e t . H e r e it is g i v e n a s a t i m e - v a r y i n g
|
signal, a l t h o u g h t h e s a m e w o u l d a p p l y t o a s p a t i a l p a t t e r n , s u c h
^
as waves in w a t e r o r s a n d ( i n w h i c h c a s e w e c o u l d m e a s u r e it as
w aveleng th, w h i c h is s i m p l y t h e r e c i p r o c a l o f f r e q u e n c y ) ,
lb) T h e s a m e s i g n a l i n a s p e c t r a l d e n s i t y p l o t . T h i s t e l l s y o u
how m u c h p o w e r is a t e a c h f r e q u e n c y . I n t h e c a s e o f t h e s i n e w a v e , all t h e s i g n a l p o w e r
frequency, ( c ) W h i t e n o i s e is a c o m p l e t e l y r a n d o m s i g n a l , s u c h as t h a t p r o d u c e d
js a t o n e
by t h e s o u n d o f
bacon fry ing . By t h e K o l m o g o r o v - C h a i r i n d e f i n i t i o n , w h i t e n o i s e is t h e m o s t c o m p l e x s i g n a l .
Again, t h is w o u l d a l s o a p p l y t o a s p a t i a l p a t t e r n , s u c h as d u s t s p r i n k l e d o n a t a b l e , ( d ) S p e c t r a l
density p l o t for w h i t e n o i s e . B e c a u s e it is c o m p l e t e l y r a n d o m , t h e r e is a n e q u a l l i k e l i h o o d o f a n y
wav elen gth o c c u r r i n g a t a n y t i m e , s o t h e s i g n a l ’s p o w e r is e q u a l l y d i s t r i b u t e d a c r o s s t h e s p e c t r u m ,
(c) In s u m m a r y , t h e K o l m o g o r o v - C h a i t i n c o m p l e x i t y m e a s u r e is s i m p l y a m e a s u r e o f r a n d o m n e s s .
Ic.c'uiivtesy R. F. Voss.)
A fr ic a n (racial mathematics
i5 6
T h e s e co rresp o n d a p p ro x im a te ly to th e th re e form al ca te g o rie s o f recu rsio n
use d in c o m p u t e r s c i e n c e , w h i c h w e w ill n o w e x a m i n e in d e t a i l .
T h r e e t y p e s o f re c u r s io n : t h e C h o m s k y h ie r a r c h y
I n a re c u r s iv e s y s te m , p r e s e n t b e h a v i o r d e p e n d s o n p a s t b e h a v i o r . I t is t h e c a p a ­
b il i t y o f th i s a c c e s s t o m e m o r y t h a t d e fin e s t h e r e l a t i v e d i f f e r e n c e in re c u r s iv e
^ pQAyer. T h e s c a l i n g c a s c a d e , fo r e x a m p l e , c o u l d n o t p r o d u c e t h e F i b o n a c c i '
I s e q u e n c e , b e c a u s e it c o u l d n o t r e c a l l p re v i o u s m e m b e r s o f t h e s e q u e n c e . S i m i \ ^ l a r d i s t i n c t i o n s a r e u se d in c o m p u t e r s c i e n c e to r a n k c o m p u t a t i o n a l p o w e r i n t o
t h r e e ty p es o f a b s t r a c t m a c h i n e s , re f e r r e d t o as “C h o m s k y ’s h i e r a r c h y . ” T h e s e
a b s t r a c t m a c h i n e s a re c o m p a r e d b y t h e i r . a b i l i t y t o r e c o g n j z e c e r t a i n c a t e g o r i e s
o f c h a r a c t e r s t r in g s . A m a c h i n e t h a t c a n r e c o g n i s e p e r i o d i c c h a r a c t e r s t r i n g s
s u c h as “a b a b n . .
o c c u r s a t t h e lo w e s t le v e l o f t h e h i e r a r c h y , t h e F i n i t e S t a t e
A u t o m a t o n ( F S A ) . A n e x a m p l e o f t h e F S A is s h o w n i n figure 10.4.
W h a t w o u ld it b e li k e t o be a n F S A ? S i n c e t h e F S A h a s n o m e m o r y s t o r ­
ag e, t h e e x p e r i e n c e w o u ld be s o m e w h a t a n a l o g o u s t o n e u r o s u r g e r y p a t i e n t s w h o
h a v e h a d b i l a t e r a l h i p p o c m n ji a X I e s io n s { M i l n e r 1 9 6 6 ) . T h e s e p a t i e n t s a r e fu lly \
■[ a w a r e a n d i n t e l l i g e n t b u t h a v e lost t h e c a p a c i t y t o t r a n s f e r k n o w l e d g e t o lo n g -
I t e r m m e m o ry . T h e ' t d . p p o c a m p a l surg~ery p a t i e n t w h o fin d s h e r s e l f a t t h e e n d o f
a b o o k c a n d e d u c e t h a t s h e h a s r e a d its c o n te n ts ., a l t h o u g h s h e d o e s n o t k n o w
y w h a t t h e p r e v i o u s c h a p t e r s w e r e a b o u t . A rtJ F S A -'h a s o n l y a n implicit m e m o r y , f 2 *
b e c a u s e its p r e s e n t s t a t e c a n n o t r e v e a l a n y t h i n g a b o u t its p a s t , o t h e r t h a n t h e }
/ f a c t t h a t it m u s t h a v e p a s s e d t h r o u g h o n e o f t h e s e q u e n c e s o f s t a t e s t h a t t e r m i-. ^
ym lL e"ur i h t p re s e n t state.
input tape
b
a
a
-
a
b
a
b
b
read only
Transition table
Current symbol
on input tape
New state
s,
n
Si
Si
h
s2
a
S2 •
S2
S2
b
Si
Current state
...
FIGURE
. , ....
.
IO .4
T h e fin ite s t a te
au to m a to n
T h e finite state a u to m a to n
(F S A ) has a list of transition
rules t h a t tell it how to change
from o n e state to th e next,
d ep e n d in g o n its curre n t state
an d th e symbol it is reading on
tire inp ut tape. It has n o mem­
ory, o th e r th a n th a t implied hy
its c u re n t state. T h is FSA will
en d up in the “a cc ep t” state S,
it th e tape ends after an even
n u m b er of h’s.
157
Complexity
T h e sec o f p a l i n d r o m i c s t rin g s (e .g ., a a b b a a ) is a g o o d e x a m p l e o f t h e l i m ­
i t a t i o n o f_ th e F S A : it l a c k s t h e a b i l i t y to m e m o r i z e t h e first h a l f o f t h e s t r in g
a n d t h e r e f o r e c a n n o t c o m p a r e it w i t h t h e s e c o n d . T h e le a s t p o w e r f u l m a c h i n e
c a p a b l e o f - t h i s m e m o r y s t o r a g e is t h e P u s h - D o w n A u t o m a t o n f(T O A )\ illus- T t ; ^
t r a t e d in figure 10.5. T h e s t a c k m e m o ry _ o f t h e P D A is u su a lly c o m p a r e d to t h e \
s p r in g -lo a d e d tray s t a c k o f t e n u sed in c a fe terias; o n c e a sy m b o l is read fro m m e m - /
'ory it is g o n e . A s a k n o w l e d g e a n a lo g y , w e m i g h t t h i n k o f a r e a d e r w h o a c c u ­
m u l a t e s s t a c k s o f b o o k s b u t g e ts rid o f e a c h b o o k a f t e r it is re a d . T h i s is a
t e m p o r a r y e x p l i c i t m e m o r y , s i n c e t h e P D A ca n ,,m a k e . tw o d i f f e r e n t t r a n s i t i o n s
g i y e n t h e s a m e s t a t e . a n d i n p u t , d e p e n d i n g o n its p a s t . It is i m p o r t a n t t o u n d e r ­
s t a n d t h a t g r e a t e r re c u r s iv e c a p a b i l i t y d o e s n o t n e c e s s a r i l y j i p p l y larger m e m ­
ory storage; it m e a n s a n im p ro v e d ability to interact w i t h m em ory. Size on ly m a tte r s
in s o fa r as it r e s t r i c t s t h e i n t e r a c t i o n .
A l t h o u g h t h e P D A c a n r e c o g n i z e all sets o f s t r in g s r e c o g n i z e d by a n F S A ,
as w e ll as m a n y o t h e r s , t h e r e a re still ( in f i n i t ely ) m a n y sets o f s t rin g s t h a t jt_ c a n n o t r e c o g n i z e / F o r exam ple;, i t j z a n n o t r e c o g n i z e t h e s e t o f all s t rin g s o f t h e form
a ^ b ^ c N ( w h e r e we h a v e N r e p e t i t i o n s o f a, fo l lo w e d by t h e s a m e for b a n d c),
b e c a u s e it h a s to w ip e o u t its m e m o r y in t h e p r o c e s s o f c o m p a r i n g t h e n u m b e r
o f a ’s a n d b ’s, le a v i n g n o i n f o r m a t i o n fo r c h e c k i n g t h e n u m b e r o f c ’s. ___^
A t t h e t o p o f t h e h i e r a r c h y (fig. 1 0 . 6 ) , t h e T u r i n g M a c h i n b t( T p I ) c a n “ t VA
re c o g n iz e a ll c o m p u t a b l e f u n c t i o n s . It is s i m p l y a P D A w i t h ' u n r e s t r i c t e d m e m ­
ory, b u t b e c a u s e o f t h i s c a p a b i l i t y i t , c a n a c h i e v e f u ll s e l f - r e f e r e n c e : t h e a b i l ­
ity tp ,g p.9iyze J.ts.,pw n p r p g r a m . A g a i n , it is n o t a d i f f e r e n c e in m e m o r y size,
b u t in m e m o r y a c c e s s — u n l i k e t h e P D A s t a c k , t h e T M m e m o r y i n t e r a c t i o n s
c a n o c c u r o v e r a n.y.past s e q u e n c e s o f a n y l e n g t h , a n d it d o e s n o t lose m e m o r y
input t a p e
a
b
a
a
b
a
b
b
F IG U R E
IO.5
T h e p u s h -d o w n
a u to m a t o n
rea d only
read/write
“S ta c k " m em o ry . T h i s a llo w s n e w s y m b o ls
to be p o s h e d d o w n o n to p o f th e s ta c k , h u t
sym bols c a n b e re a d o n ly by p o p p in g t h e m
o f f t h e t o p , a n d e a c h o n e p o p p e d is l o s t .
T he push-down automaton
(PDA) has a list of transition
rules, but these make use of an
explicit memory storage as well
as internal states.
158
A fr ic a n fra cta l m a them atics
input tape
a
b
a
a
b
a
b
FIGURE
b
1 0 .6
T h e T u rin g m a ch in e
T he Turing machine has an
unconstrained memory; it can
implement any algorithm that can
possibly exist.
a f te r it is re a d . T o c o n t i n u e t h e t e x t a n a lo g y , if t h e F S A is a p e r s o n w h o a c c o m ­
p l i s h e s t a s k s w i t h n o b o o k s , a n d t h e P D A is a p e r s o n w h o s e s i m p l e t a s k s a r e
lim ite d to b o o k s t h a t are re m o v e d a fte r th e y are read , t h e n th e T M w o u ld be
a b l e t o c o l l e c t a n d r e c a l l a ll b o o k s , i n a n y o r d e r . U n f o r t u n a t e l y t h i s d o e s n o t
s o l v e a ll o f o u r p r o b l e m s , b e c a u s e t h e u n b o u n d e d n a t u r e o f t h e T M m e a n s
t h a t it fooIishly_acceRt§...s.p.me..tasks. t h a t . r e q u i r e a n i n f i n i t e Jibrary. T h i s is c a lle d
t h e “ h a l t i n g p r o b l e m , ” a n d T u n n S h i m s e l f p r o v e d t h a t jt is u n a v o i d a b l e .
M a t h e m a t i c i a n R o zsa P e t e r s h o w e d t h a t o n e _ c a n d e f i n e a r e s t r i c t e d s e t o f p r o g ra m s _ th a t^ a r.e _ h a l.t.a b le ( w h i c h s h e c a l l e d t h e s e t_ o f _ “ p r i m i t i v e re c u r s i v e
f u n c t i o n s " ) , b u t i n d o i n g s o w e w o u l d a l w a y s s a c r i f i c e s o m e o f t h e T M ’s
c o m p u t i n g p o w e r.
T h e s e t h r e e m a c h i n e s, F S A , P D A , a n d T M , i l l u s t r a t e t h e a s c e n t u p t h e
C h o m s k v _ h i e ta r c h y . T h e y d if fe r in h a v i n g i m p l i c jj ^ m e m o r y , t e m p o r a r y e x p l i c i t
m e m o ry, a n d p e rm a n e n t^ x p l .ic j ^ t m e m o r y . By l o o k i n g a t m e m o r y as t h e basis for
t h e re c u r siv e lo o p in th e s e s y s te m s— t h a t is, as t h e e l e m e n t t h a t g o v e r n s t h e a b i l ­
ity o f th e system to p e rf o rm in t e r a c t i o n s b e t w e e n its p r e s e n t i n p u t a n d p a s t b e h a v ­
ior— w e c a n see t h a t t h e d i f f e r e n c e s in c o m p u t a t i o n a l p o w e r for th e s e m a c h i n e s
d e p e n d s o n t h e d i f f e r e n c e s in re c u r s iv e po w er.
M e n s u r i n g a n a lo g c o m p l e x i t y w i t h d ig ita l c o m p u t a t i o n
N o w l e t ’s r e t u r n to W o l f r a m a n d h is ce l l u la r a u t o m a t a . A f t e r r u n n i n g t h o u s a n d s
o f tr ials, W o l f r a m f o u r u T t h a t all c e llu la r a u t o m a t a g e n e r a l ly d i v i d e d i n t o f o u r s p e ­
cific c lasses. C la s s e s 1 a n d 2 w e r e t h o s e t h a t e i t h e r d i e d o u t , o r w e n t i n t o a p e r i ­
o d i c c y c le . C la s s 3 w as j u s t t h e o p p o s i t e : it w a s u n c o n t r o l l e d g r o w t h t h a t led to
a p p a r e n t l y r a n d o m b e h a v i o r , li k e w h i t e n o is e . B u t cla s s 4, w h i c h i n c l u d e d th e
“g a m e o f life” c e llu la r a u t o m a t o n , h a d s o m e t h i n g t h a t W o lf ra m d e sc rib e d as “c o m ­
p l e x ” b e h a v io r : n o t as ra n d o m as w h i t e no ise, b u t n o t as b o r i n g as a p e rio d ic cycle.
W o l f r a m f o u n d t h a t th is h i g h e s t c o m p l e x i t y a ls o d e m a n d e d t h e h i g h e s t c o m -
Complexity
p u t a b i li ty : w h ile p u r e o r d e r a n d p u r e d i s o r d e r c o u l d b e r e c o g n i z e d by a n F S A ,
che p acte rn s_ o f che c o m p l e x b e h a v i o r r e q u i r e d a T u r i n g m a c h i n e .
M a c h e m a c ic a l p h y sic ist J a m e s C r u t c h ffe ld (1 9 8 9 ) fo u n d a n e v e n
s i m p l e r e x a m p l e o f r e c u r s i v e c o m p l i c a t i o n in a p h y s i c a l s y s t e m . C r u t c h f i e l d
u s e d che p o p u l a t i o n e q u a t i o n m a d e f a m o u s b y b i o l o g i s t R o b e r t M a y ( 1 9 7 6 ) :
Pn + 1 = P»R (1 - Pn ) ( w h e r e P is a p o p u l a t i o n n u m b e r , sc a le d so t h a t it is b e t w e e n
0 a n d f, a n d R is t h e b i r t h r a t e ) . M a y f o u n d t h a t w h e n R is low , t h e p o p u ­
l a t i o n is s i m p l y a p e r i o d i c c y c l e , s w i t c h i n g b a c k a n d f o r t h b e t w e e n t h e
s a m e s e q u e n c e o f le v e l s . A s y o u i n c r e a s e R , ch e l e n g t h o f c h e c y c l e ( t h a t is,
th e n u m b e r o f d iffe re n t p o p u la tio n lev els you pass th ro u g h before re tu rn in g
t o t h e fir s t o n e ) i n c r e a s e s e x t r e m e l y f a s t . A t R = 3 . 1 , t h e p o p u l a t i o n is in a
tw o - l e v e l cy c le , a t R = 3 .4 in a f o u r - l e v e l c y c le , a n d a t R ~ 4 -0 t h e c y c le l e n g t h
is a t i n f i n i t y : d e t e r m i n i s t i c c h a o s . C r u t c h f i e l d w a s a b l e t o m e a s u r e t h e c o m ­
p u ta b ility o f th e s e c h a o tic flu c tu a tio n s a n d fo u n d resu lts sim ila r to th o s e of
W o lfram : b o th c o m p le te ly p e rio d ic w av es a n d c o m p le te ly d iso rd e re d w aves
w e r e c o m p u t a t i o n a l l y q u i t e s i m p l e , b u t t h o s e in b e t w e e n , w i t h a m i x o f
o rd e r a n d d iso rd e r, h a d a h ig h d e g r e e o f c o m p u t a ti o n a l c o m p le x ity . T h e
sim p le e q u a ti o n e x a m in e d by C r u tc h f ie l d re q u ire d o n ly a P D A , b u t o c h e r
research ers (B lu m , S h u b , a n d S m a le 198 9 ) d e m o n s tr a te d t h a t m o re c o m p le x
a n a l o g f e e d b a c k s y s te m s w o u l d b e c a p a b l e o f s i g n a l c o m p l e x i t y e q u i v a l e n t t o
T M c o m p u ta b ility .
F ig ure 10.7 sh o w s h o w t h e s e c o m p l e x w a v e f o r m s , c a ll e d “ 1/ F n o is e ," c o m ­
p a re to p e r i o d i c a n d w h i t e n o i s e w a v e f o r m s . T h i s is e a s i e s t t o see in t h e s p e c - j
tra l d e n s i t y p lo ts. A p e r i o d i c s i g n a l h a s all its p o w e r a t o n e w a v e l e n g t h , w h ile!
a w h i t e - n o i s e s i g n a l h a s t h e s a m e p o w e r a t a ll w a v e l e n g t h s . 1 / F n o is e is a c o m - j
•|iio'inise l)ciw ee iv t h e tw o — b ia s e d so t h a t it h a s t h e g r e a t e s t a m o u n t o f p o w er!
a t t h e lo n g e s t w a v e l e n g t h , a n d t h e le ast a t t h e s h o r te s t . For th is reas o n , I / F noise*
is fr a c ta l; it h a s f l u c t u a t i o n s w i t h i n f l u c t u a t i o n s w i t h i n f l u c t u a t i o n s . W h e n we
t h i n k o f t h e l e n g t h o f t h e s e w a v e f o r m s in t e r m s o f m e m o r y , w e c a n b e g i n to
see a c o n n e c t i o n t o c o m p u t a t i o n a l p o w e r. If a sy s te m h a d t h e s a m e b e h a v i o r ov er
a n d o v e r a g a in , it w o u ld b e coo f i x e d o n m e m o r y . If it r a n d o m l y p i c k e d a new)
b e h a v i o r e v e ry t i m e , t h e n it w o u l d b e t o o free fro m m e m o r y . B u t useful b e h a v - ]
ior is g e n e r a l l y a m i x t u r e b e t w e e n t h e t w o . F o r e x a m p l e , t h i n k o f s o m e t h i n g /
u n u s u a l you d id t o d a y — m o v i n g so c k s to a n e w side o f t h e d raw er, o r e a t i n g p r e t ;
zels i n s t e a d o f c r a c k e r s . W h a t e v e r it w as, c h a n c e s a re it w a s p r e t t y tr iv i a l. If w.e
to o k t h e s a m e w h im s ic a l a p p r o a c h t o m a j o r li f e - e v e n ts e a c h d a y — “ to d a y 1 t h i n k
I ’ll m o v e to S p a i n , o r g e t p r e g n a n t , o r b e c o m e a p o d i a t r i s t ”— w e w o u l d b e intr o u b le . O u r life is ty p ic a lly a r r a n g e d as 1 / F n o is e : h i g h - p o w e r e v e n t s s h o u l d be'
lo n g - te r m c h a n g e s , a n d lo w - p o w e r e v e n t s s h o u l d be s h o r t - t e r m c h a n g e s . 3 In fact,
159
FIGURE I O . 7
C r u t c h / i e l d - S m a l e c o m p le x ity m e a s u r e
( a -b ) P e r i o d i c noise: A simple signal, ( c -d ) White noise:
From the viewpoint o f the Crutchficlcl-Smnle measure,
this is also of low complexity. An FSA, for example,
could define this noise by making all state transitions
equally probable, ( e - f ) Fractal noise: The most complex
signals in the-Crutchfield-Sinale measure are “scaling
fractal
noises" in which there are fluctuations within fluctu­
periodic
random
noise
noise
noise
ations. These signals have the greatest amount of their
power in the lowest frequencies (longest wavelength).
Since power is rbe reciprocal of frequency, it is often referred to as 1/F noise, (g) In summary, the
Ctutchfield-Smale complexity measure is a reflection of the fractal dimension. T he “most fractal”
(e.g., dimension of 1.5) will be the most complex, and the function decreases with both higher and
lower dimensions.
(c a n d e, courtesy R. F. Voss.)
Complexity
m a n y o f t h e a n a l o g w a v e f o r m s p r o d u c e d by i n t e l l i g e n t h u m a n b e h a v i o r a p p e a r
t o b e 1/ F s ig n a ls (V oss 1 9 8 8 ; E g la s h 1 9 9 3 ) .
A s m o r e j i c i e m i s t s b e g a n t o t h i n k o f c o m p l e x i t y in te r m s o f c o m p u t a t i o n
a n d 1 / F n o is e , t h e y b e g a n t o a c c u m u l a t e e x a m p l e s t h a t s u g g e s te d t h a t t h i s w as
w h a t it m e a n t t o h a v e a " s e l f - o r g a n i z i n g ” s y s te m . In t h e e v o l u t i o n o f life, fo r
i n s t a n c e , m o s t o f t h e g e n e t i c i n f o r m a t i o n s to re s l o n g - t e r m e v e n t s , s u c h as t h e
p h y s i o lo g y t h a t u n d e r w e n t c h a n g e in l i f e ’s e v o l u t i o n fr o m w a t e r t o l a n d . M o r e
s h o r t - t e r m a d a p t a t i o n s , s u c h as s k i n co lo r, t a k e up very li t t l e o f t h e g e n e t i c m a t e ­
rial. H e r e a g a in , w e h a v e s o m e t h i n g li k e 1/ F n o is e , w i t h l o n g - t e r m e v e n t s t a k ­
in g u p t h e b u l k o f t h e s y s te m , a n d s h o r t - t e r m e v e n t s t a k i n g u p p r o p o r t i o n a t e l y
less. P h y s i c is t s P e r B a k a n d C h a o T a n g ( B a k a n d C h e n 1 9 9 1 ) f o u n d s e v e r a l
e x a m p l e s o f s i m p l e p h y s i c a l s e l f - o r g a n iz i n g s y s te m s t h a t p r o d u c e d 1 / F n o i s e . , I n
fo rest'fiT eS T 'forex'am p le, v e r y d ry w o o d s w o u ld s p r e a d fire in a n o r d e r l y c ir c le ,
w h i l e fires in w e t w o o d w o u ld b e t o o s p o r a d i c o r r a n d o m , a n d t h u s d ie o u t. B u t
i n - b e t w e e n fires s p r e a d i n a f r a c t a l p a t t e r n , w i t h m o s t o f t h e fire in l o n g - l e n g t h
p a t c h e s , less o f t h e fire in m e d i u m p a t c h e s , e v e n less in s m a l l e r p a t c h e s , a n d so
o n . In w a t e r we h a v e o r d e r l y c r y s ta ls a n d d i s o r d e r l y li q u id s , b u t in b e t w e e n w e
c a n get th e fractal p a tt e r n s o f snow flakes.
S in c e we are fam iliar w ith o u r o w n recu rsiv e in te ra c tio n s w ith m em ory,
we h a v e a _ g p o d J.n tu i t ive s e n s e f q r w h y 1 / F n o i s e s h o u l d a c c o m p a n y c o m p l e x
b e h a v i o r , a n d c l e a r l y it c a n c h a r a c t e r i z e m a n y v a r i e t i e s o f s e l f - o r g a n iz i n g sy s­
tem s— ;p erh ap s all o f t h e m if w e use t h e p r o p e r d e f i n i t i o n . B u t h o w d o e s this h a p ­
p e n ? W h a t is t h e m e c h a n i s m t h a t m a k e s it w o rk ? C o m p l e x i t y t h e o r i s t s h a v e n o t
h e s i t a t e d t o su g g e s t i m p l i c a t i o n s o f t h e i r w o r k for-/c u i t u r e ; ) h e r e I w o u l d lik e to
su g g est t h e re v e r s e : t h a t c e r t a i n a s p e c t s o f A f r i c a n c u l t u r e c a n p r o v i d e i m p o r ­
t a n t im p l i c a t i o n s for c o m p l e x i t y th e o ry . M o r e so rhan-.any o f t h e p re v i o u s e t h r . c m a t h e m a t i c s m o d e l s w e h a v e s e e n , t h i s p a r t o f m y r e s e a r c h w as m u c h m o r e o f
a c o lla b o ra tio n , m u c h clo ser to m y sen se o f th e “p a r tic ip a n t s im u la tio n ”
m e t h o d — a l t h o u g h if t r u t h b e k n o w n 1 h a d t o b e d r a g g e d k i c k i n g a n d s c r e a m ­
ing m u c h o f t h e way.
C h ristia n S in a D i a t t a : a n A f r i c a n p h y s i c i s t looks at c u l t u r e
“R h a b . ” “ P h a n t o m . ” “R h a b ! ” “ P h a n t o m !!" A s t r a n g e d ia l o g flew acro ss t h e c o m ­
p u t e r la b a t t h e I n s t i t u t d e T e c h n o l o g i e N u c l e a i r e A p p l i q u e e a t S e n e g a l ’s U n i ­
v ersity o f ..Dakar- I w as s e a t e d w i t h P r o f e s s o r C h r i s t i a n S i n a D i a t t a , d i r e c t o r o f
t h e la b ,^ w atc h in g t h e p u l s a t i n g fo r m s o f c e l l u l a r a u t o m a t a flow a b o u t t h e s c r e e n .
D r . D i a t t a l w as t h e lo c a l s p o n s o r f o r r e s e a r c h u n d e r t h e U n i t e d S t a t e s ’ F u l b r i g h t
F e llo w s h ip p r o g r a m , a n d w as e a g e r to d iscuss h is o w n id ea s. Mis p h y s i c s la b was
A fr ic a n fra c ta l m athem atics
a n i n s p ir i n g p l a c e t o be. I h a d a lr e a d y b e e n a b le t o sit in o n a g r a d u a t e s t u d e n t ’s
p r e s e n t a t i o n ; a fte r h a v i n g w i t n e s s e d t h e s a m e r i t u a l in t h e p h y s i c s d e p a r t m e n t
a t t h e U n i v e r s i t y o f C a l i f o r n i a a t S a n t a C ru z , it m a d e fo r a fascinatin.g: b i t o f crossc u lt u r a l c o m p a r is o n . I tr ie d to m a k e m y s e lf useful by s e t t i n g u p a d e m o o f a n e l e c ­
t r ic a l c i r c u i t t h a t p r o d u c e d d e t e r m i n i s t i c c h a o s ( “C h u a ’s c i r c u i t ”) a n d i n s t a l l i n g
variou s types o f so ftw are for s i m u l a ti o n s o f n o n l i n e a r d y n a m i c s . It w as o n e o f th e s e
s o f tw a r e d e m o s , R u d y R u c k e r ’s
c a l a b
,
th a t caused our m ultiling ual exchange.
A s n o t e d in c h a p t e r 7, s o m e o f R u c k e r ’s m o s t i n t e r e s t i n g p r o g r a m s a re th o s e
h e c a lls * |Z h a b o ti n s k y C A s , " w h i c h c a n p r o d u c e p a i r e d log s p i r a ls . I n a d d i t i o n
t o t h e tw o s t a t e s o f li v e c e l l a n d d e a d c e l l , t h e s e c e l l u l a r a u t o m a t a r e q u i r e a t
l e a s t o n e “gh ost..,state.” S i n c e s o m e o n e h a d p r e v i o u s l y m e n t i o n e d t h e i n d i g e ­
n o u s t e r m for. g h o s t , -rhab, it s e e m e d l i k e a n o p p o r t u n i t y fo r c r e a t i v e t r a n s l a ­
t i o n . 1 e x p l a i n e d (i n F r e n c h , t h e official la n g u a g e o f S e n e g a l ) t h a t a fte r I'eta t m ort
( t h e d e a d s t a t e ) t h e c e ll w e n t t o i ’ e ta t r h a b . T o m y s u r p r i s e , D i a t t a c o r r e c t e d
rh a b b a c k to t h e F r e n c h : “p h a n t o m . ” W e w e n t b a c k a n d f o r t h a c o u p l e o f ti m e s
b e f o r e I re a l iz e d t h a t i t ' w a s n o t j u s t m y p o o r p r o n u n c i a t i o n . O n l y l a t e r d id
I d is c o v e r m y b l u n d e r : D i a t t a was n o t fr o m t h e Is la m i c W o l o f m a j o r i t y ( i n w h o s e
l a n g u a g e rha b o c c u r s ) b u t f r o m o n e o f t h e a n i m i s t m i n o r i t y g r o u p s , t h e Jo la.
U s i n g W o l o f w as n o m o r e o f a c u l t u r a l t r a n s l a t i o n fo r h i m t h a n i t w o u l d h a v e
b e e n t o u se E n g li s h .
T h i s w as o n l y t h e s t a r t o f m y m i s t r a n s l a t i o n s . A l t h o u g h D r. D i a t t a w as
g reatly e n th u s ia s tic a b o u t my w o rk o n fractals in A fr ic a n a r c h ite c tu r e , he
s e e m e d d i s i n t e r e s t e d in t h e f r a c t a l g e n e r a t i o n s o f t w a r e . B u t h e p e r s i s t e n t l y
b r o u g h t u p A f r i c a n a r c h i t e c t u r e d u r i n g t h e c e ll u la r a u t o m a t a d e m o s . I f o u n d th is
e n t i r e l y t o o f r u s t r a t i n g : t h e w h o l e p o i n t o f rnv r e s e a r c h o n A f r i c a n f r a c t a l s was
to e x p lo re th e i n t e n t i o n a l side o f th e s e designs. C e llu la r a u to m a ta c re a te p a t ­
t e r n s n o t b y p r e p l a n n e d j d e s i g n , b u t r a t h e r by t h e i n t e r a c t i o n s o f its a g g r e g a t e
c e lls. F r o m m y p o i n t o f v iew , h a v i n g f r a c t a l a r c h i t e c t u r e a s t h e r e s u l t o f a g g r e ­
g a te s e l f - o r g a n i z a t i o n d e s t r o y e d t h e p o s s ib il it y o f i n t e n t i o n a l i t y . By fo c u s in g o n
c e l l u l a r a u t o m a t a as a n a r c h i t e c t u r a l m o d e l , D i a t t a s e e m e d t o h e u n d o i n g all
m y c a r e f u ll y p r e p a r e d r e s e a r c h . H i s e n t h u s i a s m w a s u n b e a t a b l e , h o w e v e r , a n d
1 b e g a n to s tu d y a e r i a l p h o t o s o f h i s p l a c e o f o r i g i n , t h e J o l a s e t t l e m e n t s s o u t h
o f th e C a s a m a n c e R iver. F ig ure 10.8 s h o w s t h e s e t t l e m e n t o f M l o m p , n o t far fro m
D i a t t a ’s h o m e t o w n ; its p a i r e d lo g s p i r a l s t r u c t u r e c o u l d h a v e c o m e r i g h t o u t o f
R u c k e r ’s Z h a b o t i n s k y C A s .
A t r i p to t h e C a s a m a n c e w as c le a r ly c a l l e d for. 1 w as f o r t u n a t e in fi n d in g
N fally B a d ia n e , a Jo la g ra d u a te s t u d e n t w h o h a d d o n e h is m a s t e r ’s th esis o n in d ig e ­
n o u s a r c h i t e c t u r e o f t h e s o u t h e r n C a s a m a n c e , as a g u id e . N f a l l y ’s b a c k g r o u n d is
ideal for a n a n t h r o p o lo g i s t: raised a m o n g th e I s la m ic m a j o r i ty in D akar, h e is b o t h
FIGURE 1 0 . 0
T h e J ola s e t t l e m e n t o f M l o m p , S e n e g a l
Ump. (b, Mlomp model generated by com b.n»tior.of s t o ^ s t i c and recursive process,
[nstitut Geogmlskique tie Senega!; b. courtesy o f Egondu Onyejelt.ve.)
A fr ic a n fra c ta l m athematics
s t r a n g e r t o a n d m e m b e r o f t h e Jo la so c ie ty . A s w e t r a v e l e d t h e d e l t a a r e a o f che
C a s a m a n c e R i v e t , u s i n g c a t s , tr u c k s , c a n o e s , a n d a n y t h i n g else t h a t m o v e d , h is
w a r n in g s a b o u t t h e se c re c y o f J o l a religious k n o w l e d g e w e r e re p e a t e d ly c o n firm e d .
S e c u l a r i n fo r m a t i o n a b o u t t e c h n i c a l m e t h o d s o f h o u s e c o n s t r u c t i o n , p r e c o l o n i a l
a n d p o s t c o l o n i a l s o c i a l c h a n g e s , k in s h ip , g to p p s ,.,.a .n d .jp a n y o t h e r , a s p e c t s „ p f
J o l a s o c i e ty w e r e readilY f o r t h c o m m
(E g la s h e t al. 1 9 9 4 ) . ' W e w e r e to ld t h a t t h e
c ircu lar b u ild in g c o m p le x e s w ere n o t p re p la n n e d , n o r w e re th e b ro a d cu rv es of
t h e s e c o m p l e x e s in e a c h n e i g h b o r h o o d , b u t t h a t t h e y c o u l d n o t te ll us a n y t h i n g
a b o u t th e se q u e n c e o f c o n s tr u c tio n because, u n lik e t h e W o lo f, "w e d o n o t h av e
a griot (oral h i s t o r i a n ] in J o l a so c ie ty ." T h e s p i r a l s t r u c t u r e v is ib l e in t h e p h o t o
w as m a i n l y d u e t o t h e c a r e f u ll y m a i n t a i n e d s a c r e d f o r e s t ^ L n T o u n d i n g e ^ h J _ o c a l
n e i g h b o r h o o d . B u t t h e m e c h a n i s m s for c r e a t i n g s u c h c o h e r e n t s t r u c t u r e s o v e r
s u c h a n e n o r m o u s r a n g e o f sc a le s r e m a i n e d h i d d e n . A t a n t a l i z i n g g l i m p s e o f t h e
J g] a
ry, h o w e v e r , led us t o s u s p e c t t h a t t h e r e w a s a ^ c o n sc io u s ele-
m e n t t o t h e C A - l i k e s e t t l e m e n t s t r u c t u re. F ir s t, t h e r e w a s t h e s y m b o l i s m o f t h e
c h i e f ’s d r i n k i n g vessel: a s p ir a l s h e l l . S eco n d,(K ifalfy ^h ad s e e n t h e i n t e r i o r o f o n e
o f t h e s e t t l e m e n t a lt a r s , a n d sa id t h a t it c o n s i s t e d o f a s p i r a l p assag e.
T h e b e s t c l u e w e - f o u n d was f r o m D i a t t a h i m s e l f , w h o d e s c r i b e d a lo g s p i ­
ral p a t h in c e r t a i n r i t u a l s t h a t t o o k p l a c e in t h e s a c r e d f o r e s t . B u t h o w t o r e c ­
o n c ile th is self-co n scio u s m o d e lin g w ith w h a t a p p e a re d to be th e e m e rg e n c e
o f th e s e t t l e m e n t s tru c tu r e th r o u g h ag g reg ate s e lf-o r g a n iz a tio n ? 1 finally c o n ­
f e s s e d m y d is t u r b a n c e to D i a t t a , a n d a sk e d h i m b o w I m i g h t u n d e r s t a n d t h e a p p a r ­
e n t c o n t r a d i c t i o n . H e s u g g e s t e d y e t a n o t h e r s i m u l a t i o n : t h e J o l a f u n e r a l ritu a l
(fig. 1 0 .9 a ) . 'W e h a d b e e n a le r te d to th i s c e r e m o n y as a r e s u lt o f a su spicio us d e a t h
d u r i n g c u r - v is i t, b u t w e r e n o t a l l o w e d t o a t t e n d . D i a t t a d e s c r i b e d t h e r i t u a l in
d e ta il. T h e body o f th e d eceased was p la c e d o n a p la tfo rm , a n d posts at e a c h o f
t h e f o u r c o r n e r s a r e h e l d a lo f t b y p a l l b e a r e r s . If c r i t i c a l k n o w l e d g e is t h o u g h t
to h a v e b e e n h e l d b y t h e d e c e a s e d (e.g., a s in t h e c a s e o f a m u r d e r ) , a p r i e s t asks
q u e s t i o n s . T i r e p a l l b e a r e r s , r e a c t i n g t o t h e f o r c e o f t h e d e c e a s e d , m o v e che p l a t ­
fo r m t o t h e r i g h t for yes, le ft for n o , a n d f o r w a r d fo r “ u n k n o w n . "
T h e s i m u l a t i o n for t h i s ritual..(fig. J0..9.b) is b a s e d o n a n a n a l o g f e e d b a c k
n e t w o r k . W e d o n ’t n e e d t o m a k e a n y a s s u m p t i o n s a b o u t w h e t h e r t h e p a l l b e a r ­
ers a r e e x e r t i n g f o r c e d u e t o c o n s c i o u s o p i n i o n s o r s u b c o n s c i o u s b eliefs; it is o n ly
n e c e s s a r y t o a s s u m e t h a t t h e y e x e r t f o r c e in p r o p o r t i o n t o t h i s m o t i v a t i o n S i n c e th e y c a n b o t h e x e r t fo rce a n d sen se it f r o m o t h e r s , t h i s w o u ld th e o r e t i c a l l y
a ll o w t h e s u m m a t i o n o f k n o w l e d g e a m o n g t h e p a r t i c i p a n t s t o b e e x p re s se d in t h e
m o s t e ff e c ti v e w a y p o ss ib le . I n f a c t , t h e t e c h n i q u e is m o r e e f f e c t i v e t h a n a v o t e ,
s i n c e v o t i n g c a n le a d t o t h e p a r a d o x o f a m i n o r i t y o p i n i o n w in if t h e r e a re m o r e
t h a n t w o o p t i o n s . ^ T h e i n f o r m a t i o n e m e r g e d fr o m t h e b o t t o m - u p i n t e r a c t i o n o f
Complexity
165
t h e p a rt s , y e t it w as a ls o i n t e n t i o n a l i n t h e s e n s e t h a t t h i s m e c h a n i s m fo r aggreg a te s e l f-o r g a n iz a t io n o f k n o w l e d g e h a d b e e n c o n s c i o u s ly d e s i g n e d . T h i s w a s n o t
i n t e n t i o n a l i t y as I k n e w it; it s o u n d e d m o r e lik e t h e d e s c r i p t i o n o f a n e u r a l n e t ­
w o rk j n . c o m p u t e r s c i e n c e :
If a p ro g r a m m e r has a n e u r a l n e t w o r k m o d e l o f v is io n , for e x a m p le , h e or she
c a n s i m u l a t e che p a t t e r n o f lig h t a n d d a r k fallin g o n t h e r e t in a by a c t i v a t ­
in g c e r t a i n i n p u t n o d e s , a n d t h e n l e t t i n g th e a c t i v a t i o n sp re a d t h r o u g h th e
(a) In the Jola funeral ritual four
pallbearers hold a platform aloft and
move it in response to questions. Since
the inform ation (w h e th er o n e believes it
to be of spiritual or m u n d an e origin) is
held by che pallbearers, w e c a n model che
force o f each corner as h a v in g direction
and magnitude (a vector) d eterm in ed by
the pallbearer’s conviction. Decision
making based on a co n tin u o u s range
rather th a n on yes/no is called “fuzzy
logic” in mathematics.
(b) We can th in k of the inform ation
processing in the Jola funeral as the
equivalent of a neural net (similar to th a t
in lig. 10.2) in w hich the sum o f the force
vectors of all four pallbearers are inputs to
lluee amplifiers, w ith each inverted o u tp u t
connected as negative feedback to the
oilier two. This would require pallbearers
to both exert force as well as sense it, b u t
juch force-feedback is actually quite
common in motor tasks.
unknown
FIGURE I O . 9
N e u r a l n e t m o d e l for t h e J o la f u n e r a l r itu a l
c o n n e c t i o n s i n t o th e re s t o f t h e n e tw o r k . T h e e ff e c t is a b i t lik e s e n d i n g
s h i p lo a d s o f g o o d s i n t o a few p o r t c itie s a l o n g t h e s e a c o a s t, a n d t h e n l e t t i n g
a zillion trucks c a r t t h e stu ff .along th e h ig h w ay s a m o n g t h e i n l a n d c ities. But
if t h e c o n n e c t i o n s h a v e b e e n p r o p e r ly a rr a n g e d , t h e n e t w o r k w ill so o n s e t t l e
i n t o a s e l f - c o n s is t e n t p a tc e r n o f a c t i v a t i o n t h a t c o r r e s p o n d s to a c la s sif ic a ­
t i o n o f th e sce n e . “T h a t ’s a c a t ! ”
( W a l d r o p 1 9 92 , 2 8 9 - 9 0 )
T h e t r i c k y p a r t is “ if t h e c o n n e c t i o n s h a v e b e e n p r o p e r l y a r r a n g e d . ”
C l e a r l y i t c o u l d b e a r r a n g e d fo r f o u r p e o p l e , b u t c o u l d it f o r t h i s ' c i t y o f M l o m p ,
w i t h d o z e n s o f l o c a l n e i g h b o r h o o d s a n d h u n d r e d s o f p e o p l e in e a c h ? A n d
M l o m p is n o t a n a n o m a l y . W h i l e w e sa w a m o r e e x p l i c i t f o r m a l s y s t e m in t h e
c o n s t r u c t i o n o f se v e ra l f r a c ta l s e t t l e m e n t a r c h i t e c t u r e s in c h a p t e r 2, t h e r e a re also
m a n y A f r i c a n s e t t l e m e n t s t h a t h a v e a larg e, diffuse f r a c ta l s t r u c t u r e (see D e n y e r
1978, 1 44). S elf-o rg a n izin g m e c h a n ism s t h a t arran g e su c h vast ag greg atio n s
i n t o c o h e r e n t p a t t e r n s w o u l d h a v e t o b e m o r e g l o b a l a n d less e x p l i c i t .
O n e k ey m e c h a n i s m i n c o m p l e x i t y t h e o r y is m e m o r y : t h e t h e o r y p r e d i c t s
t h a t s e l f - o r g a n iz i n g s y s te m s will u tiliz e 1 / F d.is.tpi.bu.tiQns. in m e m o r y l e n g t h . T h e
lu kasa, a visual “m e m o r y b o a r d ” d e v e l o p e d by t h e B a lu b a o f C o n g o (Z a ire ), sh ow s
ju s t s u c h f r a c ta l s c a l i n g (fig. 1 0 .1 0 ) . T h e m e m o r y s y s t e m o f t h e l u k a s a is p a rtly
b a s e d o n d i g i ta l ( t h a t is, p h y s i c a ll y a r b i t r a r y ) c o d i n g , s u c h as c o lo r , b u t R o b e r t s
( 1 9 9 6 ) n o t e s t h a t m u c h o f t h e lukasa is a “g e o m e try o f id e a s ,” m a p p i n g th e J a e a d e d
s p a t i a l s t r u c t u r e t o a n a l o g o u s h i s t o r i c a l e v e n t s . A l t h o u g h t h e r e is c o n s i d e r a b l e
i n t e r p r e t i v e a n d c o d i n g v a r i a t i o n , t h e r e is a t e n d e n c y t o h a v e s i n g l e b e a d s r e p ­
r e s e n t i n g i n d i v id u a l s , g r o u p s o f b e a d s r e p r e s e n t i n g r o y a l c o u r t s , a n d la r g e r be ad
a r r a n g e m e n t s s h o w i n g t h e s a c r e d fo r e sts t h a t h a v e b e e n g r o w i n g o v e r m a n y
g e n e r a t i o n s . T h i s v i s u a l i z a t i o n o f a 1/ F - l i k e d i s t r i b u t i o n o f m e m o r y s u g g e s ts at
le a s t t h e p o s s ib ility o f i n d i g e n o u s a w a r e n e s s o f s c a l i n g p r o p e r t i e s in m a i n t a i n ­
in g s e l f - o r g a n iz e d c o m p l e x it y .
T h e s t r o n g e s t c a n d i d a t e f o r a m e c h a n i s m u n d e r l y i n g s e l f - o r g a n i z a t i o n is
th e c o m p le m e n ta ry p air o f in d ig e n o u s-fe e d b a c k c o n c e p ts, w e e x a m i n e d j n
c h a p t e r 8. I n t h e v o d u n r e l i g i o n o f B e n i n , w e f o u n d D a n r e p r e s e n t i n g t h e s t a ­
b i l i z i n g fo r c e o f n e g a t i v e f e e d b a c k , a n d L e g b a r e p r e s e n t i n g t h e d i s r u p t i v e
fo r c e o f p o s i t i v e f e e d b a c k . S i m i l a r f e e d b a c k p a ir s w e r e f o u n d in t h e B au le
d o o r c a r v i n g s ; t h e c a i m a n s b i t i n g e a c h o t h e r ’s t a i l s a r e a s y m b o l o f n e g a t i v e
f e e d b a c k , a n d t h e fish e a t i n g e v e r l a r g e r fish r e p r e s e n t p o s i t i v e f e e d b a c k . T h i s
c o m b i n a t i o n o f o p p o s i n g f e e d b a c k l o o p s a ls o a p p e a r s t o h e a t t h e h e a r t o f th e
c o n d i t i o n s t h a t s u s t a i n s e l f-o r g a n iz i n g s t r u c tu r e s . O f c o u r s e , in o s t-s e lf-o rg a n iz in g
s y s t e m s w ill h a v e m o r e t h a n t w o lo o p s ; b u t in e v e r y c a s e 1 h a v e e x a m i n e d , at
le a s t o n e o f e a c h is p r e s e n t , a n d it is t h r o u g h t h i s i n t e r a c t i o n t h a t s u s t a i n e d
c o m p l e x i t y c a n a rise.
FIGURE 1 0 . 1 0
Lukasa
( F w m R o b e r ts a n d R o b e r ts
1996; photo
b y D i c k D e a u lie u x .)
African fra c ta l m a th em a tics
R e t u r n i n g t o t h e m o s t b a s i c e x a m p l e o f c o m p l e x b e h a v i o r , M a y ’s p o p u l a ­
t i o n e q u a t i o n , w e h a v e tw o c o m p o n e n t s . O n t h e o n e h a n d , t h e r e is p o p u l a t i o n
g r o w t h : f*n+1 = Pn R. N e x t y e a r ’s p o p u l a t i o n w ill b e th i s y e a r ’s p o p u l a t i o n tirn es^ "
t h e g r o w t h r a t e . A s l o n g as R is a p o s i t i v e n u m b e r , t h i s w ill b e a p o s i t i v e f e e d ­
b a c k lo o p . B u t t h e o t h e r p a r t o f t h e e q u a t i o n , m u l t i p l y i n g b y (1 - Pn ), w as a neg-,; Fa t i v e f e e d b a c k lo o p , a c t i n g lik e a n e p i d e m i c t h a t k ills m o r e p e o p l e w i t h la rg e r
p o p u l a t i o n size.-* T o g e t h e r th e y c r e a t e d e t e r m i n i s t i c c h a o s : t h e p o s i ti v e f e e d b a c k \
k e e p s e x p a n d i n g t h e p o p u l a t i o n , a n d t h e n e g a t i v e f e e d b a c k k e e p s it w i t h i n
j
b o u n d s . T h i s w o r k s for o t h e r c h a o s e q u a t i o n s as w e ll. F ig u r e
•'
i o
.i
j
show s a
c h a o s e q u a t i o n c a l l e d t h e " R o s s le r a t t r a c t o r ’’ m o d e l i n g a c a r w i t h tw o d riv e rs . ^
^ O n e is d r u n k a n d o v e r c o m p e n s a t e s by s t e e r i n g t o o far w i t h e a c h c o r r e c t i o n ; t h e
C
)
o t h e r is s o b e r a n d p u lls it b a c k o n t h e r o a d w h e n t h e d r u n k e n o s c i l l a t i o n s g e t \ .
{ t o o larg e. B e c a u s e it a lw ay s s t e e r s b a c k t o a s l ig h t ly d i f f e r e n t p o s i t i o n , t h e o scil-
\
) l a t i o n s n e v e r r e p e a t — d e t e r m i n i s t i c c h a o s .^
W e c a n see th e sam e c o m b in a tio n o f n e g a tiv e a n d po sitiv e feed b ack cre ­
a t i n g s e l f - o r g a n i z a t i o n in a g g re g a t e s y ste m s. T h e “ g a m e o f life" c e l l u l a r a u t o m a ­
t o n offers a p a r t i c u l a r l y c l e a r i l l u s t r a t i o n o f th i s p h e n o m e n o n . If we g iv e a ru le
s e t t h a t m a k e s b i r t h t o o easy (e.g ., t h e c e ll-g o es t o t h e “l i v e " s t a t e if t h e r e is o n e
o r m o r e n e a r e s t n e i g h b o r s a l i v e ) , t h e n t h e r e is t o o m u c h p o s i t i v e f e e d b a c k a n d
w e g e t a r a p i d ly s p r e a d i n g d is k . If w e m a k e d e a t h t o o e a sy (e.g., t h e c e l l g oes to
t h e “d e a d ” s t a t e if t h e r e is o n e o r m o r e n e a r e s t n e i g h b o r s a l i v e ) , t h e s c r e e n g oes
FIGURE 10 .1 1
R o s s le r a t t r a c t o r as fe e d b a c k i n a u t o m o b i l e d r iv in g
T he Rossler attractor is a set- of three simple equations whose output is derprminisric chaos, thar_i.s«.
a signal with variable oscillations which remain bounded but never repeat the exact same pattern.
How can such a simple system produce infinite variation? A n automobile driving model can help
us see what these equations are doing.
(a) Positive f e e d b a c k . First, there is a part of the system that provides a positive feedback loop;
this acts like a drunken driver who swerves farther and farther off the road. N ote that the car is not
properly aligned with the direction of travel; this skidding is the nonlinear relationship between
road position X and steering angle Y.
(b) N e g a t i v e f e e d b a c k . T he other part of the system is a negative feedback loop; given a swerving
input, this cautious driver steers back toward the center of the road. “C aution” is represented by
the third variable, 2.
(c) Combination o f n e g a t i v e a n d positive f e e d b a c k . Here we see the complete Rossler system at
work. T he "caution" variable Z controls the facial expression (diameter of eyes and mouth, angle of
eyebrows). Note that after the oscillation gets large enough, the negative feedback kicks in, and we
go hack toward the center of the road. Because the car never steers back to exactly the same
position on the road, the behavior never repeats. If, for example, you looked at the number of
increasing oscillations that occur before the negative feedback dampens it hack toward the center,
it would appear to he completely random, with no predictable pattern. Yet the pattern is entirely
deterministic (that is, determined only hy this set of equations); it could be predicted if you knew
the initial conditions with infinite precision.
Driver (mis)observation
noise
road
poslllon
Driver observation:
hos car been deviating?
L
YES
Move steering angle in
( Y)
new steering angle (V)
a Positive feedback
Driver ooservnllon
Increase
Driver observation:
- --- * cauttonl
is cor position >setpolnt ?
(2)
Decrease
_a steering
(V)
angle
new poslllon on road (X)
x
b Negative feedback
0.1 5 y
of negative
and positive
feedback
x
A fr ic a n fra cta l macliematics
b l a n k in a few g e n e r a t i o n s . T h e “classic” life rule set,.(found by J o h n H o r t o n C o n ­
w ay in 1 9 7 0 ) is o f t e n re f e r r e d t o as “ 3 - 4 ’’ life b e c a u s e it t a k e s 3 n e a r e s t n e i g h ­
b o rs to g iv e b ir th , b u t 4 results in d e a t h . C o n w a y d is c o v e re d t h a t th is c o m b i n a t i o n
o f n e g a tiv e a n d po sitiv e feed b ack m axim ized th e c o m p le x ity of b e h av io r. S im ­
ilarly, w h e n P e r B a k f o u n d e m p i r i c a l d a t a for s e l f - o r g a n i z a t i o n i n p h y s i c a l sys­
t e m s — fo r e st fires, e a r t h q u a k e s , a v a l a n c h e s , e t c . — h e n o t e d t h a t it o c c u r r e d o n ly
a t a “critic a l s t a t e ” in w h i c h t h e r e was a b a l a n c e b e t w e e n n o i s e - s u p p r e s s i n g m e c h ­
a n i s m s — w h i c h w o u ld c o r r e s p o n d t o n e g a t i v e f e e d b a c k — a n d t h e p o s i t i v e f e e d ­
b a c k o f n o i s e - a m p l i f y i n g lo op s.
"
It is u n f o r t u n a t e t h a t so m u c h o f th e classic re s e a rc h o n A f r i c a n social m e c h ­
a n i s m s c a m e fro m f u n c t i o n a l i s t a n t h r o p o l o g y , s i n c e t h e y m a d e a n a l m o s t e x c l u ­
siv e e m p h a s i s o n t h e r o l e o f n e g a t i v e f e e d b a c k in a c h i e v i n g e q u i l i b r i u m . W h e n
it c o m e s t o c o n s c i o u s k n o w l e d g e s y s te m s , A f r i c a n s o c i e t i e s d o n o t e x c l u s iv e l y
fo cu s o n b a la n c e , h a rm o n y , a n d stasis. T h e c o m p l i m e n t a r y roles o f D a n a n d Legba,
o f o r d e r a n d d is o rd e r, are m u c h m o r e c o m m o n , as w e s e e in t h i s passage: “I n t h e
m i n d o f t h e B a m b a r a s t h e air, w i n d a n d fire . . . a r e i n d i s p e n s a b l e e l e m e n t s o f
t h e w o r l d ’s o n w a r d m o v e m e n t . B u t as t h e s e p r i n c i p l e s m a y b e a c t i v e in a n
u n c o n t r o l l e d , t h a t is, u n r u l y a n d o f t e n e x c e s s i v e m a n n e r , N y a l e is c o n s i d e r e d
t o b e a p ro f u s e a n d e x t r a v a g a n t b e i n g . . . . S o by h e r v e r y n a t u r e N y a l e is, to a
c e r t a i n e x t e n t , a f a c t o r o f d is o r d e r . T h a t is w h y it is s a i d t h a t B e m b a . . . t o o k ,
a w a y h e r ‘d o u b l e ’ t o e n t r u s t i t t o F a r o . . . w h o s e e s s e n t i a l a t t r i b u t e is e q u i l i b ­
r i u m " ( Z a h a n 1 9 7 4 , 3 ).
A s i m i la r p a ir in g o c c u rs in t h e D o g o n r e l ig i o n , w h e r e A r n m a , th e h i g h god,
c r e a t e s t h e N u m m o t o e n a c t o rd e r , a n d a c c i d e n t a l l y c r e a t e s t h e d i s o r d e r l y
O g o ; t o g e t h e r t h e t w o g e n e t viic life as w e k n o w it. I n t h e r e p e r t o i r e o f d y n a m - •
ical c o n c e p t s o c c u r r in g in se v e r a l A f r i c a n k n o w le d g e system s, t h e r e is r e c o g n i t i o n
o f th e .u s e fu l t e n s i o n b e t w e e n e q u il ib r iu m a n d d i s e q u i l i b r i u m , t h e d a n c e b e t w e e n
o r d e r a n d c h a n c e t h a t r e s u l t s in s e l f - o r g a n iz e d c o m p l e x i t y . A n d ju s t as S t u a r t
K a u f f m a n h a s s h o w n a bias t o w a r d o r d e r in e v o l u t i o n ’s “e d g e o f c h a o s , ” t h e h i g h \
/ g o d e n s u r e s t h a t t h e tr ic k s t e r c a n a c t o n ly sp o r a d ic a lly , t h u s c r e a t i n g m o r e p o w e r j
y t o w a r d l o n g - t e r m o r d e r in t h e s e A f r i c a n c o s m o l o g i e s .
A l t h o u g h f r a c ta l s r e s u l t i n g f r o m g e o m e t r i c a l g o r i t h m s :a r e u s u a lly s e e n as
s t a t i c s t r u c t u r e s , t h e y t o o c a n b e v i e w e d as t h e c o m b i n a t i o n o f f e e d b a c k lo o p s .
A s e e d s h a p e w i t h a h u g e n u m b e r o f t i n y l i n e s e g m e n t s (fig. 1 0 . 1 2 a ) w_ii_l t e n d
t o b e s h a p e - p r e s e r v i n g u n d e r s e l f - r e p l a c e m e n t i t e r a t i o n s ; h e r e d e v i a t i o n s due_tp
r e p l a c e m e n t a r e d a m p e d —-(the d i f f e r e n c e b e t w e e n a l i n e s e g m e n t a n d t h e s e e d
s h a p e is u s u a l ly n o t i m p o r t a n t ' ( a n d t h e r e s u l t i n g g r a p h w ill h a v e a lo w f r a c ­
ta l d i m e n s i o n , i.e., t e n d i n g t o w a r d j . o ) . B u t fo r s e e d s h a p e s m a d e u p o f o n l y
a few la r g e l i n e s (fig. 1 0 . 1 2 b ) , t h e d i f f e r e n c e b e t w e e n a l i n e s e g m e n t a n d its
FIGURE l o . 1 2
F r a c t a l g r a p h ic s a s fe e d b a c k
A fr ic a n fra c ta l m athem atics
172
r e p l a c e m e n t s h a p e w ill b e v e r y i m p o r t a n t . L a r g e d e v i a t i o n s t e n d t o b e a m p l i ­
fied in a q u i c k p o s i t i v e f e e d b a c k , s o m e t i m e s e x p l o s i v e l y g r o w i n g o u t o f b o u n d s
in o n l y a few i t e r a t i o n s . F i g u r e 1 0 . 12 b h a s ' b e e n s c a l e d d o w n t o fit o n t h e p a g e ,
b u t t h e a c t u a l f r a c t a l g r a p h w ill q u i c k l y g r o w o u t o f b o u n d s a n d b l a c k e n t h e
s c r e e n e n t i r e l y (i .e . , a f r a c t a l d i m e n s i o n c l o s e t o 2 . 0 ) . F i g u r e 1 0 . 1 2 c s h o w s a
f r a c ta l d i m e n s i o n c lo s e tjo 1.5, t h e " m o s t f r a c t a l " m e a s u r e , w h i c h r e s u l t s fr o m
a b a l a n c e b e t w e e n t h e n e g a t i v e f e e d b a c k o f s m a ll s e g m e n t s h a p e p r e s e r v a t i o n
a n d t h e p o s i t i v e f e e d b a c k o f la r g e s e g m e n t r e p l a c e m e n t d e v i a t i o n .
T h e r e is n o q u a n t i t a t i v e m e a s u r e o f f r a c t a l d i m e n s i o n in p r e c o l o n i a l
A f r i c a n k n o w l e d g e sy s te m s . B u t t h e i d e a o f a s p e c t r u m p r o g r e s s i n g f r o m m o r e
o r d e r ly t o less o r d e r l y is v i v i d ly portraye r! in c e r t a i n m a t e r i a ! d e s i g n s . T h e best
e x a m p l e s a re in t h e Taffia p a l m t e x t il e s o f t h e B a k u b a (fig. 1 0 .1 3 a ). T h e se t e n d to
s h o w p e r i o d i c t i h n g a l p n g o n e axis, a n d aperiodic.t.i.ling,— o f t e n m o v i n g fr o m o rd e r
to d iso rd er— a lo n g th e o th e r. S im ila r g e o m e tric v is u alizatio n s o f th e sp e c tru m
f ig u r e
1 0 .1 3
F r o m o r d e r to d is o r d e r in a B a k u b a c l o t h
( a ) T h e B a k u b a o f t e n c r e a t e c l o t h d e s i g n s t h a t s t a y fa i rl y c o n s t a n t a l o n g t h e v e r t i c a l a x i s , b u t
g r a d u a l l y c h a n g e a l o n g t h e h o r i z o n t a l a x is . I n m a n y c a s e s, t h e h o r i z o n t a l t r a n s f o r m a t i o n s u g g e s t s
a n o rd er-d iso rd er range, (b ) C o m p u t e r sc ie n tist C liffo rd P ic k o v e r c r e a te d th is p a t t e r n to sh o w
h o w a s p e c tru m from o r d e r to d is o rd e r c o u ld b e v isu a lize d by a llo w in g a r a n d o m v a r ia b le to h a v e
i n c r e a s i n g i n f l u e n c e o n t h e g r a p h ’s e q u a t i o n . T h u s it, t o o , m a k e s u se o f p e r i o d i c r i l i n g a l o n g t h e
v e r tic a l axis a n d a p e r i o d i c a l o n g t h e h o r iz o n ta l.
(a, from
Me u r a n t 1986, b y perniis.u'on o f the author; b , f r o m Pick over 1990, b y permission o f the author.)
C o m p le xity
fro m o r d e r to d i s o r d e r h a v e b e e n u se d in c o m p u t e r s c i e n c e (fig. 1 0 .1 3 b ). A s far
as I c a n tell, t h e B a k u b a w e a v in g s n e v e r r e a c h m o r e t h a n halfw ay across t h e s p e c ­
tr u m — th e y a r e ty p ically m o v i n g b e t w e e n j a n d 1.5, t h a t is, fr o m p e rio d ic t o fractal, r a t h e r t h a n s t r e t c h i n g all t h e w ay tg vp u r e diso rd er.^
1 k n o w o f o n l y o n e A f r i c a n t e x t i l e t h a t cakes t h i s la st s t e p , a n d t h a t is t h e
b lo c k p r i n t s h o w n in figure 10.14. T h i s p a t t e r n is r e m in i s c e n t o f t h e title o f N i g e r ia n a u t h o r C h i n u a A c h e b e ’s f a m o u s n o v e l , T h in g s F a ll A p a r t. G i v e n t h e a n t i c o l o n i a l c o n t e x t o f A c h e b e ’s w r i t i n g , it m i g h t b e t e m p t i n g t o re a d it as a n
i n d i c a t i o n t h a t w h i t e n o i s e o n l y c o m e s w i t h w h i t e p e o p l e , b u t a t le ast in te r m s
fig u re
1 o . 14
B lo c k p r in t t e x tile
This print from West Africa suggests the full spectrum from order to disorder.
(From S i e b e r 1972.)
*73
A fr ic a n fra c ta l mathematics
174
o f t h e i n d i g e n o u s k n o w l e d g e s y s t e m s u c h a s s u m p t i o n s a r e u n f o u n d e d .® T h e r e
is, fo r e x a m p l e , a f o r m o f m u s ic i n d i g e n o u s t o N i g e r i a t h a t h a s s o m e t h i n g lik e
a w h i t e n o is e d i s t r i b u t i o n of-sounds. A k p a b o t (-1975) d e sc rib e s “ t h e r a n d o m m usic
o f t h e B i r o m , ” a f l u te e n s e m b l e d e s i g n e d to a llo w e a c h m u s i c i a n t o e x p r e s s i n d i ­
v i d u a l fe elin g s t h r o u g h w h a t e v e r i d i o s y n c r a t i c n o i s e ( o r e v e n s ile n c e ) _ h e o r sh e
c h o o s e s , r e s u l t i n g in " a n i n d e t e r m i n a t e p r o c e s s [in w h i c h ] t h e s o u n d s p r o d u c e d / .
b y t h e p la y e rs a re n o t o b s t r u c t e d by a c o n s c i o u s a t t e m p t t o o rg a n i z e t h e r h y t h m s ^
a n d h a r m o n i e s ” ( p . 4 6 ) . P e l t o n ( 1 9 8 0 ) re fe rs t o t h e N i g e r i a n ( Y o r u b a ) t r i c k s t e r
E s h u as t h e “lo r d o f r a n d o m , ” a n d n o t e s t h a t t h e r e is a c o u p l i n g b e t w e e n t h e
o rd erly w o rk of O lir u n a n d th is u n p r e d ic ta b le sp irit, sim ila r to t h e n e g a tiv e
fe e d b a c k /p o sitiv e feed b ack c o m b in a tio n s we n o te d earlier. T h e c h a ra c te riz a ­
ti o n o f e x tr e m e d is o rd e r m i g h t w ell b e a p p lie d to t h e e x p e r i e n c e o f c o lo n ia l
rule, b u t w e s h o u ld n o t assu m e t h a t t h e c o n c e p t was u n k n o w n b e fo re th e n . A
s u m m a r y o f s e l e c t e d A f r i c a n c o m p l e x i t y c o n c e p t s is s h o w n i n f ig u re 1 0 .1 5 ; n o t e
t h a t t h e c e n t r a ! p e a k o f s p i r i t u a l p o w e r is a n a l o g o u s t o t h e c e n t r a l p e a k o f c o m ­
p u t a t i o n a l p o w e r in t h e C r u t c h f i e l d - S m a l e c o m p l e x i t y m e a s u r e .
C o n c lu s io n
T h i s c h a p t e r is o n l y t h e b a r e o u t l i n e o f w h a t I h o p e w ill b e f u t u r e a r e a s o f
r e s e a r c h , e x a m i n i n g t h e r e l a t i o n s b e t w e e n t e c h n i c a l , c u J t u raJ j . a,p d - P 9 ]it i? ? i
s y s te m s t h r o u g h t h e n e w f r a m e w o r k s o f f e r e d b y c o m p l e x i t y , t h e o r y . F o r t h e
m o m e n t , w e w ill h a v e t o l i m i t o u r s e lv e s t o t h e few f r a g m e n t s t h a t m y S e n e g a le s e
c o l l e a g u e s p o i n t e d o u t so d i l i g e n t l y i ^ F i r s t ^ t h i s d o e s n o t . n e g a t e t h e p r e v i o u s
e x a m p l e s o f e x p l i c i t a l g o r i t h m i c d e s i g n in A f r i c a n fractals,,?..buj..j_t..docs su g g e s t
t h a t a t le a s t in t h e c a s e o f se ttle_ m en t a r c h i t e c t u r e t h e y c a n . a r i s e f r o m a n o t h e r
s o u r c e as w ell. T h e c r e a t i o n o f f r a c ta l s e t t l e m e n t p a t t e r n s t h r o u g h a g g re g a t e self­
o rg a n iz a tio n , w h ile u n li k e t h e p l a n n e d s tru c tu r e s we saw in c h a p t e r 2, d o n o t seem
t o b e t h e r e s u lt o f u n c o n s c i o u s s o c i a l d y n a m i c s (a s w e sa w j b r t h e u r b a n sp r a w l
o f E u r o p e a n c itie s in c h a p t e r 4 ). T h i s m a y b e d u e t o a differ e r v c ^ b e t w e e n A.fr i ca n
c o n c e p t s o f i n t e n t i o n , w h i c h c a n a p p ly t o a g r o u p p r p j e c t_ c r e a te c l.p v e r . s e v e r a l
g e n e r a t i o n s , v e rs u s t h e W e s t e r n fo c u s o n a n i n d i v i d u a l p e r f o r m i n g i m m e d i a t e
a c t i o n in d e f i n i n g i n t e n t i o n a l i t y . M o s t i m p o r t a n t , t h e r e a r e i n d i c a t i o n s t h a t th is
p a t t e r n c r e a t i o n t h r o u g h g r o u p a c t i v i t y is s u p p o r t e d by c o n s c i o u s m e c h a n i s m s
s p e c i f ic t o s e l f - o r g a n i z a t i o n as d e f i n e d in c o m p l e x i t y t h e o r y . B o t h t h e s c a l i n g
( ' d i s t r i b u t i o n o f i n t e r a c t i o n s w i t h m e m o r y a n d che s p e c t r u m fr o m o r d e r t o d i s ­
o r d e r h a v e a t le a s t s o m e g r a p h i c c o u n t e r p a r t s in A f r i c a n d e s i g n s . T h e b e s t c a n ­
d i d a t e for a c o n s c i o u s m e c h a n i s m is t h e c o m b i n a t i o n o f n e g a t i v e a n d p o s i t i v e
fe e d b a c k . W e d id n o t e x a m i n e e v ery possible case o f d e te r m i n is t ic c h a o s an d
^
order
fractal
disorder
Akan
(G hana):
Ananse
t h e trickster
' \ J \ S \ S \ ,
I c o n f o r '“c a l m w a t e r s ”
N y a m e ' s p o w e r o f lif e;
tu r b u le n t waters o f
Tanu
V od un
(B e n i n ,
Nigeria,
African
L e g b a , Eshu
che tricksters
diaspora):
D an
M a w u (acts th rough
lo w e r gods, e.g., th e
b ifu rca tin g doublings
o f Shango)
U'gon
(Mali):
Ogo
th e trickster
N u m m o (d raw in g based on
p h o t o o f r i t u a l s t a f f in I m p e r a t o
1 97 8)
A m i n a ( d e s c r i b e d as
a n e x p a n d i n g spiral,
lik e a w h irlw in d)
FIGURE 1 0 . 1 5
A f r i c a n c o m p l e x i t y c o n c e p ts i n r e l i g i o n
176
A fr ic a n fractal m a th em a tics
a g g r e g a t e s e l f - o r g a n i z a t i o n , b u t it w o u l d a p p e a r t h a t t h e c o m b i n a t i o n o f n e g ­
a t i v e a n d p o s i ti v e fe e d b a c k lo o p s , w h i c h fo r m t h e basis o f s e v e r a l A f r i c a n k n o w l ­
e d g e s y s te m s , a ls o f o r m ' a k e y m e c h a n i s m o f g e n e r a l s e l f - o r g a n i z i n g s y s te m s .
A s n o t e d in t h e first c h a p t e r , it is j u s t as i m p o r t a n t t o f i n d w h a t is m i s s ­
i n g as it is to f i n d w h a t is p r e s e n t . W h i l e fo u r o f t h e fiv e b a s i c c o n c e p t s o f f r a c ­
t a l g e o m e t r y — s c a l i n g , s e l f - s i m i l a r i t y , r e c u r s i o n , a n d i n f i n i t y — a r e a ll p o t e n t
aspiects o f A f r i c a n m a t h e m a t i c s , a q u a n tita tiv e m ea su re o f d im e n sio n (th e H a u sd o rfB e s ic o v itc h m e a s u r e ) is c o m p le te ly a b se n t. T h e r e is a w e a k s e n s e o f a c o m p l e x i t y
sp e c tru m o f order-disorder, w h ic h w o u ld co v ary w ith th e H a u sd o rf-B c s ic o v itc h
m e a s u r e , b u t t h a t s p e c t r u m is n e i t h e r q u a n t i t a t i v e n o r ( t o m y k n o w l e d g e ) e v e r
c o m p a r e d to a c o n c e p t o f d i m e n s i o n in any. i n d i g e n o u s A f r i c a n s y s te m . T h i s is
a n e n o r m o u s g ap in t h e A f r i c a n k n o w l e d g e o f f r a c t a l g e o m e t r y , e s p e c i a l l y s i n c e
t h e d i m e n s i o n a l m e a s u r e is o f t e n c o n s i d e r e d t h e m o s t v a l u a b l e c o m p o n e n t by
c o n t e m p o r a r y r e s e a r c h e r s i n t h e field...
O n t h e o t h e r h a n d , w e a ls o n e e d t o a p p r e c i a t e all k n o w l e d g e s y s te m s in
th e i r o w n rig h t, a n d A f r ic a n fractals h a v e a s u rp ris in g ly s tr o n g u tiliz a tio n o f
r e c u r s i o n . I n d e e d , in M a n d e l b r o t ’s s e m i n a l t e x t , T h e F ra c t a l G e o m e tr y o f N a tu r e
( 1 9 7 7 ) , t h e i n d e x lis ts “ r e c u r s i o n ” o n l y t w i c e , a n d t h e t e r m s i t e r a t i o n , self-,
r e f e r e n c e , s e l f - o r g a n i z a t i o n , a n d f e e d b a c k a r e e n t i r e l y . a b s e n t - A s w e w ill s e e ,
t h i s a b s e n c e is n o a c c i d e n t ; it.re fle c ts a E u r o p e a n h i s t o r i c a l tr e n d - B u t w h y h a v e
E u r o p e a n s t r a d i t i o n a l l y p la c e d s u c h li t t l e i m p o r t a n c e o n r e c u r s i o n , a n d w h y w as
i t so s t r o n g l y e m p h a s i z e d in A f r i c a n fracta ls? In p a r t m o f t h i s b o o k w e will t a k e
u p s u c h c r o s s - c u l t u r a l c o m p a r i s o n s in d e t a i l .
Implications
C H A P T E R
—T heoretical—
—frameworks—
------------- j n -------------
cultural studies-o f knowledge-
~J P a r t s 1 a n d 11 o f t h i s b o o k e m p h a s i z e d t h e g e o m e t r i c , s y m b o li c , a n d q u a n t i t a - ^
[ tiv e a s p e c t s o f A f r i c a n fracta ls. S o m e c ase s w e r e m o r e s p e c u l a ti v e t h a n o t h e r s — j
/ a d i f f e r e n c e t h a t 1 h o p e w as c l e a r l y i n d i c a t e d — b u t e v e n in t h e u s e 'o f m y t h i c ]
\
'
J n a r r a ti v e , 1 g e n e ra lly r e s tr a i n e d c o n c l u s i o n s to th o s e t h a t h a d g e o m e t r i c o r q u a n - i
| t i t a t i v e c o u n t e r p a r t s . In o t h e r w o r d s , t h e c l a i m s m a d e in p a r t s i a n d n s h o u l d
i
■, be fulsifiah le in t h e s e n s e o f K a r l P o p p e r ; t h e d a t a e i t h e r s u p p o r t s t h e h y p o t h e - j
\ sis o r re fu te s it.* B u t th e c h a p t e r s in th i s last s e c t i o n w ill s w i t c h t o to p ic s in c u l- '
t u r a l p o l i t i c s a n d o t h e r h u m a n i t i e s . T h e s e is s u e s a r e t o o c o m p l e x a n d
m u l t i d i m e n s i o n a l to b e r e d u c e d t o f o r m a l r e p r e s e n t a t i o n s ; t h e y c a n o n l y b e
a p p r o a c h e d b y e x p l o r i n g t h e i r j u u e r p r e t a r i y e d e p _ th s (P o e tr y ' a n r e v e a l as m u c h
t r u t h a b o u t t h e w o r l d as a n y s c i e n c e ; w e o n l y n e e d t o k e e p in m i n d t h a t it is a
d i f f e r e n t w ay o f g o i n g a b o u t it. W h i l e t h e p h i l o s o p h y , p o l i t i c s , a n d p o e t i c s o f
c u l t u r e a r e n o t s t r ic t ly fa jsjfia b le, t h e y c a n o f t e n a p p r o a c h t h e j i r e a s o f l i f e j h a t
P o p p e r i a n p o s i t i v i s m c a n n o t - — a r e a s w e c a n n o t li v e w i t h o u t .
G i v e n t h a t o n e c a n m a k e a g o o d c a se fo r a t lea st f o u f o f t h e five b a sic e le - ^
m encs o f f r a c t a l g e o m e t r y in A f r i c a n m a t h e m a t i c s , w h a t s h o u l d we m a k e o f it \
in te r m s o f c u l t u r e ? T o ask t h i s q u e s t i o n e f f e c t i v e l y w e n e e d t o a v o j d tw o p i t ­
falls. T h e first is t h e p o s s ib il it y o f “o v e r d e t e r m i n e d ” e x p l a n a t i o n s fo r A f r i c a n
fractals, e x p l a n a t i o n s t h a t s e e m t o b e w a i t i n g f o r us b e f o r e w e ’v e e v e n b e g u n
i8 o
Im plications
t o e x a m i n e t h e e v i d e n c e . T h e s e c o n d is t h e d i f f i c u l t y o f s u s t a i n i n g s k e p t i c i s m
in a r a c i a l l y c h a r g e d e n v i r o n m e n t . j t h e p o s s i b i l i t y t h a t w e m i g h t sh y a w a y fr o m
c r i t i q u e o v e r fe ars t h a t e x p r e s s i n g a n e g a t i v e v i e w c o u l d b e t a k e n <is h a v i n g a n
e t h n o c e n t r i c o r r a c i s t m o t i v a t i o n ^ B o t h fa ilin g s a r e e q u a ll y d a m a g i n g . R e c e n tl y ,
re s e a rc h e rs h a v e d r a w n a t t e n t i o n to t h e w ays t h a t th e o r ie s o f k n o w le d g e
( e p i s t e m o l o g y ) c a n s n e a k u n e x a m i n e d i n t o c u l t u r a l p o r t r a i t s . If we a r e to a v o id
t h e t r a p o f s e e i n g A f r i c a n f r a c t a l s as a n i n d i c a t i o n t h a t A f r i c a n s a r e “c l o s e r to
n a t u r e , ” o r c o n c r e t e r a t h e r t h a n a b s t r a c t t h i n k e r s , o r u n i f i e d in a s i n g l e h o m o ­
g en eo u s c u ltu re, t h e n we n e e d to k n o w ..a .b it a b o u t th e orig in o f th ese m is­
c o n c e p t i o n s . T h e fir s t s t e p i n th a t\p v o c e s s ,.is t o e x a m i n e t h e e p i s t e m o l o g i c a l
f r a m e w o r k s t h a t a re a p p l i e d t o t h e s t u d y o f c u l t u r e .
T h e u n i t y j d i v e r s i t y d e b a t e a n d t h i n d e s c r ip t io n
A c c o r d i n g to M u d i m b e j r i 9 8 8 ) , t h e c o n c e p t o f a u n i t a r y , t r a d i t i o n a l “A f r i c a n c u l ­
t u r e ” is a n i n v e n t i o n c r e a t e d first b y c o l o n ia l is ts , w h o s o u g h t t o r a t i o n a l i z e t h e i r
c o n q u e s t w i t h t h e m y t h o f th e p rim itiv e, a n d s u b s e q u e n tly by a n ti c o lo n i a li s ts s e e k ­
ing to c o n so lid a te t h e i r o p p o sitio n . A s im ilar c ritiq u e is p ro v id e d b y (A p p ia h (1 9 9 2 ) ,
’w h o su gg ests t h a t t h e d if fe re n c e s a m o n g v a r i o u s A f r i c a n s o c i e t i e s w e re m u c h to o
b r o a d to a llo w a n y g e n e r a l i z a t i o n s (p . 2 5 ) : “S u re l y d i f f e r e n c e s in re lig io u s o n t o l ­
ogy a n d r i tu a l , in t h e o rg a n i z a ti o n o f p o l i t i c s a n d t h e fam ily, in r e l a ti o n s b e t w e e n
t h e sex es a n d in a rt , in sty les o f w a rfa re a n d c u is in e , in la n g u a g e — surely all th e s e
are f u n d a m e n t a l k i n d s o f d if f e r e n c e s ? ”
A p p i a h a n d M u d i m b e p r o m o t e v a r i o u s k i n d s o f s o l i d a r i t y in c o n t e m p o r a r y
A f r i c a ( a s w e ll as i n t e r n a t i o n a l l y in t h e d i a s p o r a ) ; t h e y o n l y c a u t i o n t h a t t h i s
c u l t u r a l u n i t y is o f r e l a t i v e l y r e c e n t o r i g i n , a n d t h a t a t t e m p t s to s e e a n A f r i c a n
‘‘e s s e n c e ” o r a u n i f i e d A f r i c a n c u l t u r e p r e c e d . i n g . m a j o r E u r o p e a n i n t e r v e n t i o n
(i.e., p re v io u s to t h e First W o r l d W a r ) will e v e n t u a l l y h a v e t o fall b a c k o n racjaMy
d e f i n e d c a t e g o r i e s , w h i c h is c e r t a i n l y a s e l f - d e f e a t i n g b a s i s f o r a n t i r n c i s t
m o v e m e n t s . F r o i n A p p i a h ' s a n t i e s s e n t i a l i s t p o i n t o f v i e w o n e c a n n o t d is c u s s
p r e c o l o n i a l “A f r i c a n c u l t u r e ; " o n l y “A f r i c a n c i i j t u r e s . ”
O n t h e o t h e r e x t r e m e o f t h e u n i t y v e r s u s d i v e r s i t y d e b a t e 1ies t h e A f r o c e n t r i c p o s i t i o n . W h i l e its p r o p o n e n t s a ls o a g r e e t h a t t h e r e was_n.o_.single,
homogerie.0115 A fricarT o T lT u T e, t h e y e m p h a s i z e ^ t h e ^ h j p x e d - . e l e m e n t s . A s a n t e
a n d A s a n t e ’s A fr ic a n C u l t u r e : R hythm s o f U n ity ( 1 9 8 5 ) , f o r e x a m p l e , b e g i n s by
ti.
s t a t i n g t h a t w h ile b la c k u n it y c a n n o t be b ased o n g e n e t i c g r o u n d s , b ro a d ly s h a r e d
c u l t u r a l u n d e r c u r r e n t s w e r e f o u n d t h r o u g h o u t t h e d i v e r s e s o c i e t i e s o f p re -
/
co lo n ia l A frica:
T heoretical fra m e w o rk s in cultural studies
A l t h o u g h th e p r e c is e a c tio n s an d id eas m a y d iffe r w ith in th e a c c e p ta b le ra n g e
a n d still r e m a in s q u a r e ly in t h e c a t e g o r y o f A f r i c a n c u ltu re , th e r e are s o m e b e h a v ­
io rs a m o n g s o m e A f r i c a n e t h n i c g r o u p s w h i c h m a y h a v e t h e o p p o s it e m e a n ­
i n g a m o n g o t h e r s . T w i n n e s s is c o m m o n l y c O A s i d e r e d a p o s i t i v e c h a r a c t e r i s t i c
in A f r i c a n s o c i e t i e s , y e t t h e r e a r e s o m e e t h n i c g r o u p s w h i c h a c c e p t t w i n n e s s
as a n e g a t i v e c h a r a c t e r i s t i c . . . .Y e t t h is p a r t ic u l a r is t ic e m p h a s i s w o u l d n o t m a k e
th e e th n ic g ro u p u n re la te d to th e o th e rs . P a tte rn e d b e h a v io rs by A f r ic a n e t h ­
n ic g ro u p s are c u ltu r a l, n o t rigid o r fix e d , b u t rela ted to h is to r y a n d e x p e r ie n c e .
C u l t u r e c a n v a r y o v e r t i m e , b u t in t h e c a s e o f A f r i c a n c u l t u r e , it w i l l a l w a y s
b e a r t i c u l a t e d in t h e s a m e w ay.
T h e r e is a lo t g o in g o n in th i s p a ra g r a p h , b u t t h e cru cia l p o i n t fo r m y a n a l y ­
sis is A s a n c e a n d A s a n t e ’s d i s t i n c t i o n b e t w e e n t h e .sur/ace p a r t i c u l a r i t i e s o f
v a r i o u s e t h n i c g ro u p s , w h i c h m a y d iffer, a n d d eep er c u l t u r a l s e n s i b i l i t i e s o r p a t ­
tern s o f a r tic u la tio n (w h ic h th e y la te r illu strate w ith “ th e th r e e tr a d itio n a l
v a lu e s : h a r m o n y w i t h n a t u r e , 2 h u m a n e n e s s , a n d r h y t h m ” [p. 7]). In t h i s A f r o c e n t r i s m , it is o n ly a t t h e d e e p le v e l in w h i c h we fin d i m p o r t a n t c u l t u r a l a t t r i b ­
u te s h e l c H n c o m m o n .
p p p i a h also m a k e s t h i s d i s t i n c t i o n b_etween m v i a l su rfa c e ^ a n d t h e ‘T u nd a m e n t a T ‘ d e p t h s . T h e o n l y d i s a g r e e m e n t b e t w e e n h i m a n d t h e A s a n t e s is
w h e t h e r o r n o t t h e d e p t h s r e v e a l d i f f e r e n c e s . O n e way a r o u n d t h i s q u e s t i o n is
in t h e “t h i c k d e s c r i p t i o n ” p r o p o s e d by a n t h r o p o l o g i s t C l i f f o r d p e e r t z O1973).
G e e r t z w a s m o t i v a t e d in p a r t b y h i s d i s s a t i s f a c t i o n w i t h t h e w ay s t h a t C l a u d e
L e v i - S t r a u s s ’s s t r u c t u r a l i s m s e e m e d t o r e d u c e s y m b o li c c u l t u r e to a flat, m e c h a ­
n is ti c s y n t a x . F o r p e e r t j , c u l t u r a U y m b o l s s h o u l d b t ^ i i i ^ , l g n i ^ f j j ^ n a m i c play,
a n d t h e e t h n o g r a p h e r s h o u l d s h o w t h e i r t u r b u l e n t e x p a n s i o n t h r o u g h layers o f
m e a n i n g , n o t t h e i r r e d u c n o n to jpsinj^le fixed s t ru c tu r e . G e e r tz d e fin e d th e s e d e e p
e l e m e n t s , w h i c h t e n d to b e m o r e s u b j e c t i v e a n d literary, as specific to a p a r t i c ­
u lar c o m m u n i t y . F o r h i m j t w o u ld b e e x t r e m e l y d ifficult _to c o m p a re d e e p e l e ­
m e n t s fro m o n e lo c a t i o n to t h e n e x t , b e c a u s e t h e d e e p e l e m e n t s a re the_resu.lt
o f lo cal i n t e r p r e t a t i o n s . T a k e n t o t h e e x t r e m e , G e e r t z ’s c h ic k d e s c r i p t i o n w o u ld
sim p ly reply t h a t t h e q u e s t i o n A p p i a h a n d t h e A s a n t e s a r e a s k i n g c a n n o t b e
a n sw e re d .
T h e f r a m e w o r k 1 h a v e u s e d in p a r t s 1 a n d u o f th i s b o o k , w h i c h is t h a t o f
e t h n o m a t h e m a t i c s in g e n e r a l , m i g h t b e re f e r r e d to as ch in d e s c r i p t i o n ^ stu d y
o f th e su rfa c e p a r t i c u l a r i t i e s , s u c h as m a t e r i a U l e s i g n s a n d s y m b o li c f o r m u la s . A s
th e A s a n t e s p o i n t o u t , a m a t h e m a t i c a l e l e m e n t like d o u b l i n g (“ tw in n e s s " in t h e i r
q u o t a t i o n ) is ju s t a su rfa c e fe a tu re . W h e t h e r o r n o t it h a s d e e p e r m e a n i n g s — a n d
th u s t h e e n t i r e A f r o c e n t r i s m / a n t i e s s e n t i a l i s m d e b a t e — is a q u e s t i o n o u t s i d e o f
t h i n d e s c r i p t i o n . F o r t h i s r e a s o n , t h e t h i n d e s c r i p t i o n use o f A f r i c a n i c o n s to
181
Implications
r e p r e s e n t s p e c i f ic m a t h e m a t i c a l c o n c e p t s o r s t r u c t u r e s ( e .g ., t h e t r i c k s t e r =
d i s o r d e r ) is n o t n e c e s s a r i l y in c o n f l i c t w i t h t h e j h i c k d e s c r i p t i o n o f t h e s e s y m ­
b o ls i n t h e i r d e e p s e m i o t i c d y n a m i c s . P e l t o n ( 1 9 8 0 ) s e t s u p j u s t s u c h a c o n f l i c t ,
a n d p e r h a p s r i g h t l y s o — t h e r e h a s i n d e e d b e e n - a t e n d e n c y fo r s t r u c t u r a l i s t s to
c l a i m t h a t t h e y h a d r e d u c e d c u l t u r e t o its t r u e e s s e n c e . T h e i r e r r o r w as to
in s is t t h a t t h e s e b a r e - b o n e s s t r u c t u r e s w e r e t h e t r u l y d e e p m e c h a n i s m s o f c u l ­
t u r e , a n d t h a t t h e d i s c u r s i v e p la y o f m e a n i n g s h o u l d b e d i s r e g a r d e d as s h a l l o w
d i s t r a c t i o n . A s l o n g as w e k e e p t h e t h i c k s t u f f as t h e d e e p , a n d t h e p a r e d d o w n s t r u c t u r e s as t h e s u r f a c e , t h e r e is n o c o n f l i c t .
W h i l e t h e l a c k o f A f r i c a n u n i t y in “ t w i n n e s s ” is n o t a p r o b l e m fo r t h o s e
. c o n c e r n e d o n l y w i t h d e e p e r m e a n i n g s , w o u l d n ’t it p r e s e n t a p r o b l e m f o r t h i n l y
"7
d e s c r ip ti o n ? T h a t is, if d o u b l i n g is s u p p o s e d to b e a n i m p o r t a n t f e a t u r e o f A f r i c a n y
m a t h e m a t i c s , t h e n h o w d o e s o n e e x p l a i n t h e A f r i c a n s o c i e t i e s t h a t d o n o t u se
it? I n d e e d , h o w is f r a c ta l g e o m e t r y s u p p o s e d to b e a n _ A f r i c a n k n o w l e d g e s y s te m
if t h e e x a m p le s o f its use are so d isparately s c a tte r e d across t h e c o n t i n e n t ? T o a n sw er
th is q u e s tio n , we n e e d to c o n sid e r w h a t W i t t g e n s te in c alled a “family re s e m b la n c e .”
W h e n w e lo o k a t t h e p h o t o g r a p h o f a la r g e fa m ily w e c a n s e e t h a t e v e r y o n e is
r e l a t e d , e v e n if t h e r e is n o s i n g l e c h a r a c t e r i s t i c t h a t t h e y all s h a r e ( s o m e h a v e
b ig n o s e s a n d s o m e s m a ll, s o m e l i g h t h a i r a n d s o m e d a r k , e t c . ) . I n t h e s a m e way,
it is n o t u n c o m m o n for a g ro u p o f m a t h e m a t i c a l id e a s t o s h a r e m a n y c o m m o n ­
a l i ti e s w i t h o u t a s i n g u l a r e s s e n c e . I n J a m e s G l e i c k ’s ( 1 9 8 7 ) h i s t o r y o f c h a o s
th e o r y , fo r e x a m p l e , h e sh o w s t h a t t h e e m e r g e n c e o f n o n l i n e a r d y n a m i c s as a dis­
c i p l i n e w as d u e t o a s lo w g a t h e r i n g o f m a n y d i f f e r e n t s t r a n d s o f m a t h e m a t i c s —
s t r a n g e a t t r a c t o r s , f r a c ta l g e o m e t r y , c e l l u l a r a u t o m a t a , a n d s o o n . I n o r d e r for
s c i e n t i s t s t o c o l l a b o r a t e o n t h i s d e v e l o p m e n t , t h e r e w a s a l o n g p e r i o d in w h i c h
s e v e r a l r e s e a r c h e r s w o r k e d h a r d to p o i n t o u t t h e fa m ily r e s e m b l a n c e o f t h e s e d i s ­
p a r a t e m a t h e m a t i c a l to o l s , a n d m a n y a s p e c t s o f t h e i r r e l a t i o n s h i p s a r e still
u n c e r t a i n today. Sim ilarly, A fr i c a n fractal g e o m e try is n o t a s i n g u la r b o d y o f k n o w l ­
ed g e , b u t r a t h e r a p a t t e r n o f r e s e m b l a n c e t h a t c a n b e s e e n w h e n w e d e s c r i b e a
w i d e v a r i e t y o f A f r i c a n m a t h e m a t i c a l i d e a s a n d p r a c t i c e s . A n d as w e sa w in t h e
c a s e o f B a n n e k e r ' s q u i n c u n x , it is n o t t h e o n l y p a t t e r n p o ss ib le .
P a rticip a n t s i m u la tio n
W h e t h e r o n e b e l i e v e s i n G e e r t z 's t h i c k d e s c r i p t i o n o r in s o m e o t h e r m e t h o d for
re s e a r c h in g t h e d e e p e r m e a n i n g s o f a lo cal c u ltu re , a n d n o p o l o g i s t s g e n e r a l ly agree
t h a t it re q u ire s lo n g - te r m local e t h n o g r a p h i c study. M y thin, d e s c r i p t i o n field w ork
^ l a s t e d o n l y a y e a r a n d m o v e d t h r o u g h S e n e g a l , M a l i, B u r k i n a F a so , T h e G a m ­
b ia , C a m e r o o n , B e n i n , a n d G h a n a . T h i s d is p e r s e d i n v e s t i g a t i o n is q u i t e u n l i k e
T heoretical jYameivorks m cultural studies
w h a t is u n d e r t a k e n b y m o s t a n t h r o p o l o g i s t s , w h o o f t e n s p e n d a c o u p l e o f years
in o n e v illa g e a l o n e , u sin g “p a r t i c i p a n t o b s e r v a t i o n ” to tr a v e r s e t h e d e p t h s o f th e
lo c a l c u l t u r e by a c t i v e l y l i v i n g it. T h e r e is, h o w e v e r , a n i m p o r t a n t j h f f e r e n c e :
I w as n o t t r y in g t o u n d e r s t a n d h o w t h e Y o ru b a e x p e r i e n c e grief, o r t o d e t e r m i n e
' t h e i n n e r m e a n i n g o f c o m m u n a l sp irit a m o n g t h e B ak a. M y i n t e r e s t was p rim a rily
in t h e f o r m a l p r o p e r t i e s o f d e s i g n , in m e t h o d s o f c o n s t r u c t i o n , a n d in o t h e r
t e c h n i c a l q u e s t i o n s t h a t c o u ld o f t e n b e a n s w e r e d in a d i r e c t a n d s i m p l e fa s h io n .
M a n y o f t h e A f r i c a n s 1 s p o k e w i t h w e r e c l e a r l y r e l i e v e d t o h e a r t h a t 1 w as a
m a t h e m a t i c i a n . - * O f c o u r s e 1 w as s t il l f a c e d w i t h s e v e r a l o f t h e s a m e p r o b l e m s
i n v o l v i n g e t h n o g r a p h i c a c c u r a c y a n d a u t h o r i t y ( s e e C l i f f o r d 1 9 8 3 ). B u t e v e n
th e s e w ere s o m e tim e s d ifferen tly p o sed. In p a rtic u la r, 1 b e g a n to t h i n k o f my
m e t h o d o l o g y n o t as p a r t i c i p a n t o b s e r v a t i o n , b u t r a t h e r as p articip an t simulation,
se e k in g to c o l l a b o r a t i o n m a th e m a tic a l a n aly sis a n d v ir tu a l r e c o n s tr u c tio n
w ith m y A fricari co lleag u es.
P a r t i c i p a n t s i m u l a t i o n w as c a r r i e d o u t to c o n c l u s i o n o n ly in t h e r e s e a r c h
w ith C h r i s t i a n S i n a D i a t t a , h u t 1 tr ie d t o m a i n t a i n t h e p ra c t ic e a t s o m e level w ith
e v e r y o n e I h a d t h e o p p o r t u n i t y t o w o r k w i t h . T h a t m e a n t h a u l i n g d ia g r a m s of
f r a c ta l g r a p h i c s w i t h m e i n t o t h e e q u a t o r i a l r a i n f o r e s t a n d a c ro s s t h e s a v a n n a h ,
a n d d i s r u p t i n g r e s e a r c h t i m e w i t h m a t h le c t u r e s , b u t in t h e e n d it w as w ell
w o r t h i t 7 T h e r e w as t h e p o t e n t i a l p r o b l e m t h a t s o m e o n e w h o k n e w w h a t 1 was
afte r m i g h t f a b r i c a t e w h a t 1 w a n t e d t o h e a r (as in Sc. L o uis, S e n e g a l , w h e n o n e
o f t h e local c h i l d r e n h e a r d nJe t a l k i n g a b o u t B e n j a m i n B a n n e k e r a n d c l a i m e d
t o k n o w h i m p e r s o n a l l y ) , / v ’m o r e p r e s s i n g p r o b l e m w as m y r e s i s t a n c e t o t h e i r
sugg estio ns, as o c c u r r e d in m y in itia l d i s a p p o i n t m e n t w i t h tire la c k o f p la c e v a lu e
n o t a t i o n in che B a m a n a d i v i n a t i o n c o d e , o r h e a r i n g che d e s c r i p t i o n o f t h e o s c i l ­
la to ry s n a k e as “ D a n a t w o r k ” (a ll 1 c o u l d t h i n k o f a t t h e t i m e w as a r o a d c o n ­
s tr u c ti o n s ig n ). O f c o u rs e , th e r e are a lw a y s th e a fte re ffe c ts — S e n e g a le s e
s o c i o lo g i s t F a to u S o w sa id “ if t h e r e a r e n o t f r a c ta l s in A f r i c a n o w , t h e r e surely
will h e by t h e t i m e y o u l e a v e ”— h u t t h e n t h a t is a fe a t u r e o f all e t h p o g r a p h y ; j u i d
p a r t i c i p a n t s i m u l a t i o n is a b o ur t u r n i n g thac_inco _ a n .a d v a n t a g e .
T h e r e a s o n c o l l a b o r a t i v e a p p r o a c h e s ..like p a r t i c i p a n t s i m u l a t i o n w e r e n o t
t r a d i t i o n a l l y u s e d in e t h n o g r a p h y c o m e s f r o m c o n c e r n s o v e r a c c u r a c y — t h e
desire to-ol 5 t a i n a n o b j e c t i v e a c c o u n t — a n d c o n c e r n s o v e r a u th o r it y , a s u s p ic io u s
m o t iv e in tj^e c o l o n i a l c o n t e x t o f m o s t t r a d i t i o n a l a n th r o p o lo g y . C liffo rd ( 1 9 8 3 )
d e s c r ib e s t h e m o v e t o w a r d c o l l a b o r a n v e ’t e c h n i q u e s as b o t h t h e a n t h r o p o l o g i s t s ’
o w n s e l f - c r it iq u e o f a u t h o r i t y a n d as a g r o w in g r e c o g n i t i o n t h a t s i n c e t h e e t h n o ­
g r a p h e r h a s as m u c h m o t i v a t i o n as t h e i n f o r m a n t d o e s, a c c u r a c y a n d o b j e c t i v ­
ity c a n b e b e t t e r a p p r o a c h e d hy s h a r i n g a u t h o r i t y w i t h i n d i g e n o u s v o i c e s t h a n
by u sing t h e m in a k i n d o f v e n t r i l o q u i s t a c t. S i m p l y p r o c l a i m i n g a c o l l a b o r a t i v e
Implications
184
a p p r o a c h is o f c o u r s e n o g u a r a n t e e t h a t y o u w ill h a v e o n e , a n d j i a r t i c i p a n t s i m ­
u l a t i o n is p e r h a p s e v e n m o r e s u s c e p tib l e t o m a n i p u l a t i o n d u e t o . t h e ro le o f te c h n o lo g i c a l^ e x p e r tj s e ,
O n t h e o t h e r h a n d , s i n c e d i e c re a t i o n o f v i r t u a l w o r l d s — s i m u l a t i o n s — is
irk so m e w ays t h e p r o d u c t i o n o f s o m e t h i n g fake, p a r t i c i p a n t s i m u l a t i o n d o e s h a v e
t h e a d v a n t a g e o f a v o i d i n g s o m e o l d - f a s h i o n e d c o n c e p t s o f a u t h e n t i c i t y . It w as, ^
a f t e r all, t h e c r e a t i o n p f a n “a u t h e n t i c n a t i y e ” (s e e A p p a d u r a i 1 9 9 5 ) t h a t h e l p e d
/ c o l o n i s t s t o ja il r e b e l s a m o n g b la c k S o u t h A f r i c a n s a n d N a t i v e A m e r i c a n s ; a n d
! o n e c o u ld e v e n h e a r t h e o c c a s i o n a l g u il t- r id d e n l a m e n t a m o n g t h e c o l o n i a l rulers
|
t h a t t h e y t h e m s e l v e s w e r e to b l a m e for h a v i n g a c c i d e n t a l l y p o l l u t e d t h e n a t u r a l
|
p u r i t y o f th e s e “c h i l d r e n o f t h e f o r e s t” w i t h t h e i r o w n t r o u b l i n g a rtific e ( s e e t h e
^ a p a r t h e i d c u l t u r e c o m e d y , T h e G o d s M u s t Be C r a ^ ) . L o c a t i n g i n d i g e n o u s a c ti v 1ity in v ir tu a l w o rld s c a n , if d o n e p roperly, c o u n te r ^ th i s h a b i t u a l t e n d e n c y to p la c e
a r t ifi c ia l o n t h e W e s t e r n s i d e a n d n a t u r a l o n t h e i n d i g e n o u s sid e. .
D o in g it p r o p e r ly re lie s o n t h e o t h e r ro o t, w h ic h c o m e s fro m t h e o ld f a s h i o n e d — a n d , 1 t h i n k , s t il l c r u c i a l — m e t h o d o f p a r t i c i p a n t o b s e r v a t i o n .
P a r tic ip a n t o b s e r v a tio n re c ru its a k in d o f re s p o n sib ility t h a t c a n be sadly la c k ­
in g in v i r t u a l e t h n o g r a p h i e s . T a k e , f o r e x a m p l e , t h e g r o w i n g fie ld o f c y b e r ­
e t h n o g r a p h y , in w h i c h a n t h r o p o l o g i s t s s t u d y t h e v i r t u a l c o m m u n i t i e s o f t h e
I n t e r n e t . S i n c e “l u r k i n g " ( o b s e r v i n g t h e e l e c t r o n i c e x c h a n g e s w i t h o u t p a r t i c i ­
p a t i n g ) is s o easy, t h e r e h a v e b e e n a n u m b e r o f s t u d i e s in w h i c h t h e e t h n o g ­
r a p h e r is r e d u c e d t o e a v e s d r o p p e r o r spy, w i t h n o a t t e m p t t o w o r k w i t h t h e
c o m m u n i t y in e i t h e r o f f - l i n e o r o n - l i n e liv e s . O n t h e o t h e r h a n d , r e c r u i t s c a n
in c lu d e b o th d ra fte e s, w h o h a v e little real in te re s t in w o rk in g c o lla b o ra tiv e ly ,
a n d f a n a t i c s , 'w h o a t e all t o o i n t e r e s t e d in w h a t G a y a t r i S p i v a k ( 1 9 8 7 ) c a ll s t h e
“b e n e v o l e n c e o f t h e w e s t e r n g a z e .”
T h u s p a r t i c i p ^ ^ s i m u l a d o n . js_an_atternpt to t a k e t h e b e s t o f bo th ^ a p p ro a ch es,
a n d t o u se t h e m in a k i n d o f c h e c k s - n n d - b a l a n c e s s y s te m . By in s i s t i n g o n p a r ­
t i c i p a t i o n w e .c a n h e l p a y o i d _ g l i b j r e s p o n s i b i l i t y ; and . by u s i n g s i m u l a t i o n w e c a n
strive to a v e rt th e p o lic in g o f b o u n d a rie s a r o u n d c o n s m ic t ip n s o f a u th e n tic ity
a n d re a lis m . F r o m t h i s p o i n t o f v ie w w e d o n o t n e e d t o e m p h a s i z e t r a d i t i o n o v e r
i n v e n t i o n ; t h e m a t h e m a t i c a l c r e a t i o n s o f a s i n g l e i n d i v i d u a l a re s till e x a m p l e s
o f i n d i g e n o u s m a t h e m a t i c s , e v e n if s h e is t h e o n l y o n e w h o k n o w s t h e y e x is t.
In ten tto n aliP y a n d e th n o m a th e m a tic s
T h e r e a r e c l e a r a d v a n t a g e s to a m e t h o d o l o g y t h a t c a n c r e d i t t h e i n v e n t i o n s o f
a single in d iv id u a l, b u t w h a t a b o u t th o s e c r e a t io n s t h a t d o n o t h a v e a single i n v e n ­
tor? A s we s a w in t h e c a s e o f c o m p l e x i t y in c h a p t e r i o , it is p o s s ib le t o e r r o n
T heoretical fra m e w o rk s in cu ltu ra l studies
I
che o c h e r s i d e by i n s is t in g c h a t c o n s c i o u s c r e a t i o n s c a n o n l y c o m e f r o m s i n g u ­
lar i n v e n t o r s . A b e t t e r u n d e r s t a n d i n g o f tht's p r o b l e m £ a n b e g a i n e d t h r o u g h t h e
c o n tr ast b e tw e e n e th n o m a c h e m a tic s a n d m a th e m a tic a l, an th ro p o lo g y . M a th e .rn.atical a n t h r o p o l o g y is g e n e r a l ly fo c u s e d o n r e v e a l i n g p a t t e r n s t h a t are n o t c o n ' sc io u sly d e t e c t e d by its s u b j e c t s o f s tu d y . I n p a r t t h i s is d u e t o a c o n v i c t i o n t h a t
m a n y o f t h e u n d e r p i n n i n g s o f s o c i e ty a re f o r c e s u n n o t i c e d by its m e m b e r s — n o t
o n ly b e c a u se s u c h forces o p e r a t e d a t le v e ls b e y o n d in d iv id u a l a w a ren ess; b u t a ls o „
b e c a u s e r e g u l a to r y m e c h a n i s m s w o u l d h a v e t o be c o v e r t , o b s c u r e d , o r o th e r w i s e '
p r o t e c t e d fr o m m a n i p u l a t i o n a n d c o n s c i o u s r e f l e c t i o n . For th e s e rea s o n s, m a t h e ­
m a t i c a l a n t h r o p o l o g y m a k e s g o o d s e n s e , a n d it h a s i n d e e d p r o d u c e d w o n d e r f u l
i n s ig h t s . B u t its e m p h a s i s o n u n c o n s c i o u s p r o c e s s a ls o aro se fr o m i m i t a t i o n o f
t h e r e s e a r c h e r - o b j e c t r e l a t i o n in t h e n a t u r a l s c i e n c e s : if a n t h r o p o l o g i s t s w e re
s i m p l y r e p o r t i n g i n d i g e n o u s d i s c o u r s e , t h e n t h e y w o u ld n o t c o u n t as s c i e n t i s t s .
T h i s p ro b le m o f m e re re p o r tin g is in d e e d t h e c a se for “ n o n - W e s t e r n m a t h e m a t i c s , ”
w h i c h is m a i n ly fo c u s e d o n d i r e c t t r a n s l a t i o n s f o r C h i n e s e , H i n d u , a n d M u s lim ,
m a t h e m a t i c s a n d t h u s c o n s i d e r e d a s u b j e c t for h i s t o r i a n s . H e n c e m a t h e m a t i c a l
a n t h r o p o l o g y ’s t e n d e n c y t o a v o i d i n t e n t i o n a l i c y c a n b e p r o b l e m a t i c .
T h e T n t e n t i o n a l i c y p r o b j e m in m a t h e m a t i c a l a n t h r o p o l o g y c a n b e s e e n in
K o l o s e i k e ’sM-J-9.74) m o d e l fo r m u d t e r r a c e c o n s t r u c t i o n in t h e lo w h i l l s o f
E c u a d o r. ^ o l o s e i k t J b e g a n w i t h t w o h y p o t h e s e s : e i t h e r t h e I n d i a n s l e a r n e d fro m
th e ln c a s t o n e te rra c e s in t h e h i g h m o u n t a i n s a b o v e , o r th e y w e re u n i n t e n t i o n a l
b y - p r o d u c ts o f c u l t i v a t i o n o n h il ls id e s . H e t h e n m a d e a list o f n i n e o b s e r v a t i o n s
t h a t w e re r e l e v a n t t o d e c i d i n g bet< veen t h e tw o . O f p a r t i c u l a r i n t e r e s t a re t h e
fo llo w in g :
. 3. T h e sam e hillside soil is u sed in r a m m e d - d i i t b ouses a n d fen ce walls, a n d
th e s e sta n d for years.
4. B u t 1 n e v e r saw a terra ce b e in g c o n s t r u c t e d , n o r did peo ple talk a b o u t su c h
a project.
5. S m a ll cav es are o fte n d u g in t o t h e te r ra c e face for s h e l te r d u ri n g r a i n ­
storms- T h a t this p o ten tially w e a k e n s t h e terrace face does n o t seem to c o n ­
c e rn people.
(i9 7 4 , 29-30)
K olo seik e c o n c lu d e s th a t th e s e te rra c e s are th e u n i n t e n ti o n a l re s u lt o f \
/ a n a c c r e t i o n p r o c e s s fr o m t h e c o m b i n a t i o n o f c u l t i v a t i o n a n d e r o s i o n , a n d
\
th e n p ro c e e d s to d e v e lo p a m a t h e m a t i c a l m o d e l for th e ra te o f t e r r a c e
g r o w th . M y p o i n t is n o t in q u e s t i o n i n g t h e a c c u r a c y o f t h e m o d e l , b u t r a t h e r
th e way t h a t i n d i g e n o u s i n t e n c i o n a l i t y is p o s i t i o n e d as a n o b s t a c l e t h a t m u s t
be o v e r c o m e b e f o r e m a t h e m a t i c s c a n b e a p p l i e d . E v e n a s m a l l d e g r e e o f
a w a re n e s s — b e i n g a w a r e t h a t a c a v e d u g i n t o a t e r r a c e fa c e m i g h t w e a k e n it—
m u s t be e l i m i n a t e d .
.
Jmpliaiticms
i8 6
I n a d d i t i o n , it rev eals a p a r t i c u l a r c u lt u r a l c o n s t r u c t i o n o f t h e s u p p o s e d un iI
v ersal a t t r i b u t e o f “ i n t e n t i o n . ” A s a W e s t e r n e r , K o l o s e i k e is u s e d t o a s o c i e t y in
a h u rry . P r o j e c t s t o b e . d o n e m u s t g e t d o n e , a n d a lw a y s w i t h s o m e o n e in c h a r g e .
!
T h e i d e a o f a l o n g - t e r m i n t e n t i o n a l p r o j e c t , p e r h a p s ‘e x t e n d i n g o v e r s e v e r a l
■ g e n e r a t io n s , o r t h e c o n s t i t u t i o n o f c o ll e c ti v e i n t e n t i o n a l i t y r a t h e r t h a n in d iv id u a l
i n t e n t , is n o t b r o u g h t u n d e r c o n s i d e r a t i o n . I t m a y w e ll b e t h a t t h e m a t h e ­
m a t i c a l m o d e l K o lo s e i k e o ff e re d w a s n o t o n l y a c c u r a t e , h u t a ls o h a d a n in d i g e 1
nous co u n terp art.
E t h n o m a t h e m a t i c s , in c o n t r a s t , h a s e m p h a s iz e d t h e p o s s ib il it ie s fo r i n d i g e ­
n o u s i n t e n t i o n a l i t y in m a t h e m a t i c a l p a t t e r n s . "For e x a m p l e , G e r d e s ' . ( i 9 9 i ) used
t h e L u s o n a s a n d d r a w i n g s o f t h e C h o k w e p e o p l e o f n o r t h e a s t e r n A n g o l a to
d e m o n s t r a t e i n d i g e n o u s m a t h e m a t i c a l k n o w l e d g e . W h i l e it w o u ld h a v e b e e n
p ossible to a t t r i b u t e this p r a c t ic e to a n u n c o n s c i o u s so c ia l p ro cess, s u c h as t h e reg­
u l a t i o n o f a u th o r it y , G e r d e s c h o s e to focus o n t h e i r propertjes_as.£p.Qscjous_indigen o u s i n v e n t i o n s . A s c h e r ( 1 9 9 1 ) n o t e s t h e s a m e cype o f E u l e r i a n p a t h d ra w in g s
in t h e S o u t h Pacific, a n d s h o w s t h e m t o be p r i m a r i l y m o t i v a t e d hy s y m b o l i c n a r ­
ra t iv e s , in p a r t i c u l a r t h e i r use b y t h e M a l e k u l a i s la n d e r s as a n a b s t r a c t m a p p i n g
o f k i n s h i p r e l a ti o n s . A g a i n , t h i s is in s t r o n g c o n t r a s t t o t h e t r a d i t i o n o f m a t h e ­
m a t i c a l a n t h r o p o l o g y , w h e r e k i n s h i p a lg e b r a w as c o n s i d e r e d a t r i u m p h o f W e s t ­
e r n a n a ly s is (arid e v e n a s o u r c e o f m a t h e m a t i c a l s e l f - c r i t i q u e ; K ay t i 9 7 1 ] h a r s h ly
n o t e s t h e a n t h r o p o l o g i s t s ’ t e n d e n c y to i n v e n t a n e w " p s e u d o - a l g e b r a ” for v a rio u s
k i n s h i p sy s te m s r a t h e r t h a n a p p ly o n e u n i v e r s a l s t a n d a r d ) .
, A sch er); d e sc rip tio n o f th e N a tiv e A m e r ic a n g a m e of D is h show s th is
c o n tr a s t Tn a m o re su b tle form, i n t h e C a y u g a v e rs io n o f t h e g a m e , six p e a c h stones,
b l a c k e n e d o n o n e sid e, a re to s se d , a m X th e n u m b e r s l a n d i n g b l a c k sid e o r b r o w n
sid e u p w e re re c o r d e d . T h e t r a d i t i o n a l C a y u g a p o i n t s c o r e s fo r e a c h o u t c o m e are
( t o t h e n e a r e s t i n t e g e r v a l u e ) in v e r s e ly p r o p o r t i o n a l t o t h e p r o b a b i li ty . A s c h e r
d o e s n o t p o s i t a n i n d i v i d u a l C a y u g a g e n iu s w h o d i s c o v e r e d p r o b a b i l i t y th e o ry ,
n o r d o e s s h e e x p l a i n t h e p a t t e r n as m e r e ly a n u n i n t e n t i o n a l e p i p h e n o m e n o n o f
r e p e a t e d a c ti v it y . R a t h e r , h e r d e s c r i p t i o n (p . 9 3 ) is f o c u s e d o n h o w t h e g a m e is
e m b e d d e d in c o m m u n i t y c e r e m o n i a l s , s p i r i t u a l b e l i e f s , a n d h e a l i n g ri tu a l s ,
s p e c ific a lly t h r o u g h t h e c o n c e p t o f “c o m m u n a l p l a y i n g ” i n w h i c h w i n n i n g s 3 re
a t t r i b u t e d t o t h e g ro u p r a t h e r t h a n t o t h e i n d i vidual''p.lnve.r. J u x t a p o s i n g th i s c o n t e x t w7 t h T l 'e t a il e e r a t te n t io n t o a b s t r a c t c o n c e p t s . o f r a n d o m n e s s a n d p r e d i c ta b i li ty
in a s s o c i a t i o n w i t h t h e g a m e — in p a r t i c u l a r rite id e a o f “e x p e c t e d .v alu es " a s s o ­
c i a t e d w i t K s u c c e s s iv e to sses— h a s t h e e f f e c t o f a t t r i b u t i n g t h e i n v e n t i o n o f
p r o b a b i l i t y a s s i g n m e n t s to c o l l e c t i v e i n t e n t .
A t th e s k e p tic a l e x t r e m e in e t h n o m a t h e m a t i c s , D o n a l d C r o w e has re frain ed
fr o m m a k i n g a n y i n f e r e n c e s a b o u t i n t e n t i o n a l i t y a n d in s is t s t h a t h is s t u d ie s o f
Theoretical fra m e w o rk s in cultural studies
187
s y m m e t r y in i n d i g e n o u s p a t t e r n c r e a t i o n s (s e e W a s h b u r n a n d C r o w e 1 9 8 8 ) are
sim ply e x am p les o f ap p lied m a th e m a tic s . B ut sin ce C ro w e has re s tric te d h is /
/’ w o rk to o n ly t h o s e p a t t e r n s w h i c h c o u ld b e a t t r i b u t e d to c o n s c i o u s d e s ig n ( p a i n t -
ing, c a r v i n g a n d w e a v i n g ) , it c re a t e s t h e o p p o s i t e '^ f f e c t o f m a t h e m a t i c a l a n t h r o - )
.-1 p o l o g y ’s a t t e m p t to e l i m i n a t e i n d i g e n o u s i n t e n t . T h i s is e v i d e n c e d b y C r o w e ’s /
d e d i c a t i o n t o t h e use o f th e s e p a t t e r n s in m a t h e m a t i c s e d u c a t i o n , p a r t ic u l a r ly h is C
/
!
C, t e a c h i n g e x p e r i e n c e in N i g e r i a d u r i n g t h e la t e 1 960s, w h i c h g re a t ly c o n t r i b u t e d , ' ’
' | t o Z a s la v s k y ’s ( 1 9 7 3 ) s e m i n a l t e x t , A fr ic a C o u n t s .
W h i l e n o n - W e s t e r n m a t h e m a t i c s is e x c l u s i v e l y f o c u s e d o n d i r e c t t r a n s ­
l a t i o n s ( s u c h as H i n d u a lg e b r a o r M u s l i m g e o m e t r y ) , e t h n o m a t h e m a t i c s c a n b e
o p e n t o a n y s y s t e m a t i c p a t t e r n d i s c e r n a b l e t o t h e r e s e a r c h e r . In f a c t , e v e n t h a t
d e s c r i p t i o n is t o o r e s t r i c t i v e : , b e f o r e G e r d e s ’s s t u d y t h e r e . w a s n o W e s t e r n c a t e ­
gory o f “re c u r siv e ly g e n e r a t e d E u le r i a n p a t h s ” ; it was o n ly in t h e a c t o f t h e i r p a r - C "
t i c i p a n t s i m u l a t i o n t h a t G e r d e s — a n d t h e C h o k w e — C re a te d t h a t h y b r i d . A n d
u n lik e m a th e m a tic a l a n th ro p o lo g y , e t h n o m a th e m a ti c s p uts a n e m p h a s is o n
t h e a t t r i b u t i o n o f c o n s c i o u s i n t e n t t o t h e s e p a t t e r n s . A t t h e s a m e t i m e , it
d e m a n d s q u a n t i t a t i v e o r g e o m e t r i c c o n f i r m a t i o n t h a t is l a c k i n g i n t h e p u r e l y
i n t e r p r e t i v e a p p r o a c h o f N e w A g e m y s ti c is m , s u c h as t h a t o f F r i t j o f C a p r a ’s T a o
o f P h y sic s (s e e c r i t i q u e s in R e s t i v o 1 9 8 5 ) . C l a i m s t h a t a n c i e n t k n o w l e d g e sy s­
t e m s r e v e a l t h e s t r u c t u r e o f t h e a t o m o r t h e e q u i v a l e n c e o f m a .tte r a n d e n e r g y
d o m o r e h a r m t h a n g o o d — first b e c a u s e th e y a re w ro n g , a n d s e c o n d b e c a u s e t h e r e
is n o m e a n s by w h i c h su c h k n o w le d g e c o u ld be o b t a i n e d . S u c h m y s tific a tio n d a m ­
ages c r e d i b l e r e s e a r c h in i n d i g e n o u s k n o w l e d g e sy s te m s , a n d r e m o v e s t h e a ttr.b
b u t i o n o f i n t e n t i o n a l i t y a n d i n t e l l e c t u a l l a b o r fr o m t h e p u t a t i v e k n o w e r s .
Lvehiflev. -is o. brish a n d n o t a la d d e r:
the c u ltu r a l lo c a tio n o f A f r i c a n fr a c ta ls
W e a re in c r e a s i n g ly s u r r o u n d e d by e x p l a n a t i o n s j a a s e d o n b io l o g i c a l d e t e r m i n ­
ism, a n d t h e r e is n o n e m o r e v i r u l e n t t h a n r a c i s m } E v e n in t h e s u p p o s e d li b e ra l
c lim a te o f U .S . a c a d e m ja , m y le ctu re s o n fractiilsTn A fric a are f r e q u e n tl y follow ed
by a qu ,estio n .ab o q t n e u ro s c ie n C e rT y p ic a lly this is a n i n n o c e n t re m a rk c o n c e r n i n g
N o a m ( b h o m s k y ’s id e a s o n u n i v e r s a l c o g n i t i v e s t r u c t u r e , b u t e v e n so, it is q u i t e
te llin g t h a t a l e c t u r e o n E u r o p e a n f r a c ta l s in v o k e s q u e s t i o n s a b o u t t h e g e n i u s o f
in d iv id u a ls , w h i l e A f r i c a n fr a c ta ls a r e c o m p u l s i v e l y a t t a c h e d to bio logy.
T h e m y t h o l o g y o f r a c e is to o c o m p l e x t o r e c o u n t h e r e (s e e ^ n o te 6 ) , b u t it
is useful to d i s t i n g u i s h b e t w e e n t w o c a t e g o r i e s o f racism .( P r i m i t i v i s t racism..^
o p erates by m a k i n g a g ro u p o f p e o p le t o o c o n c r e t e , a n d j . h u s “clo ser_to n a t u r e ”—
n o t re ally a c u l t u r e a t all, b u t r a t h e r b e in g s o f u n c o n t r o l l e d e m o t i o n a n d d i r e c t
’
LT.l T I ?! rrrw^i •»
Implications
b o d ily s e n s a tio n , r o o t e d in a n e d e n ic ecology. O r i e n t i a l i s t ra c ism o p e r a t e s by m a k - /
in g a g r o u p o f p e o p l e t o o j r b s t r a c t , and__thus “a r a b e s q u e ”— n o t r e a l ly a n a t u r a l ]
h u m a n , b u t o n e w h o is d e v o id o f e m o t i o i v c a r i n g only, f o r m o n e y o r- a n in s c ru ta b le
sp iritu al tra n sc e n d e n c e .
T h e a l t e r n a t i v e t o b i o g e n e t i c e x p l a n a t i o n is s o c i o c u l t u r a l , a n d h e r e t h e
c a t e g o r i e s o f p r i m i t i v e a n d o r i e n t a l c a n b e m u c h m o r e c o m p l e x . H is t o r i c a l l y ,
m a n y research ers w h o stro n g ly o p p o se d b o th racism a n d e th n o c e n tr a c is m h av e
b e e n l o c a t e d in i n s t i t u t i o n s w i t h t i t l e s l i k e “ M u s e u m o f P r i m i t i v e A r t s ” or
“ D e p a r t m e n t o f O r i e n t a l i s t S t u d i e s , ” a n d it w o u l d b e u n w i s e to s i m p l y s n e e r a t
t h e i r w o rk , p a r t ic u l a r ly c o n s i d e r i n g t h e a n t i r a c i s t c o n t r i b u t i o n s by b l a c k a n t h r o ­
po lo gists s u c h as Z o ra N e a l e H u r s t o n o r J o m o K e n y e tta . T h e r e is v a lu e t o b e f o u n d ;
in e v e n t h e w e a k e s t o f t h e s e o p p o s i t i o n a l t h e o r i e s , a n d p r o b l e m s in e v e n t h e i
strongest.
I n g e n e r a l t h e s e t h e o r i e s c a n j j e g r o u p e d i n t o t w o s t ra te g i e s : s a m e n e s s a n d
^ d iffe re n c e . S a m e n e s s c a n u su a lly a v o i d o r i e n t a l i s m a n d p r i m i t i v i s m , s i n c e it
a r g u e s t h a t w h a t is i m p o r t a n t a b o u t a n o n - W e s t e r n c u l t u r e a r e th o s e t h i n g s h e l d
i n c o m m o n w i t h t h e E u r o - A m e r i c a n s , a n d w h a t is d i f f e r e n t is ( i n t h i s c o n t e x t )
tr iv i a l. C l a u d e L e v i-S t ra u s s , fo r e x a m p l e , a r g u e d t h a t t h e “s a v a g e m i n d ” is b a s e d
o n system s o f sy m b o lic s tru c tu r e s , j u s t.l ik e t h e E u ro p e a n m in d , so t h a t a n
A f r i c a n w o r k i n g w i t h a s y s te m o f m y t h o l o g i c a l s y m b o l s is p e r f o r m i n g t h e s a m e
c o g n i t i v e o p e r a t i o n s as a E u r o p e a n w o r k i n g w i t h a s y s te m o f c o m p u t e r c o d e s y m ­
bo ls. O n e . d r a w b a c k o f s a m e n e s s is t h a t w e b e c o m e p la y e r s i n a g a m e c r e a t e d by
s o m e o n e else: “ 1 a m w o r t h w h i l e o n ly in s o far as 1 a m t h e s a m e as y o u . ” D iff e re n c e
c a n a v o id t h i s tr a p , a l t h o u g h it h a s m o r e t j p u b l e a v o i d i n g p r i m i t i v i s m a n d o r i ­
e n t a l i s m . F o r e x a m p l e , A i'm e C e s d i r e ’s n e o l o g i s m " n e g r i t u d e ”'bega'!i a s a w ay o f '
s p e a k i n g a b o u t t h e d i f f e r e n c e o f A f r i c a n c u l t u r e in o p e n - e n d e d , d y n a m i c , c r e ­
a t i v e te r m s , b u t la tp r ( i n t h e h a n d s o f o t h e r s ) t h e c o m p a r i s o n ' w a s fr o z e n i n t o a
s e t o f b i n a r y o p p o s i t i o n s ( i n t u i t i v e vs. a n a l y t i c , c o n c r e t e vs. a b s t r a c t , e t c .) .^ In
o t h e r w o rd s , b o t h s a m e n e s s a n d d i f f e r e n c e h a v e m o m e n t s o f f a ilu re as w ell as
m o m e n t s o f su c c e s s .__________________ _
T h e r e c e n t focus o n a n c ie n U E g y p t) in c e r t a i n c ir c le s o f A f r i c a n s t u d i e s h a s
ce rta in ly s e e n _ b o t h ju o m e n ts . M o t iv a t e d by c o n s id e ra b le scho larly w ork (e.g., D rak e
1 9 8 4 ) , it h a s also b e c o m e a t t a c h e d t o s o m e d i s r e p u t a b l e a n d q u e s t i o n a b l e cla im s
(s e e c r i t i q u e s in O r i t z d e M o n t e i l a n o 1 9 9 3 ; M a r t e l 1 9 9 4 ; L e fk o w itz 1 9 9 6 ) . It is
w o r t h n o t i n g , h o w e v e r , t h a t s o m e o f t h e c r i t i q u e s h a v e b e e n e q u a l l y l a c k i n g in
t h e i r r e s t r a i n t . In his r e v i e w o f t h e P o r t l a n d B a s e l i n e Essays, for e x a m p l e , R o w e
( 1 9 9 5 ) — w h i l e ri g h tl y p o i n t i n g t o a n u m b e r o f u n s u p p o r t e d a s s e r t i o n s — im p lie d
t h a t c l a i m s for a n a n c i e n t E g y p t i a n g li d e r s h o u l d h e d is m i s s e d b e c a u s e t h e
a u t h o r w as m e r e l y a n a e r o d y n a m i c s t e c h n i c i a n r a t h e r t h a n a P h . D . R o w e was
T heoretical fra m e w o rk s in cu ltu ra l studies
I
q u i t e r i g h t in o b j e c t i n g t o t h e w ild le a p f r o m e m p i r i c a l te s ts o f a s m a ll w o o d e n
c a r v i n g to t h e a u t h o r i t a t i v e c la i m s fo r a n c i e n t E g y p t i a n s flying fr o m p y ra m id s ;
b u t to im ply t h a t sim p le e x p e r i m e n t s a re a u t o m a t i c a l l y s u s p e c t b e c a u s e t h e y w e re
m a d e by a t e c h n i c i a n is n o t h i n g h u t cln ssist p r e j u d i c e . O n t h e o t h e r h a n d , th e
fact t h a t t h i s r e s e a r c h e r w as a t e c h n i c i a n r a t h e r t h a n a P h D sp e a k s to t h e u n d e r ­
lyin g c a u s e for th e s e p ro b l e m s : t h e l a c k o f i n s t i t u t i o n a l r e s o u rc e s a n d p re c a r io u s
e c o n o m i c s a m o n g m a n y b la c k e d u c a t i o n a l c o m m u n i t i e s .
A p p e a ls to a n c i e n t E gypt c a n also e n c o u n t e r p ro b le m s as a s trateg y o f s a m e ­
ness. O n t h e o n e h a n d , a n c i e n t E g y p t ’s s t a t u s as a s t a t e e m p i r e d i r e c t l y o p p o s e s
p r i m i t i v i s t a s s u m p t i o n s t h a t A f r i c a c o n s i s t s o f n o t h i n g b u t tr i b a l v illa g e s . O n
t h e o t h e r h a n d , it r e i n f o r c e s t h e v ie w t h a t t h e k n o w l e d g e s y s te m s o f n o n s t a t e
in d ig en o u s societies are n o t c o m p a r a b le to th o s e o f s t a te societies. T h i s vie w cotn es
fr o m t h e o ld id e a o f c u l t u r a l e v o l u t i o n as a l a d d e r , a u n i l i n e a l .. p r o g r e s s io n fr o m
“p r i m i t i v e ” t o “a d v a n c e d . ” I n t h e l a d d e r m ^ d e L i h g „ s m a l l - s c a l e d e c e n t r a l i z e d
(“b a n d ”) s o c i e ti e s w o u ld b e o n t h e b o t t o m ru n g , t h e m o r e h i e r a r c h i c a l ( “ tr i b a l ” )
s o c ie tie s w o u ld be o n t h e n e x t r u n g , a n d t h e m o s t h i e r a r c h i c a l ( “st a te " ) s o c ie tie s
w o u ld b e o n th e t o p ru n g . O f c o u rs e , sim p ly p o s i ti n g t h a t t h e s o c ie tie s w i t h c o m ­
p l e x so c ia l o r g a n i z a t i o n (e.g., l a b o r s p e c i a l i z a t i o n a n d p o l i t i c a l h i e r a r c h y ) h a v e
g r e a t e r t e c h n o l o g i c a l c o m p l e x i t y is n o t i n h e r e n t l y d e m e a n i n g ; b u t it is n o t
/"'entirely a c c u r a t e . A n t h r o p o l o g i c a l r e s e a r c h h a s p e r s i s t e n t l y s h o w n t h a t n e i t h e r ^
I s o c i a l s t r u c t u r e s n o r t h e i r k n o w l e d g e s y s t e m s c a n b e c o n s i s t e n t l y r a n k e d in a '
J u n i l i n e a ! s e q u e n c e ; f o r e x a m p l e , m o n o t h e i s t i c r e l i g i o n s t e n d t o o c c u r in b a n d ^
a n d s t a t e s o c i e t i e s m o r e t h a n in t r i b a l . J u s t as b i o l o g i c a l e v o l u t i o n h a s b e e n
re v is e d f r o m Lo.vejQ.y’j “g r e a t c h a i n o f b e i n g ” to. G o u l d ’s “c o p io u s ly ,j> ra n c h i n g
b u s h ,”^ so t o o c u l t u r al e v o l u t i o n is n o w ty p ic a lly p o r t r a y e d as a b r a n c h i n g d i v e r ­
sity o f fo rm s. T h e r e is n 6 r e a s o n t o f o c u s - o n s t a t e s o c i e t i e s o v e r n o n s t a t e s o c i ­
e ti e s in t h e p u r s u i t o f a n t i p r i t m t i v i s r p o r t r a i t s .
T h e d if fic u ltie s o f t h e o r e t i c a l f r a m e w o r k s in the. e p i s t e m o l o g y o f n o n s t a t e
s o cieties h a v e b e e n m u c l i m q r e mixecl. A p p i a h ( 1 9 9 2 ) p ro v i d e s a n e x t e n s i v e d is ­
c u s s io n o f t h i s i n t e r s e c t i o n , s t a r t i n g w i t h e th n o p h i l o 's o p h y . H is a n a ly s is w e a v e s
( b e t w e e n t h e positions, o f W ire d 'q ( 1 9 7 9 ) , w h o c r i t i q u e s t h e fo cu s o n c o m p a r i s o n
to W e s t e r n s c i e n c e r a th e r'tlV a n r e l i g i o n ( n o t i n g t h a t it le a v e s t h e s u p e r s t i t i o n s
[ a n d folk p h i l o s o p h i e s o f t h e W e s t u n e x a m i n e d ) , a n d E ^ u n t o n d | i ) ( i 9 8 3 ) , w h o (
\ a r g u e s a g a in s t a n y m i m e t i c c o m p a r i s o n , s u g g e s tin g t h a t etKiTO'pFfilosophy a n d i t s \
laities a re d r e s s in g E u r o p e a n m o t i v a t i o n s in a u t o c h t h o n o u s g a rb . B o t h c r i t i q u e s )
^could c e r t a i n l y b e a p p l i e d to A f r i c a n f r a c ta l s . B u t lik e M u d i m b e ’s ( 1 9 8 8 ) F o u cau ltian analysis o f A fric a n epistem ology, a n d G i l r o y ’s (1 9 9 3 ) fractal history ( w h ic h
we wit! e x a m i n e in t h e f o l lo w in g c h a p t e r ) , A p p i a h ' s d i a l e c t i c a l c o n t o u r m a p s
A f r i c a n e p i s t e m o l o g y as a n h i s t o r i c a l p r o c e s s r a t h e r t h a n a n o b j e c t o f s t r i c t l y
Implications
190
p re - o r p o s t - W c s t e r n p r e s e n c e . T h e c a u t i o n s o f W i r c d u , H o u n t o n d j i , a n d o t h ­
ers a re serio u s r e m i n d e r s t h a t A f r i c a n fra ctals c a n o n l y s u c c e e d as p a r t i c i p a n t s i m ­
u l a t i o n , n o t as I n d i a n a - J o n e s d i s c o v e r i n g - a n o t h e r l o s t t e m p l e .
( G i v e n th o s e p r e c a u t io n s , it m a k e s s e n s e to see A f r i c a n fr a c ta ls as ju s t
a n o t h e r m o m e n t in a h i s t o r i c a l s e q u e n c e . O n e c o u l d , f o r e x a m p l e , p l a c e
t h e m i n M u d i m b e ’s h i s t o r y o f e t h n o p h i l o s o p h y , o r Z a s l a v s k y ’s ( 1 9 7 3 ) h i s t o r y
o f research o n A fric a n m a th . B ut th e re are o th e r rese a rc h e rs w h o h a v e p o in ted
o u t so m e o f th e fractal c h a ra c te ris tic s o f A f ric a n d e sig n s a n d p ra c tic e s , a n d
i t is u s e f u l t o e x a m i n e t h e m as a g r o u p , e v e n if t h e y l a c k t h e c l e a r h i s t o r i c a l
tr a je c to ry o f o t h e r c ate g o ries. W e h a v e a lre a d y m e n t i o n e d t h e o b s e r v a tio n of
n o n l i n e a r s c a l i n g by B n t i s h ’a r t h i s t o r i a n W i l l i a m ^ a g g ^ c h a p t e r 6 ) , a n d tjhe
i n t e r p r e t a t i o n o f s c a l i n g d e s i g n s as s i g r d f i e r s o f t j T f i m . t y j j i th e .,w p rk .-.o f
C a m e y p o m a n t h .^ o lo g tj p u E n g e l b e r t M v e n g ( c h a p t e r 9 ) . L e o p o l d S e n g h o r ,
t h e d i s t i n g u i s h e d m a n o f l e t t e r s w h o b e c a m e S e n e g a l ’s first p r e s i d e n t , a ls o h a d
a n e y e fo r A f r i c a n f r a c t a l s . H is t e r m w a s “d y n a m i c s y m m e t r y , ” w h i c h h e t o o k
f r o m arc h i s t o r i a n s . B u t S e n g h o r ’s m o t i v a t i o n w a s p r i m a r i l y id e o l o g y ; d e f i n ­
in g a “ n e g r i t u d e ” t h a t w o u l d e n c o m p a s s t h e k i n d o f c u l t u r a l p o l i t i c s h e saw
as n e c e s s a r y t o i n d e p e n d e n c e . M o s t r e c e n t l y , H e n r y L o u i s ^ a t f i ^ ( 1 9 8 8 )
e x p l o r e d t h e d o u b l i n g p r a c t i c e s o f v o d u n d i v i n a t i o n in t e r m s o f j a j i t e r a r y v e r ­
sion o f d e te rm in is tic ch ao s; here th e re c u rsio n g e n e ra te s a c u ltu ra l u n c e rta in ty
tK a tT re e s g e n d e r i d e n t i t y f r o m s t a t i c b o u n d a r i e s : “T h e F o n a n d Y o ru b a e s c a p e
t h e W e s t e r n v e rs io n o f d iscu rsiv e se x ism th r o u g h t h e a c ti o n o f d o u b lin g th e
d o u b l e ; t h e n u m b e r 4 a n d its m u l t i p l e s a r e s a c r e d in Y o r u b a m e t a p h y s i c s . E s u ’s '
t w o s i d e s ‘d i s c l o s e a h i d d e n w h o l e n e s s , ’ r a t h e r t h a n c l o s i n g o f f u n i t y , t h r o u g h
t h e o p p o s i t i o n , t h e y s i g n if y t h e p a s s a g e f r o m o n e t o t h e o t h e r as s e c t i o n s o f
a subsum ed w h o le .”
W h i l e all four h a v e h i t u p o n m a t h e m a t i c a l a s p e c ts o f A f r i c a n fractals, n o n e
o f th e s e a u th o r s h a v e focused o n re p r e s e n ta t io n s o f m a t h e m a t ic a l k n o w led g e.
M v e n g , t h e t h e o l o g i a n , p r o v i d e s a t h e o l o g i c a l i n t e r p r e t a t i o n . Fagg, t h e a r t i s t ,
c o n c l u d e s w i t h a c o m p a r i s o n t o D ’A r c y T h o m p s o n ’s f a m o u s n a t u r e d r a w in g s .
S e n g h o r , t h e s t a t e s m a n , se e s h is d y n a m i c s y m m e t r y as a s i g n o f c u l t u r a l — a n d
t h u s n a t i o n a l — id e n tity . A n d G a t e s , as a li te r a ry c r i t i c , sees it as d is c u rs iv e t e c h ­
n i q u e . S u r e l y m y in s i s t e n c e o n i n d i g e n o u s m a t h e m a t i c s is n o less a n i m p o s i t i o n
o f s e e i n g t h e w o r ld t h o u g h m y o w n le n s e s , b u t s i n c e t h a t is n o d i f f e r e n t f r o m
t h e o t h e r e x p l a n a t i o n s , w h y d o e s e t h n o m a t h e m a t i c s a p p e a r t o b e so m u c h
m o r e c o n t r o v e r s i a l ? I t is b e c a u s e a p o r t r a i t o f m a t h e m a t i c a l s o p h i s t i c a t i o n in
n o n s ta te societies creates a strong c o n flic t w ith th e o ld lad d er m odel of cu ltu ral
e v o l u t i o n , a m o d e l t h a t is itse lf o v e r d u e fo r e x t i n c t i o n .
Theoretical fra m e w o rk s in cu ltu ra l studies
ConclMsion
S o fa r w e h a v e o u t l i n e d s e v e r a l t h e o r e t i c a l f r a m e w o r k s t h a t c o u l d r a i s e p r o b ­
l e m s f o r A f r i c a n f r a c t a l s . O n t h e o n e h a n d , t h e r e ^ a r e t h e o r i e s in w h i c h t h e
d e s i g n s c o u l d b e d is m i s s e d as u n c o n s c i o u s b i o l o g i c a l o r s o c i a l p r o c e s s . O n t h e
o t h e r h a n d , g reat care m u st be t a k e n to a v o id e it h e r in flated claim s or a
m a th e m a tic a l v e rsio n o f n e g ritu d e . W i t h th e e x c e p tio n o f b io lo g ical d e te r ­
m in is m , n o n e o f th e fra m e w o rk s re v ie w e d h e r e are necessarily g o o d o r bad.
T h e r e a r e c a s e s in w h i c h m a t h e m a t i c a l a n t h r o p o l o g y is m o r e a p p r o p r i a t e
t h a n t h e e t h n o m a t h e m a t i c a l a p p r o a c h , o r w h e n s a m e n e s s is a b e t t e r s t r a t e g y
th a n d ifferen ce, o r w h e n a t t e n t i o n to a n c ie n t Egypt n e e d s to su p ersede a t t e n ­
t i o n t o s u b - S a h a r a n A f r i c a — j u s t as t h e r e a r e c a s e s in w h i c h t h e o p p o s i t e is
tr u e . O u r g o a l is n o t t o f i n d t h e o n e t r u e f i n a l f r a m e w o r k — it d o e s n o t e x i s t —
. b u t t o k e e p a w e l l - s t o c k e d t o o l b o x a n d k n o w h o w t o p i c k t h e r i g h t t o o l fo r
t h e r i g h t j o b . N o w t h a t w e a r e w e l l p r e p a r e d f o r c o n s t r u c t i v e ta s k s , it is t i m e
to m o v e t o p o l i t i c s .
CHAPTER
— T h e ---politics o f
—A frican—
—fractals™
G i v e n t h e p o s s ib le d a n g e r s in m i s i n t e r p r e t i n g A f r i c a n f r a c ta l s , h o w c a n w e p u t
t h e m t o g o o d use? S o c ia l t h e o r i s t s fr o m m a n y d i f f e r e n t d i s c i p l i n e s h a v e u se d tw o
m a t h e m a t i c a l c o n c e p t s w e h a v e d is c u s s e d , r e c u r s i o n a n d t h e a n a l o g - d i g i t a l
d ic h o t o m y , in c o n s t r u c t i n g t h e i r ideologies. M a n y th e o r ie s o f c o m m u n i c a t i o n h a v e
a s s u m e d t h a t t h e r e is s o m e k i n d o f u n iv e r s a l e t h i c a l o r s o c ia l d i f f e r e n c e b e t w e e n
u s i n g a n a l o g s ig n a ls a n d u s i n g d i g i ta l s y m b o ls . O t h e r t h e o r i e s h a v e m a i n t a i n e d
t h a t re c u rsio n h a s so m e k in d o f u n iv ersal e th ic a l o r so cial valu e. B o th are u lti­
m a t e l y fa ilu re s in t h e s e n s e t h a t e t h i c s a n d v a l u e s d o n o t lie w i t h i n m a t h e ­
m a t i c a l d i s t i n c t i o n s . Yet t h e y a re a ls o o n t h e r i g h t t r a c k in t h a t s u c h a s s o c ia ti o n s
c a n b e locally fo rm ed — it is ju s t t h a t d iffere n t lo c a t io n s w ill re s u lt in d if fe re n t social
m e a n i n g s . S u c h lo c a lly s p ecifie d s o c i a l a t t a c h m e n t s t o f r a c t a l s c a n b e u se fu l for
u n d e r s t a n d i n g c u l t u r a l p o l i t i c s in A f r i c a a n d b e y o n d .
T h e p o litic s o f t h e a n a lo g -d ig ita l d i s t i n c t i o n
J e a n J a c q u e s R o u s s e a u is o f t e n c r e d i t e d as a f o u n d e r o f “ o r g a n i c r o m a n t i c i s m , ”
t h e t h e o r y t h a t t h e N a t u r a l is i n h e r e n t l y b e t t e r t h a n t h e A r ti f ic i a l. W h e t h e r or
n o t t h i s is d e s e r v e d , J a c q u e s D e r r i d a ( 19 7 4 ) ta k e s h i m t o ta s k fo r p r o p o s i n g t h a t
a natu ral/artificial difference c a n be fo u n d b e tw e e n d iffe ren t languages. J e a n Jacqu es
T h e politics o f A fr ic a n fractals
R o u s s e a u p r o p o s e d t h a t t h e “n a t u r a l crys o f a n i m a l s , ” m u s ic , a n d “ a c c e n t u a t i o n ”
( t h a t is, p i t c h i n t o n a t i o n in t h e h u m a n v o i c e ) a r e all a s i m i l a r ty p e o f c o m ­
m u n i c a t i o n . I n t h i s I w o u ld t e n d t o a g re e , s i n c e i n s t r u m e n t a l m u s ic a n d h u m a n
p i t c h i n t o n a t i o n a re f o r . t h e m o s t p a r t a n a l o g r e p r e s e n t a t i o n s a n d s i n c e h e was
pro bably th in k in g o f a n a lo g ex am p les o f a n im a l c o m m u n ic a tio n (a lth o u g h
m a n y a n im a ls , fo r e x a m p l e v e r v e t m o n k e y s , use d ig i ta l c o m m u n i c a t i o n as w ell).
R o u s s e a u c o n t r a s t e d t h i s t o “a r t i c u l a t i o n ” i n t h e h u m a n v o i c e , by w h i c h h e
m e a n t t h e li n g u i s t i c ( a n d h e n c e d i g i t a l ) p a r t s o f s p e e c h . B u t i n s t e a d o f s e e i n g
t h e d i s t i n c t i o n as tw o d i f f e r e n t types o f r e p r e s e n t a t i o n , o n e a n a l o g a n d th e o t h e r
d i g i ta l , R o u s s e a u c l a i m e d t h a t a n a l o g s i g n a l s w e r e n o t a fo rm o f r e p r e s e n t a t i o n
a t all. In his view, dig ital versus a n a lo g was r e p r e s e n t a t i o n versus T h e R eal. M usic,
a n im a l cries, an d e m o tio n a l i n t o n a ti o n w ere s o m e h o w m o re n a tu r a l and
a u t h e n t i c . W o r s e y e t, h e i n f l a t e d t h i s i n t o a c u l t u r a l d i f f e r e n c e , m a i n t a i n i n g
t h a t w h i l e E u r o p e a n l a n g u a g e s w e re la r g e ly b a s e d o n ( d i g i t a l ) a r t i c u l a t i o n , t h e
l a n g u a g e o f t h e n o b e l s a v a g e w as c l o s e r t o n a t u r e .
O n e m i g h t h o p e t h a t D e r r i d a w o u ld c o r r e c t t h e m a t t e r a n d p o i n t o u t t h a t
a n a l o g s ig n a ls a re ju s t as m u c h a r e p r e s e n t a t i o n — ju s t as m u c h fakes, j u s t as easy
t o lie o r te l l t r u t h w i t h , a n d j u s t as a r t if i c ia l— as d i g i t a l s y m b o ls are. B u t h e t o o
fa iled t o p r o d u c e a b a l a n c e d p o r t r a i t . D e r r i d a d i d in s ist t h a t all h u m a n lin g u is ­
tics is f u n d a m e n t a l l y dig ital ( q u ite tr u e ) , b u t h e d id n o t b o t h e r to say a w o rd a b o u t
o t h e r i n o d e s o f v o c a l r e p r e s e n t a t i o n . T h i s e r r o r is d u e t o D e r r i d a ’s c o n c e r n o v e r
che a u t h o r i t a r i a n id e o lo g y t h a t o r g a n i c r o m a n t i c i s m c a n p r o d u c e . For e x a m p l e ,
h is t o r y is full o f d i c t a t o r s w h o c l a i m e d t h a t t h e i r e t h n i c g r o u p w as t h e re a l o r
n a t u r a l o n e , a n d t h a t o t h e r s w e re a r t i f i c i a l p o l l u t a n t s in t h e i r E d e n . R o u s s e a u
h i m s e l f d id n o t h a v e s u c h fa s c istic t e n d e n c i e s , b u t D e r r i d a is r i g h t in p o i n t i n g
o u t t h a t o r g a n i c rc m a n ti-c is r.v c a r. alv/ay s b e u s e d in t h a t w a y , n o m a t t e r w h o it
is c o m i n g fro m . * O n e n e e d n o t p a n i c so m u c h , h o w e v e r , a n d b a n i s h a n a l o g sig­
n a ls fr o m e x i s t e n c e ; it is e n o u g h t o g iv e t h e m t h e s a m e e p i s t e m o l o g i c a l s t a t u s
as d i g i ta l s y m b o l s — n o m o r e a n d n o less.
1 h a v e f o u n d th is e g a l i t a r i a n v ie w o f t h e a n a l o g / d ig i ta l d i s t i n c t i o n v ery d if­
ficult to p ro m o te ; it seem s t h a t e v e r y o n e h a s t h e i r o w n fav o rite view. W h e n 1 sp o k e
to c h a o s t h e o r i s t R a l p h A b r a h a m , for e x a m p l e , h e e x p l a i n e d t h a t a n a l o g system s
w ere in h i s view t h e r e a l m o f spirit, t h e v i b r a t i o n s o f A t m a n . P o s t m o d e r n th e o r y
m a v e n J a m e s C liffo rd , t o t h e c o n tr a r y , i n s is t e d t h a t o n ly d ig i ta l r e p r e s e n t a t i o n
is c a p a b l e o f t h e flexible r e a r r a n g e m e n t s t h a t c o n s t i t u t e h u m a n t h o u g h t . T h i s
sa m e b a t t l e h a s b e e n p la y e d o u t in t h e h i s t o r y o f A f r i c a n c u l t u r a l s tu d ie s . D u r ­
ing th e ) 960s, realism was in vogue, a n d w h a t c o u ld h a v e b e e n a w o n d e rfu l e x p l o ­
r a t i o n o f t h e a n a l o g r e p r e s e n t a t i o n t e c h n i q u e s in A f r i c a n c u l t u r e w as o f t e n
r e d u c e d t o r o m a n t i c p o r t r a i t s o f t h e “ r e a l ” a n d “n a t u r a l , ” w h ile A f r i c a n sy m b o l
193
/mpficacions
194
s y s t e m s s u f fe re d f r o m n e g l e c t . D u r i n g t h e l a t e 1 9 7 0 s , t h i s b e g a n t o r e v e r s e
itself—r w i t h t h e a d v e n t o f p o s t m o d e r n i s m , A f r i c a n c u l t u r a l p o r t r a i t s b e c a m e
increasin gly focused o n d iscou rse a n d sy m b o l systems, e v e n a t t h e e x p e n s e o f ig n o r­
in g a n a l o g r e p r e s e n t a t i o n s .
I t is i m p o r t a n t , h o w e v e r , t o see h o w th e s e r e s t r i c t i o n s h a v e b e e n c o n t e s t e d ,
p a r t ic u l a r ly in b la c k i n t e l l e c t u a l c o m m u n i t i e s . H o o k s ( 1 9 9 1 , 2 9 ) s u m m a r i z e s h e r
o w n r e a c t i o n to r o m a n t i c o r g a n i c i s m : “T h i s d is c o u r s e c r e a t e d t h e id e a o f ‘p r i m ­
i t i v e ’ a n d p r o m o t e d t h e n o t i o n o f a n ‘a u t h e n t i c ’ e x p e r i e n c e , s e e i n g as ‘n a t u r a l ’
t h o s e ex p re s sio n s o f b la c k life w h i c h c o n f o r m e d t o a p re - e x is t in g p a t t e r n o r s t e r e o ­
ty p e .” R o se (1993.) describ es th e h is to ry o f rap m u sic, also arising in t h e m id -1 9 7 0 s,
as n o t j u s t a r e s i s t a n c e to o r g a n i c r o m a n t i c i s m , b u t as a t e c h n o c u l t u r a l r e b e l l i o n
t h a t m a k e s D e r r i d a lo o k like G u t e n b e r g . C o r n e l W e s t , H o u s t o n B ak er, H o r t e n s e
S p il le r s , a n d H a z e l C a r b y h a v e m a d e i n t e r v e n t i o n s in A f r i c a n A m e r i c a n i n t e l ­
l e c t u a l d is c o u rs e in s i m i l a r w ay s, as h a v e w o rk s o f b l a c k s c i e n c e f i c t i o n s u c h as
G e o r g e S c h y l e r ’s Black N o M o r e , R a l p h E l l i s o n ’s invisible M a n , 2 T o n i C a d e B a m b a r a ’s T/ie S a lt E aters, S a m u e l R . D e l a n y ’s D h alg ren , a n d O c t a v i a B u t l e r ’s Xenogenesis trilogy. A n e g a l i t a r i a n v ie w o f t h e n a tu r a l/ a r ti f i c ia l d i c h o t o m y c a n b e s e e n
in b l a c k i n t e l l e c t u a l h i s t o r y r u n n i n g f r o m G e o r g e W a s h i n g t o n C a r v e r ' s c o n c e p t
o f “ G o d ’s K i n g d o m o f t h e S y n t h e t i c ” t o M u d i m b e ’s “ I n v e n t i o n o f A f r i c a . ” -*
In d e e d , C a r v e r a n d M u d i m b e ’s c o n c e p t s .a r e q u it e sim ilar; it is n o t M u d i m b e ’s c o n ­
t e n t i o n t h a t A f r i c a n u n i t y la c k s a s p i r i t u a l b o n d , b u t r a t h e r a c e l e b r a t i o n o f t h e
s p i r it o f i n v e n t i o n , w h i c h r e q u i r e s r e s i s t a n c e t o t h e E u r o p e a n c l a i m t h a t s p i r i t
c a n ex is t o n ly in c a te g o rie s o f t h e n a t u r a l . A f r i c a n a n i m i s m is m a r k e d b y a n e x t r a ­
o r d i n a r y a c c e p t a n c e o f t h e relig io u s s i g n i f i c a n c e o f artifice,'* fr o m gris-gris t o t h e
m o j o h a n d , a n d its t e c h n i q u e s fo r p a s s i n g i n f o r m a t i o n t h r o u g h t h e p h y s i c a l
d y n a m i c s o f s o u n d a n d m o v e m e n t s h o w t h a t t h i s faith in t h e p o w e r o f a n a l o g r e p ­
r e s e n t a t i o n is n o t m i s p l a c e d .
T h e p o litics of re c u rs io n
W h i l e D e r r i d a was t r a s h i n g o r g a n i c r o m a n t i c i s m , M i c h e l F o u c a u l t w a s a t t e m p t ­
in g t o d o t h e s a m e for h u m a n i s m . H is h is to r i c a l s t u d ie s d e m o n s t r a t e t h a t h u m a n ­
ist g o als o f r e c u r s i o n — t o b e s e l f - g o v e r n e d , s e l f - c o n t r o l l i n g i n d i v i d u a l s — a r e n o t
i n n o c e n t ; b u t r a t h e r d e v e l o p h i s t o r i c a l l y in c o m b i n a t i o n w i t h v a r i o u s t e c h ­
n i q u e s o f s o c i a l c o n t r o l . I n a n e ra w h e r e " s e l f - m a n a g e m e n t ” u s u a l ly m e a n s t h a t
t h e c o r p o r a t i o n y o u w o r k for h a s d e v e l o p e d i m p r o v e d t e c h n i q u e s fo r selfe x p l o i t a t i o n , it is n o t h a r d t o see w h a t F o u c a u l t is g e t t i n g a t . A s in t h e c a s e o f
D e r r i d a ’s w a r n i n g s a g a i n s t c l a i m s t h a t a n a l o g r e p r e s e n t a t i o n w ill a u t o m a t i c a l l y
le a d t o m o r e e t h i c a l li v in g , F o u c a u l t w a r n s a g a i n s t s e e i n g r e c u r s i o n as a m o r a l
The politics o f A fr ic a n fractals
f o r m u l a . 5 W h i l e A f r i c a n a n a l o g s y s te m s raise t h e p r o b l e m o f s o m e o n e m a k i n g
cla im s a b o u t w h a t is m o r e real o r m o r e n a t u r a l , A f r i c a n r e c u r s io n — espe cially t h e
r e c u r s iv e a r c h i t e c t u r e o f A f r i c a n s e t t l e m e n t s — raise s t h e p r o b l e m o f h u m a n i s t
c la im s . T o see h o w th i s c a n be a p r o b l e m , c o n s i d e r t h e f o l lo w in g tw o case s tu d ie s
o f A f r i c a n a r c h i t e c t u r e . C a p l a n ( 1 9 8 1 ) s t u d i e d t h e r e l a t i o n b e t w e e n 'h o u s i n g a n d
w o m e n ’s a u t o n o m y in T a n z a n i a . S h e d e s c r i b e d h o w t h e flex ibility o f h o u s i n g
a ll o w e d w o m e n to c r e a t e n e w h o m e s if t h e y w a n t e d a d i v o r c e , o r t o e x t e n d o ld
h o m e s if th e y w a n t e d t o s h ift t h e fa m il y s t r u c t u r e . A s in m a n y A f r i c a n s e t t l e ­
m e n ts, this self-organized h o u s in g c r e a t e d a self-sim ilar s tru c tu r e — fractals— w h i c h
a ll o w e d g r e a t e r s o c ia l s e l f - c o n t r o l fo r w o m e n . W h e n s o c i a li s m b r o u g h t m o d ­
e r n i z a t i o n p ro g r a m s , t h i s a u t o n o m y w as t h r e a t e n e d by t h e “ i m p r o v e d ” h o u s i n g
design, w h i c h s o m e tim e s re s em b le d c o n c r e t e a rm y b arracks.^ H e r e o n e w o u ld c o n ­
c l u d e t h a t f r a c ta l is b e tte r.
S t o i l e r ( 1 9 8 4 ) d e s c r ib e d a S o n g h a i t o w n in w h i c h a c a s t e s y s te m e n s u r e d
t h a t t h e b e s t la n d w as v o l u n t a r i l y g i v e n o v e r t o t h e h i g h e s t c a s t e m e m b e r s . It
was n o t a m a t t e r o f forcing p e o p le a g a in s t t h e i r will, b u t simply u n q u e s t i o n e d c o m ­
m o n s e n s e t h a t o n e s h o u l d w a n t t o b e l o c a t e d in t h e i r p r o p e r p la c e . T h i s fractal, s e lf-o rg an iz ed a r c h i t e c t u r e w as a fo r m o f s e l f - e x p l o i t a t i o n . E v e n tu a l ly sev eral
m e m b e r s o f t h e c o m m u n i t y d e c i d e d to b r e a k o u t o f th i s o p p r e s s iv e s t r u c t u r e by
b u i l d i n g h o u s e s a l o n g t h e n e w h i g h w a y . T h u s l i b e r a t i o n in th i s c a s e m e a n t
leaving th e fractal geometry, a n d lin in g u p in s tra ig h t E u clid ean fo r m a tio n — exactly
th e o p p o s ite o f th e T a n z a n ia n villa ge s t u d ie d by C a p l a n . S to l le r ’s w o rk rficely illus­
tr a t e s M i c h e l F o u c a u l t ’s w a r n i n g a g a i n s t s i m p l i s t i c h u m a n i s t fo r m u la s : selfd e t e r m i n a t i o n is n o t n e c e ss a rily l i b e r a t i n g ; ir c a n se r v e t o s u p p o r t so c ia l c o n t r o l
"
r a t h e r -t h a n resist it. M e i t h e r - f r a c t a l - n c r E u c l i d e a n g e o m e t r i e s h a v e a n y i n h e r ­
e n t e t h i c a l c o n t e n t ; s u c h m e a n i n g s arise fr o m t h e p e o p l e w h o use t h e m .
C o l o n i a l i s m a n d a r c h i t e c t u r a l fr a c ta ls
R e n e D escartes was n o t m u c h o f a h u m a n i s t ; ' i n his view self-organized a r c h i te c t u r e
is j u n k . H e m a k e s th is c le a r in h i s f a m o u s D iscourse o n Metfiodolog)1;
(TJhere is less p e rfectio n in w orks m a d e o f several pieces an d in w o rks m ade
by th e h a n d s o f several m a s te rs t h a n in t h o s e works o n w h ic h b u t o n e m a s te r
h as w orked. T h u s o n e sees chat b u il d in g s u n d e r t a k e n a n d c o m p l e te d by a
single a r c h i te c t are c o m m o n l y m o r e b eau tifu l a n d b e tt e r o rd e re d t h a n th o s e
t h a t several a rc h ite c ts h a v e tried to p a t c h up. . . . T h u s 1 im ag ined t h a t peo ple
w h o , h a v in g o n c e b e e n h a l f savages a n d h a v in g b e e n civilized o n ly gradually,
have made their laws only to th e e x te n t th a t th e in c on ven ien ce caused by crimes
195
Implications
a n d q u arre ls forced t h e m to d o so, w ould n o t b e as w ell o rd e r e d as th o s e w h o,
from th e very beg in nin g o f their c o m in g together, h a v e followed th e fundam ental
p recep ts o f so m e p r u d e n t legislator. .
(1 6 7 3 , 12)
F o r D e s c a r t e s , “se l f - o r g a n iz e d ” is s y n o n y m o u s w i t h sa v a g e s , t h e im p e r f e c ­
tio n o f b o th m aterial a n d social stru ctu re. L ack o f c o m p le te E u c lid e a n reg u lar­
ity m e a n s r a n d o m n e s s : fo r “s t r e e t s c r o o k e d a n d u n e v e n , o n e will say t h a t it is
c h a n c e m o r e t h a n t h e will o f s o m e m e n u s i n g t h e i r re a s o n c h a t h a s a r r a n g e d t h e m
th u s ” (p. 12). T h e lack o f C a r te s ia n c o o r d i n a te s in m a n y A f r i c a n s e t tl e m e n t s w ould
th u s e v id e n c e t h e i r n e e d for t h e g u id a n c e o f c o l o n i a l reas o n . A s H u l l ( 19 7 6 ) no tes,
h u g e c e n t e r s o f u r b a n life in A f r i c a w e r e i n d e e d d i s r e g a r d e d b y E u r o p e a n s as
“u n s t r u c t u r e d b u s h c o m m u n i t i e s ” o n j u s t t h e s e p r i n c i p l e s . W h i l e T i m b u k t u was
g r a n t e d c i t y h o o d d u e to its grid p a t t e r n o f s t r e e t s , t h e Y o r u b a c it ie s o f e q u a l p o p ­
u l a t i o n size a n d e c o n o m i c , t e c h n i c a l , a n d l a b o r s p e c i a l i z a t i o n h a v e b e e n d i s r e ­
g a r d e d as m e r e l y g i a n t v illa g e s d u e t o t h e i r la c k o f C a r t e s i a n reg u la rity .^ T h u s
f r a c t a l a r c h i t e c t u r e w as u s e d as c o l o n i a l p r o o f o f p r i m i t i v i s m . T h i s d e b a t e o v e r
t h e u r b a n s t a t u s o f n o n - E u c i i d e a n s e t t l e m e n t s c o n t i n u e s in t h e p o s t c o l o n i a l era
(see S c h w a b 1 9 6 5 ; L lo y d 1 9 7 3 ).
T h e o c c a s i o n a l C a r t e s i a n l i n e a r i t y in A f r i c a n a r c h i t e c t u r e t h r e w a h i t c h
i n t o t h i s c o l o n i a l j u s t i f i c a t i o n , i n 1871 t h e G e r m a n g e o l o g i s t C a r l M a u c h “d i s ­
c o v e r e d ” t h e r u i n s o f G r e a t Z i m b a b w e . S t u n n e d by t h e e v i d e n c e o f p re c ise s t o n e
c u t t i n g o n a m a s s i v e s c a le , h e p r o p o s e d t h a t t h e b u i l d i n g s w e r e n o t o f A f r i c a n
d e s i g n , b u t w e r e i n s t e a d d u e to t h e Q u e e n o f S h e b a ' s v i s i t t o S o l o m o n . T h e
R h o d e s i a n g o v e r n m e n t u se d t h i s e x p l a n a t i o n a s a p a r t o f its p r o p a g a n d a a g a in s t
B la c k r u l e ( M a c i n t o s h a n d M a c i n t o s h , 1 9 8 9 ) . A c t u a l l y , t h e y h a d m u c h less to
f e a r in t h e t r u t h : t h e s t o n e w as n o t c u t , b u t it n a t u r a l l y b r o k e i n t o l i n e a r
s h e e t s ( a f t e r h e a t i n g ) d u e t o its g e o lo g i c p r o p e r t i e s . M o r e o v e r , m o s t o f t h e o u t ­
sid e w a lls w e r e o r i g i n a l l y c o v e r e d w i t h s m o o t h clay, c r e a t i n g a n o n l i n e a r s e t
o f s c a l i n g s h a p e s ( w h i c h C o n n a h [ 1 9 8 7 ] r e f e r s t o as " r a n d o m c u r v e d f o r m s " ) .
T h i s is n o t t o d i m i n i s h t h e r e m a r k a b l e t e c h n o l o g i c a l sk ill o f t h e c o n s t r u c t i o n ,
h u t t o p o i n t o u t t h a t o n e c u l t u r e ’s s i g n for “a r t if i c ia l " c a n b e a n o t h e r ’s sign for
" n a t u r a l . ” E u c l i d e a n v ersus f r a c ta l d o e s n o t n e c e s s a r il y m e a n artificial versus n a t ­
ural; t h a t , t o o , is c u l t u r a l l y i n f l u e n c e d .
D u rin g th e d e v e lo p m e n t of co lo n ia l cities, th e c h a o s o f A frican a rc h ite c ­
t u r e w a s u se d as b o t h s y m b o l a n d s y m p t o m o f E u r o p e a n fears o v e r s o c ia l c h a o s .
P e n n a n t (1 9 8 3 ) provid es a n e x a m p l e o f th is c o n c e r n a b o u t p r o p e r s e t tl e m e n t g e o m ­
e tr y i n his e x a m i n a t i o n o f c o l o n i a l d e v e l o p m e n t in M a la w i: " T h e lan g u a g e o f th is
1930s policy discourse is significant. M e d i c a l e x p e r t s w r o te o f ‘in v e s tig a tio n s ’ s h o w ­
in g ' u n q u e s t i o n a b l y ’ a n d o f ‘a b u n d a n t p r o o f . ’ . • • L a y E u r o p e a n s s h o w e d ‘c o n ­
c e r n , ’ ‘a l a r m , ’ a n d ‘h o r r o r . ’ A f r i c a n s , w i t h t h e i r ‘p r i m i t i v e h a b i t s , ’ o f ‘p r o m is c u o u s
The politics o f A/riccm fractals
d e f e c a t i o n ’ f o r m e d a ‘f l o a t i n g ’ o r ‘s c a t t e r e d ’ p o p u l a t i o n in n e e d o f ‘c o n t r o l ’ a n d
' s u p e r v i s i o n ' in a ‘p r o p e r l y l a i d - o u t v il la g e o r l o c a t i o n . ’ ”
I n t h e a b o v e c a s e w h e r e “p r i m i t i v e ” m i x e s - w i t h m o d e r n , t h e f r a c t a l t r a d i ­
t i o n was a t h r e a t . B u t k e p t in w h a t c o l o n i a l i s t s t h o u g h t o f as its n a t u r a l ro le , it
c o u ld m a k e fractal s e t t l e m e n t s a p p e a r to b e n e f i t th e c o lo n i a l e n te r p ris e . T h e n o v ­
elist K a r e n B l i x e n ( l s a k D i n e s e n ) , i n O u t o f A fr ic a ( 1 9 8 8 ) , d e s c r ib e d h e r a t t e m p t s
to lay o u t grids for A f r i c a n w o r k e r s ’ h o u s e s o n h e r r a n c h . T h e y refu sed t o fo llo w
t h e s e l i n e a r i n s t r u c t i o n s a n d fit t h e i r h o u s e s in p a t t e r n s m a t c h i n g t h e ir r e g u la r
c o n fig u ra tio n o f t h e la n d . T h a t s u c h e c o lo g i c a l fit c o u ld b e q u i t e efficient was n o t,
h o w e v e r, lost o n t h e c o lo n ists. “T h e s q u a t t e r s ’ la n d was m o r e in te n s e ly a liv e t h a n
t h e re s t o f t h e fa rm , a n d w a s c h a n g i n g w i t h t h e s e a s o n s ” (p . 9 ). A r c h i t e c t u r a l
f r a c ta ls c o u l d b e p a r t o f c o l o n i a l r o m a n t i c i s m as lo n g as t h e y e n s u r e d a s u p p ly
o f s e l f - s u p p o r t i n g w o rk e rs .
E v e n in th e case o f so cial c o n tr o l, in d ig e n o u s fractals co uld be utilized. British
c o l o n i a l po lic y , fo r e x a m p l e , a t first f a i le d in cases w h e r e t h e r e w as a d e c e n t r a l ­
ized n e t w o r k r a t h e r t h a n a la r g e h i e r a r c h y . T h i s w as a p p r o a c h e d in t h e c ase o f
t h e I b o w i t h a sy s te m o f “ i n d i r e c t r u l e ” b a s e d o n “w a r r e n t c h ie f s " ( I s ic h e i 1 9 7 6 ).
T h e Ib o a u t o n o m y o f s e l f - o r g a n iz a t io n w as t u r n e d a g a in s t t h e m ; in a s e n s e it was
g ra s s -ro o ts c o l o n i a l i s m . T h e a r c h i t e c t u r a l e q u i v a l e n t o f t h i s s y s te m c a n b e seen,
i n a m a n u a l for c o l o n i a l - e r a h o u s i n g d e s i g n s f r o m t h e A g e n c y for I n t e r n a t i o n a l
D e v e lo p m e n t ( H in c h c l if f 1946, 3 1 ). H e r e t h e Ib o s’ fractal s e t t l e m e n t p a tt e r n (radial
houses a r o u n d a c e n t e r in e a c h v illa g e , r a d i a l v illages a r o u n d t h e s e t t l e m e n t c e n ­
t e r ) is t i d i e d u p t o s u i t E u r o p e a n c o n c e p t i o n s o f s y m m e t r y w h i l e r e t a i n i n g th e
o v e ra ll i n d i g e n o u s f r a c ta l s t r u c t u r e .
F r a c t a l s a n d r a c ia l r e d is tr ic tin g
In th e i n t r o d u c t i o n t o his s e m in a l F ra cta l G e o m e tr y o f N a tu r e , B e n o i t M a n d e l b r o t
e x a m i n e s s o m e o f t h e d i s p a r a g i n g c o m m e n t s t h a t w e r e m a d e a b o u t t h e e a rly
fractal fo rm s o f G e o r g C a n t o r , H e l g e v o n K o c h , a n d o th e r s . R e j e c t e d as “b iz a r r e ”
a n d " t o r t u r o u s , ” t h e s e “d r a g o n s ” w e r e c o n s i g n e d co t h e o d d i t i e s s e c t i o n a t t h e
e n d o f t h e few m a t h t e x t s t h a t w o u l d e v e n c o n s i d e r t h e m . S t r i k i n g l y s i m i l a r
la n g u a g e h a s b e e n u s e d t o r e j e c t t h e o u t l i n e s o f v o t i n g d i s t r i c t s t h a t w e r e
a lt e r e d t o i n c l u d e la rg e r A f r i c a n A m e r i c a n p o p u l a t i o n s , a n d t h e s e d o i n d e e d
a p p e a r t o b e f r a c ta l s (fig. 1 2 .1 ) .8 W e r e t h e c o u r t s as m i s t a k e n l y h a s t y t o d i s r e ­
gard f r a c ta l s as m a t h e m a t i c i a n s w e re ?
T h e E u c l i d e a n s h a p e o f v o t i n g d i s t r i c t s is n o t a n a r b i t r a r y s a m p l i n g — th is
Jj?
could o n ly b e d o n e by r a n d o m l y s e l e c t i n g v o t e r s fr o m e v e r y w h e r e in t h e s ta te .
A c c o r d i n g to t h e 199 3 S u p r e m e C o u r t r u l i n g in Shaur v. R en o , it is m e a n t to
198
Implications
Q eo rg ia c o n g r e s s io n a l d i s t r i c t x i
trr 1 9 9 2
S i m i l a r i t y o f i r r e g u l a r r e d i s t r i c c i n g p a r c e r n c a n b e s e e n a t m u l t i p l e s c a le s .
{O r ig in a l
maps
c o u r te s y o f
die
I
|
C a r l V i n s o n I n s t i t u t e , p r o [/ h ie h ig h lig h t s b y t h e a u t h o r . )
d e s i g n a t e a g e o g r a p h i c lo c a l e in w h i c h " s h a r e d i n t e r e s t s ” i n f o r m t h e v o te . T h e
;j
o b j e c t i o n to c r e a t i n g a d i s t r i c t in w h i c h c o n t o u r s a re “p r e d o m i n a n t l y m otivated"
b y r a c e is t h a t it c r e a t e s a b ia s in t h e s a m p l i n g o f t h e g e o g r a p h i c l o c a t i o n . T h is |
w o u ld c e r t a i n l y b e t h e c a s e if w e w e r e to t a k e a r a n d o m s a m p l i n g , s e p a r a te d vot- .jf*
ers by e t h n i c i t y , a n d t h e n d e s i g n a t e d th o s e e t h n i c g r o u p s as t h e v o t i n g districrs. f„
H o w e v e r , if s o m e e t h n i c g ro u p s a re d i s t r i b u t e d in E u c l i d e a n s e t t l e m e n t patterns, L*
a n d o t h e r s in f r a c t a l s e t t l e m e n t p a t t e r n s , t h e n w h y c o n s i d e r E u c l i d e a n district >
s h a p e s t o b e u n b i a s e d , a n d f r a c t a l d i s t r i c t s h a p e s t o b e b i a s e d ? I d o n o t know if«
A f r i c a n A m e r i c a n h o u s i n g d i s t r i b u t i o n s a r e m o r e f r a c t a l t h a n o t h e r s — niuT
e v e n if th e y are, I w o u ld n o t n e c e s s a r il y a s s u m e a c u l t u r a l c o n n e c t i o n to African,'
f r a c ta ls — b u t t h e fa c t t h a t we n o w k n o w o f s o c i e t i e s in w h i c h f r a c ta l s e t t l e m e n t
p a t t e r n s a re b e a u t i f u l f u s i o n s o f f o r m a n d f u n c t i o n s u g g e s ts t h a t w e m i g h t
r e c o n s i d e r t h e i r p o t e n t i a l ro le in A m e r i c a n p o li ti c s .
A f r i c a n fr a c ta ls f r o m c u l t u r a l v is io n a rie s
F ra c t a ls a n d c h a o s t h e o r y h a v e b e e n i n c r e a s i n g l y m e n t i o n e d in che h u m a n i t i e s
as e i t h e r a to o l o r a n o b j e c t o f c u l t u r a l a n a ly s is , b u t t o o o f t e n t h e a p p r o a c h o f
t h e s e s t u d i e s h a s le ft t h e im p r e s s i o n o f m a t h e m a t i c a l i n k b lo ts a l l o w i n g w rite rs
t o see w h a t e v e r t h e y p le a s e . L y o ta rd ( 1 9 8 4 ) sa w f r a c ta l g e o m e t r y as c o n t r i b u t ­
in g t o a “p o s t m o d e r n c o n d i t i o n ” w h o s e c o n t r a d i c t o r y n a t u r e w o u l d d i s r u p t
a u t h o r i t a r i a n c e r t a in t y ; a m o r e c a u t i o u s v e r s i o n o f t h i s th e s is is floated in D eleuze
a n d G u a t t a r i ( 1 9 8 7 ) . A t le a s t tw o a u t h o r s ( S t e e n b u r g 19 91; A r g y ro s 1 9 9 1 )
h a v e a rg u e d t h a t fractals a n d o t h e r b r a n c h e s o f c h a o s th e o r y h a v e c re a te d a d ir e c t
c h a l l e n g e t o p o s t m o d e r n i s m , i n t e g r a t i n g t h e d i s r u p t i o n s it c r e a t e d . P o r u s h
( 1 9 9 1 ) a n d o t h e r s insist t h a t " d e t e r m i n i s t i c c h a o s " is a t t e m p t i n g t o s u b s t i t u t e
a f e e l i n g o f free will for fa ta lity .^ S o b c h a c k ( 1 9 9 0 ) s u g g e s ts t h a t it im p li e s “a n
e m b r a c e o f ir r e s p o n s ib il it y in a w o r l d a l r e a d y b e y o n d c o n t r o l . ” W h e n S o b c h a c k
c it e s P e i t g e n a n d F r e e m a n in h e r c o n d e m n a t i o n o f c h a o s t h e o r y as a d e n i a l o f
" t h e sp e c ific ity o f h u m a n e m b o d i m e n t a n d h i s t o r i c a l s i t u a t i o n , ” I c a n ’t h e l p b u t
t h i n k o f P e i t g e n ’s fractal g e o m e t r y c o u rs e a t t h e U n i v e r s i t y o f C a l i f o r n i a at S a n t a
C ruz, w h e re h e c o m m e n te d o n G e r m a n m a t h e m a t ic i a n s w h o a lte re d th e ir
c a r e e r s t o o p p o s e N a z i a n t i - S e m i t i s m o r s u p p o r t p e a c e effo rts ; o r o f F r e e m a n ’s
( 1 9 8 1 ) u se o f M a r t i n L u t h e r K in g in h is d is c u s s i o n o f c h a o s in n e u ro p h y s io lo g y .
H o w c a n w e c r i t i q u e t h e w o rk o f c h a o s t h e o r i s t s as la c k i n g h i s t o r i c a l sp e c ific ity
a n d e m b o d i m e n t if w e ig n o r e t h e i r o w n h i s t o r i e s a n d b o d ie s ?
H a y le s ’s C h a o s B ound ( 19 9 0 ) t o o k a m o r e s u b t le a p p r o a c h . L ike P o ru s h a n d
o t h e r s , H a y l e s ’s li te r a ry m e t h o d a ll o w s h e r t o g lid e far t o o easily b e t w e e n u n ­
r e l a te d ideas; by t h e ti m e s h e h a s to s s e d t o g e t h e r q u a n t u m th e o r y , e n t r o p y , a n d
G o d e l ’s t h e o r e m w i t h d e c o n s t r u c t i o n a n d “h o l i s m , ” o n e c a n o n ly c o n c l u d e t h a t
a n y c o m p l i c a t e d idea c a n b e a m e t a p h o r for a n y o t h e r c o m p l i c a t e d id ea. B u t h e r
d e ra i le d a n a ly s is o f li te r a ry w ork s, s h o w i n g d e e p p a r a l le l s b e t w e e n se lf-re fle x iv e
w ritin g a n d s e l f-r e fe re n tia l m a t h e m a t i c s , su g g es ts t h a t w h e n g r o u n d e d in specific
l o c a t io n s t h e fu s io n o f f r a c ta l g e o m e t r y a n d c u l t u r a l i n t e r p r e t a t i o n c a n h e p r o ­
fo u n d ly r e w a r d i n g .
P a u l G i lr o y make's e x p li c it use o f f r a c ta l s in his p o r t r a i t o f t h e d iv e r s ity a n d
d y n a m i c is m w i t h w h i c h b o t h t r a d i t i o n a l A f r i c a a n d t h e A f r i c a n d i a s p o r a h a v e
o rg a n iz e d t h e i r c r o s s - c u l t u r a l flow s. T h e r e c u r s i v e c o n s t r u c t i o n o f h is B la c k
A t l a n t i c c a n b e s e e n , fo r e x a m p l e , in t h i s q u o t e fr o m J a m e s B ro w n o n a v is it to
Implications
h e a r F e la K u t i in N i g e r i a : “[B]y t h i s t i m e h e w a s d e v e l o p i n g A f r o - b e a t o u t o f
A f r i c a n m u s ic anc! f u n k . H is b a n d h a d a s t r o n g r h y t h m ; I t h i n k C l y d e p i c k e d u p
o n it in h is d r u m m i n g , a n d B o o tsy d u g it t o o . S o m e o f t h e id e a s m y'bancl w as g e t­
t i n g fr o m t h a t b a n d h a d c o m e fro m m e i n t h e first p l a c e , b u t t h a t w as o k a y w i t h
m e . I t m a d e t h e m u s ic t h a t m u c h s t r o n g e r ” ( 1 9 9 3 , 199)G i l r o y c i t e s t h e i m p a c t o f t h e V i r g i n i a J u b i l e e S i n g e r s o n t o u r in S o u t h
A f r i c a in 1 8 9 0 , t h e r e t u r n o f sla v e s f r o m B razil t o N i g e r i a , t h e R a s t a f a r i c u l t u r e
in Z i m b a b w e , a n d o t h e r e x a m p l e s o f “m u t a t i o n s p r o d u c e d d u r i n g its c o n t i n g e n t
l o o p s a n d f r a c ta l tr a j e c t o r i e s ." P e r h a p s h i s m o s t r a d i c a l m o v e is a c l a i m for d ias p o r i c m i x i n g w i t h J e w is h c u l t u r e — W .E .B . D u B o is p a s s i n g fo r a Je w t o m a i n ­
t a i n safety in E a s te r n E u ro p e , t h e u se o f t h e E x o d u s t h e m e in M a r t i n L u t h e r K ing,
J r., a n d M a r c u s G a r v e y , a n d E. W . B l y d e n ’s c h i l d h o o d in a J e w is h c o m m u ­
n i t y . 1^1 T h e f r a c ta l i m a g e r y w o r k s i n m a n y d i f f e r e n t w ay s fo r G i l r o y — fr o m t h e
tu r b u le n t m e t a p h o r o f h y b rid ity to th e c o n c r e t e d e s c r ip ti o n o f s h ip s ’ p a th s
a n d t r a v e l e r s ' r o u t e s ( o r “r o o t s / r o u t e s ” as h e p u t s i t ) . 11 W h i l e m u s ic is, w i t h ­
o u t d o u b t , G i l r o y ’s s t r o n g e s t e x a m p l e , h e d o e s s l ip i n t o t h e p r o b l e m a t i c la b e ls
o f r e p r e s e n t a t i o n a l v e r s u s “n o n r e p r e s e n t a t i o n a l ” r a t h e r t h a n d i g i t a l v e r s u s
a n a l o g , 12 b u t h e m a k e s it c l e a r t h a t t h e m u s i c r e v e r b e r a t i n g a c ro s s h i s B la c k
A t l a n t i c is n e i t h e r p u r e n o r n a t u r a l .
W h i l e G i l r o y is p r i m a r i l y f o c u s e d o n f r a c t a l s as s p a t i a l r e p r e s e n t a t i o n s o f
b l u r r e d b o u n d a r i e s , h e a ls o b ri e fly m e n t i o n s t h e i r p o t e n t i a l for “ a s t r i k i n g
im a g e o f t h e s c o p e o f a g e n c y w i t h i n r e s t r i c t e d c o n d i t i o n s ” ( 1 9 9 3 , 2 3 7 0 2 8 ) ; t h a t
is, t h e a b i l i t y f o r g e o m e t r i c e x p a n s i o n w i t h i n b o u n d e d s p a c e b e c o m e s a n a n a l ­
ogy for o p p o s i t i o n a l p o litic a l e x p a n s i o n in h u m a n b o n d a g e . T h e m e t a p h o r is c a r ­
r i e d to a m o r e e x a c t i n g r e l a t i o n s h i p i n G a r y V a n W y k s s t u d y o f S o t h o - T s w a n a
m u r a l s u n d e r t h e a p a r t h e i d s y s te m o f S o u t h A f r i c a . V a n W y k ( 1 9 9 3 ) f o u n d t h a t
t h e litem a , o r t h e h o u s e p a i n t i n g p a t t e r n s o f t h e S o t h o - T s w a n a w o m e n , u tiliz e
a l t e r n a t i o n s o f ir r e g u l a r i t y a n d r e g u l a r i t y a t s e v e r a l s c a le s , s o m e t i m e s re s u lti n g
in a r e s e m b l a n c e t o f r a c ta l p a t t e r n s . N o t i n g t h a t t h e s c a l i n g is a s s o c i a t e d w i t h
t h e g e o m e t r i c s t r u c t u r e o f flo w e rs, a n d f lo w e r s w i t h t h e r e g e n e r a t i v e p o w e r of
w o m e n ( b o t h s p i r i t u a l l y a n d in s o c ia l s t r u g g l e s ) , V a n W y k ’s e t h n o g r a p h y c o n ­
c l u d e s t h a t t h e m u r a l s e x p r e s s e d p o l i t i c a l o p p o s i t i o n to a p a r t h e i d h y p r o v i d i n g
a v is u a l a n a l o g in w h i c h “ a w o m a n c a n b e s e c r e t i v e w h i l e a t t h e s a m e t i m e h i d ­
in g n o t h i n g ” ( D e l e u z e a n d G u a t t a r i 1 9 8 7 , 2 8 9 —2 9 0 ) .
A l t h o u g h t h e w o rd “ fr a c ta l " is n o w h e r e m e n t i o n e d in h is t e x t , A n t o n
S h a m m a s ’s n o v e l A ra b esq u es ( 1 9 8 8 ) is a n e x e m p l a r o f n o n l i n e a r in s ig h t a n d re c u r­
s iv e c u l t u r a l c o m m e n t a r y . H e a v e r ( 1 9 8 7 ) a n a ly z e s t h e n o v e l t h r o u g h t h e N o r t h
A f r i c a n a r t i s t i c fo rm o f t h e “a r a b e s q u e , ” a n d s h o w s h o w S h a m m a s h a s used th is
f r a c ta l t o s u s t a i n t h e c y c lic t i m e a n d m u l t i p l e i d e n t i t i e s r e q u i r e d to a r t i c u l a t e a
FIGURE I 2.2
R e l i g i o n s in s t i t u t i o n s in t h e m a p o f C a iro , 1 8 9 8
202
Jm/)Iicauons
p o li ti c a l flexibility c ru c ia l t o th e su rv iv al o f c u lt u r a l diversity. “A s a n 'Israeli A r a b , ’
S h a m m a s is a m e m b e r o f a m i n o r i t y g r o u p — b u t as a C h r i s t i a n , h e falls o u t s i d e
t h e Is la m i c m a i n s t r e a m o f t h e m i n o r i t y . . .• . O n t h e o t h e r h a n d , 'fie w r it e s in
H e b r e w , t h e la n g u a g e o f t h e d o m i n a n t l y J e w is h c u l t u r e , w h i c h is it s e l f a m i n o r ­
ity w i t h i n t h e p r e d o m i n a n t l y A r a b M i d d l e E a s t ” (p . 4 9 ) .
S u c h r e c u r s i v e n e s t i n g is e m p h a s i z e d t h r o u g h o u t t h e t e m p o r a l flow ,
n a r r a t i v e s t r u c t u r e , a n d c o n c e p t u a l d y n a m i c s o f t h e n o v e l . H e a v e r su g g es ts t h a t
t h e “ n o n m i m e t ic g e o m e tric a l a b stra c tio n s o f th e a ra b e s q u e ” are a spatial m o d el
f o r S h a m m a s . H e n o t e s t h a t in p a r t t h e s e c y c li c r e e n t r i e s a c t t o n e g a t e o n e
a n o t h e r ; u n d e r m i n i n g , fo r e x a m p l e , t h e fr u itle s s a r g u m e n t o f “1 w a s h e r e f i r s t .”
B u t n e g a t i o n is n o t t h e o n l y m e a n i n g b e h i n d t h e a r a b e s q u e , as H e a v e r p o i n t s
o u t in a p a s s a g e t h a t tie s I s l a m i c s o c i a l s t r u c t u r e t o a n a l o g r e p r e s e n t a t i o n ,
r e c u r s i o n , a n d t h e s c a l i n g p r o p e r t i e s o f fra ctals.
T h e a ra b e s q u e does n o t serv e o n ly a n e g a tiv e , c r itic a l f u n c t i o n ; it also bears a
p o sitiv e , u t o p i a n message. It acts as a n a n a lo g u e , in t h e a re a o f visual arts, to
t h e p o s itio n o f Islamic “c o n t r a c t u a l i s m ” in t h e so c ia l s p h e r e . . . . In c o n tr a s t
t o w e ste r n c o rp o r a tiv is m , w ith its p re f e re n c e for h i e r a r c h i c a l s tru c tu r e s in
w h i c h a lim ite d n u m b e r o f c o n c lu s io n s are d r a w n from a lim ite d n u m b e r of
premises (o n th e m odel o f geometry), th e cyclical rh y th m s o f th e arabesque could
w ell be said to c h a r a c t e r i z e a n “ i n d e f i n i t e l y e x p a n d a b l e ” s t r u c t u r e . T h e
arabesque pro vides a fram ew ork w ith in w h ic h it b e c o m e s possible to reduce the
a p p aren tly “c h a o tic variety o f life’s reality” to m an a g e a b le p ro po rtio ns, yet w i t h ­
o u t “arb itra rily s e t ti n g b o u n d s to it.”
( H e a v e r 1987, 61)
C l e a r l y , w h e n H e a v e r re f e rs t o t h e l i m i t i n g d a n g e r s o f a " m o d e l o f g e o m ­
e t r y ” h e is t h i n k i n g o f E u c l i d e a n s t r u c t u r e s ; i t - i s t h e f r a c t a l geo m et-r.y-of-the
a r a b e s q u e w h i c h c o n v e y s t h e h o p e f u l m e s s a g e o f S h a m m a s . I n c h a p t e r 2 we
e x a m i n e d t h e a r a b e s q u e b r a n c h e s o f s t r e e ts t h a t a p p e a r in a m a p o f C a i r o , E gypt.
I n a n o t h e r s e c t i o n o f t h i s m a p (fig. 1 2 . 2 ) , a w i d e d i v e r s i t y o f r e l i g i o u s i n s t i ­
tu tio n s flo w er at th e en d s o f th e se b ra n c h e s, a tte s tin g to th e p o sitiv e p o t e n ­
t i a l o f f r a c t a l s in c u l t u r a l p o l i t i c s .
CHAPTER
J3
— Fractals
-------------------------------------in the----------------------------------------------—European---------------------------------- ---------—history of--------------------------------------------mathematics------------------------------------------
A n t h r o p o l o g i s t s h a v e r e c e n t ly t a k e n a n in c r e a s i n g i n t e r e s t in t h e c u l t u r a l a n a l y ­
sis o f E u r o - A m e r i c a n s o c i e ti e s . I n p a r t t h i s is a r e a c t i o n to t h e m a n y d e c a d e s o f
focus o n i n d i g e n o u s so cieties, as if t h e i r b e h a v i o r re q u ire d e x p l a n a t i o n w h ile t h a t
o f E u r o p e a n s w as s e l f - e v i d e n t . A t first t h i s “ re fle x iv e e t h n o g r a p h y ” s o u n d e d like
a n in g e n i o u s way t o t u r n t a b l e s o n s o m e v e ry t r o u b l i n g a s p e c t s o f a n t h r o p o l o g ­
ical a u t h o r i t y , b u t it t o o h a s d r a w b a c k s . O c c a s i o n a l l y o n e s u s p e c ts a h i d d e n s ig h
o f re l ie f fro m a n t h r o p o l o g i s t s w h o d e c i d e t h e y c a n p l a c e t h e m s e l v e s o n t h e c u t ­
tin g ed g e by “stu d y in g th e i r o w n t r ib e ” (ju st as c y b e r e t h n o g r a p h y s o m e tim e s seem s
s u s p ic io u s ly c o n v e n i e n t ) . N e v e r t h e l e s s , t h e r e is a n i m p o r t a n t p l a c e f o r a n t h r o ­
po log ical stu dies o f E u ro - A m e r ic a n s . It w o u ld b e a n u n b a l a n c e d p o r t r a it if we w ere
to see A f r i c a n fra c ta ls in n e e d o f c u l t u r a l an a ly s is, a n d W e s t e r n fractals as m e re ly
self-evident m ath em atics.
A c u l t u r a l h i s t o r y o f E u r o p e a n fr a c ta ls
A n c i e n t G r e e k p h i l o s o p h y is o f t e n r e m e m b e r e d fo r P l a t o ’s r a t i o n a l r e a l m o f
u n c h a n g i n g , s t a t i c fo rm s. B u t in t h e h i s t o r y o f m a t h e m a t i c s , it is i m p o r t a n t to
c o n s i d e r o t h e r i n t e l l e c t u a l c u r r e n t s in t h a t so cie ty, in p a r t i c u l a r t h e p a r a d o x e s
o f t h e p h i l o s o p h e r Z e n o o f E le a a n d d i e d is c o v e r y o f i r r a t i o n a l n u m b e r s by t h e
P y th a g o r e a n s .
203
Implications
204
A c c o rd in g to a n c ie n t h isto rian s, P y thagoras o f S a m o s g a th e re d k n o w led g e
in E g y p t a n d B a b y l o n in t h e s i x t h c e n t u r y
b .c
.e .
a n d e sta b lish e d a s e c r e t s o c i­
e t y in M a g n a G r a e c i a { w h a t is n o w s o u t h e a s t e r n I t a l y ) . H i s d is c ip les,...in c!u d in g o n e o f t h e first r e c o r d e d w o m e n m a t h e m a t i c i a n s , T h e a n a ? s w o r e a n o a t h
t o m a i n t a i n s t r i c t d i e t a r y r e g u l a t i o n s , se c r e c y , a n d a re l ig i o u s f a i t h i n n u m b e r s .
T h e P y th a g o re a n co sm o lo g y w as a h a r m o n io u s u n ity based o n w h o le n u m b e r s
( i , 2, 3 . . .) a n d t h e i r r a t i o s ( f r a c t i o n s s u c h as 2 /3 , 5 /2 , e t c . ) . F r o m t h e m o t i o n
o f h e a v e n l y b o d ie s to t h e laws o f m u s ic , t h e y f o u n d in c r e a s i n g e v i d e n c e fo r t h e i r
a r i t h m e t i c r e l i g i o n . B u t a t s o m e p o i n t — a n d m u c h i n k h a s b e e n s p i l l e d in t h e
d a t e d e b a t e — c a m e t h e d i s c o v e r y o f w h a t t h e y t e r m e d alogos, t h e “ i r r a t i o n a l ”
n u m b e r s ( a n a m e t h a t w e h a v e k e p t t o t h i s d a y ) . U n l i k e w h o l e n u m b e r r a t io s ,
w h i c h e i t h e r t e r m i n a t e ( 5 / 2 = 2 . 5 0 0 0 0 . . .) o r r e p e a t ( 1 3 / 1 1 = 1 . 1 8 1 8 1 8 ... .),
i r r a t i o n a l n u m b e r s , s u c h a s t h e s q u a r e r o o t o f t w o ( 1 . 4 1 4 2 1 3 5 6 . . •), c o n t i n u e
to c h a n g e fo rev er. T h e y c a n n o t b e e x p r e s s e d as t h e r a t i o o f t w o fi n it e i n t e g e r s ;
as g e o m e t r i c m a g n i t u d e s t h e y a r e “ i n c o m m e n s u r a b l e li n e s .” T h e m o s t p la u s ib l e
o r i g i n for t h e P y t h a g o r e a n k n o w l e d g e o f i r r a t i o n a l s is i n a n a t t e m p t t o d e t e r ­
m i n e t h e d i a g o n a l o f a p e n t a g o n . If y o u w is h to d e t e r m i n e t h e r a t i o o f d i a g o ­
n a l t o sides fo r a r e g u l a r h e x a g o n , it is q u i t e easy, b e c a u s e all d i a g o n a l s i n t e r s e c t
in t h e c e n te r . B u t t h e d i a g o n a l s o f a p e n t a g o n ju s t form a s m a ll e r p e n t a g o n . S i n c e
t h e s a m e o p e r a t i o n c a n b e r e p e a t e d a g a i n a n d a g a i n , a n i r r a t i o n a l n u m b e r is
e x p o s e d . 1 T h i s “ ir ra tio n a lity " in th e h e a r t o f t h e i r s p iritu al p r a c t ic e w as t o o m u c h ,
a n d m em b ers o f t h e g ro u p ag re e d n o t to rev eal th is s e c re t o n p a in o f d e a th .
Z e n o o f E le a (fl. c a . 4 5 0
b .c.e.),
a d iscip le o f P a rm e n id e s , p ro v id e d a
•series o f p a r a d o x e s t h a t a ls o c o n f l i c t e d w i t h t h e n u m e r i c a l f a i t h o f t h e day. H is
m o s t f a m o u s e x a m p l e is a r a c e b e t w e e n A c h ilE e s,_ th e f l e e t e s t o f r u n n e r s , a n d a
t o r t o i s e . A l l o w i n g t h e t o r t o i s e a s p o r t i n g c h a n c e , A c h i l l e s g iv e s it a c o n s i d e r ­
a b l e le a d ( l e t ’s sa y
jo o
f e e t ) . B u t by t h e t i m e h e c a u g h t u p t o t h e p l a c e w h e r e
t h e t o r t o i s e b e g a n , it h a d a l r e a d y a d v a n c e d 10 fe e t. By t h e t i m e h e g a i n e d t h a t
d i s t a n c e , t h e t o r t o i s e h a s c r e p t f o r w a r d o n e fo o t. Z e n o c o n c l u d e d t h a t a l t h o u g h
e x p e r i e n c e p r o v e s o t h e r w i s e , lo g ic a lly t h e t o r t o i s e s h o u l d w i n t h e r a c e . B a c k in
450
b.c.e.,
t h e s e p a r a d o x e s o f i n f i n i t y ( a n d i n f i n i t y ’s flip s id e , t h e i n f i n i t e s i m a l )
w e r e u n n e r v i n g , e v e n s h o c k i n g to p h i l o s o p h e r s w h o d e p e n d e d o n r a t i o n a l i t y as
t h e g a t e w a y t o re l ig i o u s p e r f e c t i o n .
I n P la t o 's p h i l o s o p h i c c o sm o lo g y , s p iritu a l p e r f e c t i o n w as s e e n as t h e h i g h e r
le v e l o f t r a n s c e n d e n t stasis, a n d il lu s io n a n d i g n o r a n c e w e r e t h e r e s u l t o f life in
o u r l o w e r r e a l m of. c h a n g i n g d y n a m i c s ( “f l u x , ” w h i c h in a n c i e n t G r e e k also
m e a n s “d i a r r h e a ” ). S e v e r a l o f P l a t o ’s s t u d e n t s a t t e m p t e d t o i m p r o v e t h e m a t c h
b e t w e e n t h e c h a r a c t e r i s t i c s o f m a t h e m a t i c s a n d t h e r e q u i r e m e n t s o f che s t a t i c
realm . E u d o x u s p ro p o s e d to e lim in a te irra tio n a ls hy r e d e f in in g “ r a tio ,” an d
Fractals in the European history o f mathematics
205
X en o c ra c e s in t r o d u c e d a d o c t r i n e o f in d iv is ib le s t o o p p o s e Z e n o ’s p ara d o x es . A r i s ­
t o t l e , n o t i n g t h a t in f in ity + in f in ity = in fin ity , s u g g e s te d t h a t t h i s “s e l f - a n n i h i ­
l a t i n g ” c h a r a c t e r i s t i c c o u l d b e e l i m i n a t e d b y r e s t r i c t i n g r e f e r e n c e t o in f in ity as
a l i m i t to b e a p p r o a c h e d , r a t h e r t h a n as a t h i n g itself, a p r o p e r o b j e c t o f m a t h e ­
m a t i c a l 'inquiry.2
'
T h e P l a t o n i c re f o r m w as q u i t e s u c c e s sfu l, a n d as a r e s u l t m a t h e m a t i c i a n s
in t h e f o l l o w i n g c e n t u r i e s p a id li t t i e a t t e n t i o n t o t h e k i n d s o f r e c u r s i o n t h a t led
to Z e n o ’s t r o u b l i n g i n f in i te regress. O n e e a r l y e x c e p t i o n w as t h a t o f L e o n a r d o
F i b o n a c c i in t h e t w e l f t h c e n tu r y . H e i n t r o d u c e d t h e first re c u r s iv e series s h o w n
to b e o f use in m o d e l in g th e n a t u r a l w orld. I n c h a p t e r 7 w e saw t h a t th e F ib o n a c c i
series a p p e a r s t o h a v e b e e n u tiliz e d in t h e t e m p l e a r c h i t e c t u r e a n d w e i g h t b a l­
a n c e s o f a n c i e n t E g y p t. T h e r e m a y a c t u a l l y b e a c o n n e c t i o n b e t w e e n t h e tw o.
W h i l e li t t l e b i o g r a p h i c a l m a t e r i a l is a v a i l a b l e , G i e s a n d G i e s ( 1 9 6 9 ) a n d o t h e r
s o u r c e s h a v e p u t t o g e t h e r a g o o d a c c o u n t o f w h a t life w as p r o b a b l y lik e for
y o u n g L e o n a r d o o f P isa . F o ll o w i n g s c h o o l i n g in P isa , in w h i c h a r i t h m e t i c was
larg ely b a s e d o n t h e L a t i n w r i t i n g s o f B o e t h i u s ( c i r c a 5 0 0
c
.e .),
L e o n a r d o ’s
fath er s e n t for h im from th e N o r t h A fric a n city o f Bugia (B ou gie). T h e r e h e learned
t h e I n d i a n p l a c e - v a l u e n o t a t i o n ( p r o b a b l y t h r o u g h A r a b i c so u r c e s ). H e was
i n s p ir e d by t h i s i n n o v a t i o n a n d t r a v e l e d a l o n g t h e M e d i t e r r a n e a n t o C o n s t a n ­
t i n o p l e , E g y p t, S y ria, S icily, a n d P r o v e n c e , c o l l e c t i n g m a t h e m a t i c a l k n o w l e d g e
fr o m b o t h s c h o l a r s a n d o r d i n a r y m e r c h a n t s .
T h e r e s u l t i n g t e x t , Liber A b a c i ( B o o k o f t h e A b a c u s ) , h a s a s t r o n g Is lam ic
in f lu e n c e . L e v e y ( 1 9 6 6 ) , for e x a m p l e , s h o w s t h a t m a n y o f a b u K a m i l ’s s i x t y - n i n e
p r o b l e m s c a n b e f o u n d in L e o n a r d o ’s t e x t . B u t t h e F i b o n a c c i series, i n t r o d u c e d
u n o b t r u s i v e l y as t h e s o l u t i o n to a p r o b l e m in r a b b i t p o p u l a t i o n g ro w th , d o e s n o t
h a v e a k n o w n Is la m i c c o u n t e r p a r t . P e r h a p s it is s i m p l y a n i n d e p e n d e n t i n v e n ­
t i o n , b u t if t h e w e i g h t b a la n c e 's y s te m " w a s in use a t ' t h a t t i m e , L e o n a r d o c o u ld
h a v e easily p i c k e d it u p fr o m a m e r c h a n t d u r i n g h is t r a v e ls in E gypt. A n d it is
p o ss ib le t h a t t h r o u g h its re lig io u s use in a n c i e n t E g y p t t h e se rie s h a d r e t a i n e d
so m e sig n ific a n c e as a n it e m o f s a c r e d o r m y s ti c a l k n o w l e d g e a n d was t h u s t r a n s ­
m i t t e d t h r o u g h s c h o l a r ly c o n t a c t .
G ie s a n d G i e s (7 9 6 9 , 6 1 ) n o t e t h a t L e o n a r d o ’s p r a c t i c e o f r e d u c i n g all fra c­
tio n s to ] in che n u m e r a t o r “w e n t b a c k t o a n c i e n t E g y p t, a n d p e r h a p s d e r i v e d
from t h e fa c t t h a t fr a c tio n s w e re r e g a r d e d less as n u m b e r s in t h e i r o w n r i g h t t h a n
as sig ns o f d i v i s i o n .” B o y er (1 9 6 8 , 2 8 1 ) suggests t h a t t h e Liber A b a c i p r o b l e m w ith
re c u rsiv e n e s t i n g o f s e v e n s ( “ S e v e n o ld w o m e n w e n t t o R o m e , e a c h w o m a n h a d
se v e n m u le s; e a c h m u l e c a r r ie d s e v e n s a c k s . . . ’’) o r i g i n a t e d in its a n c i e n t E g y p t­
ia n c o u n t e r p a r t ( R h i n d M a t h e m a t i c a l P a p y ru s p r o b l e m # 7 9 ) . A n d F ib o n a c c i d oes
provide a n a rra tiv e s ta te m e n t of th e recu rsiv e c o n s tru c tio n ,^ h ig h lig h tin g th e
Implications
206
s a m e s e l f - g e n e r a t i n g a s p e c t o f t h e series t h a t w o u ld b e e m p h a s iz e d b y t h e a n c i e n t
E g y p t i a n b e l i e f s y s te m .
I f tliis i n f l u e n c e , ( w h e t h e r m e r e l y c o n t e x t u a l o r d i r e c t ) d o e s in f a c t e x i s t ,
it s h o u l d n o t d e t r a c t f r o m t h e g e n i u s o f L e o n a r d o ’s w o rk . H i s ‘’g e n e r a l s o l u t i o n
fo r f i n d i n g “c o n g r u e n t n u m b e r s ” f o r s q u a r e s h a s b e e n h a i l e d as “ t h e f i n e s t p i e c e
o f r e a s o n i n g in n u m b e r t h e o r y o f w h i c h w e h a v e a n y r e c o r d b e f o r e t h e t i m e o f
F e r m a t . ” B u t w h e n it c o m e s t o t h e u s e o f t h e F i b o n a c c i s e r ie s in t h e c o n t e m ­
p o r a r y h i s t o r y o f m a t h e m a t i c s (cf. B r o o k e 1 9 6 4 ) , t h e r e is a c t u a l l y n o e v i d e n c e
o f a d i r e c t c o n t r i b u t i o n f r o m F i b o n a c c i h i m s e l f . By a ll a c c o u n t s , G e r m a n
a s t r o n o m e r J o h a n n e s K e p le r r e d i s c o v e r e d t h e series i n d e p e n d e n t l y in 1611, a n d
it w as o n l y in t h e m i d - i 8 o o s , w i t h t h e f o r m a l p u b l i c a t i o n o f L ib e r A b a c i , t h a t
F r e n c h m a t h e m a t i c i a n E d o u a r d L u c a s f o u n d t h e P is a n h i s t o r i c a l p r e d e c e s s o r a n d
n a m e d it ac c o rd in g ly . T h i s fa c t h a s r e c e i v e d litt le a t t e n t i o n , a n d m o s t t e x t s p r e s ­
e n t F i b o n a c c i ’s d i s c o v e r y as if i t w e r e i n a d i r e c t i n t e l l e c t u a l l i n e o f d e s c e n t
ra th e r th a n an h o n o ra r y title g iv e n to a w ell-d e se rv in g b u t d is c o n n e c te d a n t e ­
c e d e n t. F ib o n a c c i h im self seem ed u n h e s ita n t a b o u t th e m u ltic u ltu ra l c o n tr i­
b u t i o n s t o h i s w o r k ; t h e f i r s t s e n t e n c e o f L iber A b a c i s t a t e s , “T h e n i n e I n d i a n
fi g u r e s a r e . . .
N o d o u b t h e w o u ld h a v e b e e n q u ite c o n t e n t a ttr ib u tin g th e
s e r ie s t o o r i g i n a t o r s o f a n y h e r i t a g e .
F i b o n a c c i ’s s e r ie s was s i m p l y u n b o u n d e d g r o w t h ; t h e r e w a s n o i n t r o d u c ­
t i o n o f t h e i n f i n i t e e x c e p t in w ay s t h a t A r i s t o t l e w o u ld h a v e a p p r o v e d . T h e s e v ­
e n t e e n t h c e n tu r y b r o u g h t a t t e n t i o n to th e c o n c e p t o f t h e " in f i n ite s im a l”
{ r e v i v e d f r o m its G r e e k b a n i s h m e n t in K e p l e r ’s S te r e o m e tr ia [ 1 6 1 5 D , a n d c o n ­
v e r g e n c e t o a l i m i t . a s i n f i n i t y is a p p r o a c h e d ( e .g ., t h e a l g o r i t h m s fo r g e n e r a t ­
in g p i ) ; b u t i n f i n i t y w o u l d s till e x i s t o n l y as a n e v e r - r e a c h e d o r i e n t a t i o n r a t h e r
t h a n a l e g i t i m a t e o b j e c t o f study- T h e A r i s t o t e l i a n v o i c e c o u l d s till b e h e a r d in
1831, w h e n m a t h e m a t i c i a n C a r ! F ried ric h G au ss (1 7 7 7 - 1 8 5 5 ) c a u t i o n e d his friend
S c h u m a c h e r a g a i n s t in f in i ty : " I m u s t p r o t e s t m o s t v e h e m e n t l y a g a i n s t y o u r
use o f t h e in f in ite as s o m e t h i n g c o n s u m m a t e d , as th is is n e v e r p e r m i t t e d in m a t h e ­
m a t ic s . T h e i n f i n i t e is b u t a. f a ^ o n d e p a r l e r , m e a n i n g a l i m i t t o w h i c h c e r t a i n
r a t i o s m a y a p p r o a c h as c lo s e ly as d e s i r e d w h e n o t h e r s a re p e r m i t t e d t o in c r e a s e
i n d e f i n i t e l y . ” B u t G a u s s ’s d i s t i n c t i o n w a s s h o r t - l i v e d . A s w e s a w in c h a p t e r 1,
t h e w o r k o f G e o r g C a n t o r , w h i c h h a d p r o d u c e d t h e first f r a c t a l , t h e C a n t o r set,
e n d e d t h e A r i s t o t e l i a n v ie w o n i n f in i ty . L ik e F i b o n a c c i , C a n t o r t o o m a y h a v e
h a d s o m e n o n - E u r o p e a n i n f l u e n c e i n h i s w o rk .
T h e C a n t o r s e t (fig. i- 3 -ia ) w a s h is v i s u a l i z a t i o n o f t r a n s f i n i t e n u m b e r
t h e o r y . I t s h o w s t h e i n t e r v a l o f ze ro t o o n e o n t h e re a l n u m b e r li n e , a n d i n d i ­
c a t e s t h a t t h e n u m b e r o f p o i n t s is n o t d e n u m e r a h l e — t h a t is, g r e a t e r t h a n i n f i n ­
ity. B u t a t t h e t i m e , p u r e m a t h e m a t i c s w a s o n ly o n e o f C a n t o r ’s c o n c e r n s . H is
Fractals in the European Ktstor^y 0 / mathematics
real fa s c in a t io n w as in t h e th e o l o g ic a l im p lic a tio n s ; t h e in c r e a s in g classes o f in f in ­
ity h e d is c o v e re d s e e m e d to p o i n t to w a rd a religio us t r a n s c e n d e n t a l . C a n t o r ’s b io g ­
r a p h e r s differ g reatly o n t h e c u l t u r a l s i g n if i c a n c e o f t h i s p o i n t . E. T. B ell felt t h a t
C a n t o r ’s J e w is h e t h n i c o r i g i n r u l e d h is life, a n d h e m a d e s e v e r a l r e m a r k s a b o u t
t h e i n h e r i t a n c e o f p e r s o n a l i t y t r a i t s — p a r t i c u l a r l y d i s t u r b i n g in l i g h t o f h is
II
II
II
II
II
II
1111 1111
FIGURE
T h e
II II
1111 1111
I3 .I
C a n to r set
(a) T h e first f r a c t a l , c r e a t e d b y G e o r g C a n t o r i n 1 8 7 7 . ( b ) T h i s d e s i g n is f o u n d o n t h e t o p o f
co lu m n s in t h e t e m p l e s o f a n c i e n t E g y p t . G e o r g C a n t o r ’s f t o s i c r u c i a n b e l i e f s a n d h i s c o u s i n M o r t i z
C a n t o r, a n e x p e r t o n che g e o m e t r y o f E g y p t i a n a rt , m a y h a v e p u t h i m in c o n t a c t w i t h t h i s E g y p t i a n
design.
( b, f r o m F u n r ie r 1 8 2 1 .)
I m p li c a t io n s
r e m a r k s o n C a n t o r ’s a r c h r i va l , t h e J e w i s h m a t h e m a t i c i a n L e o p o l d K r o n e c k e r :
“T h e r e is n o mo r e v i c i o u s a c a d e m i c h a t r e d t h a n t h a t o f o n e J e w for a n o t h e r w h e n
t he y disagree o n purel y sc i enti fic ma t t e r s . W h e n t w o in t e l l e c t u a l J ews fall o u t t he y
di s a g re e al l ove r , t h r o w r e s e r v e t o t h e dogs, a n d d o e v e r y t h i n g in t h e i r p o w e r t o
c u t o n e a n o t h e r ’s t h r o a t o r s t a b o n e a n o t h e r i n t h e b a c k ” ( Be l l 1 9 3 9 , 5 6 2 - 5 6 3 ) .
A n o t h e r C a n t o r b i o g r a p h e r , J. W. D a u b e n , says t h a t s i n c e C a n t o r ’s m o t h e r
was R o m a n - C a t h o l i c “s h e was by d e f i n i t i o n n o n -J e w i s h , t h u s it fol lows t h a t G e o r g
C a n t o r was n o t Jewi s h, c o n t r a r y t o t h e v i e w w h i c h h a s p r e v a i l e d in p r i n t for m a n y
y e a r s ” ( D a u b e n 1 9 7 9 ) . N a z i s c h o l a r s s o l v e d t h e i r w o r r i e s by s p r e a d i n g a s t o r y
t h a t C a n t o r wa s f o u n d a b a n d o n e d o n a s h i p b o u n d for S t . P e t e r s b e r g ( G r a t t a n G u i n e s s 1971, 352).
A c t u a l l y C a n t o r ’s J e w i s h i d e n t i t y wa s q u i t e c o m p l e x . H i s fami l y h a d i n d e e d
c o n v e r t e d t o C h r i s t i a n i t y , b u t h e was we l l a w a r e o f h i s h e r i t a g e . H e r e f e r r e d t o
hi s g r a n d m o t h e r as “ t h e I s r a e l i t e ” a n d w r o t e a re l i g i ou s t r a c t t h a t a t t e m p t e d t o
s h o w t h a t t h e r e wa s n o v i r g i n b i r t h , a n d t h a t t h e r eal f a t h e r o f J e s us C h r i s t wa s
Jo s e p h o f A r im a th e a . C a n t o r e v e ntua lly jo ined the Rosicrucians, w h o s e mysti­
c a l/ s c i en t i f i c a p p r o a c h t o a s u p p o s e d E g y p t i a n o r i g i n f o r all r e l i g i o n s p r o b a b l y
a p p e a l e d n o t o n l y t o his i n t e l l e c t u a l i n t e r e s t s , b u t al s o t o h i s s y n c r e t i c e t h n i c ­
ity. C a n t o r c h o s e a H e b r e w l e t t e r as h i s n e w s y mb o l : t h e a l e p h , b e g i n n i n g o f t h e
a l p h a b e t , was u s e d t o r e p r e s e n t t h e b e g i n n i n g o f t h e n o n d e n u m e r n b l e sets.
W h i l e His b i o g r a p h e r s a r g u e d J e w o r n o t - J e w, off o r o n , z e r o o r o n e , C a n t o r h i m ­
self p r o v e d t h a t t h e c o n t i n u u m f r o m zero t o o n e c a n n o t b e d e l i m i t e d b y a n y s u b ­
d i v i s i o n process, n o m a t t e r h o w l o n g its a r g u m e n t s .
G i v e n C a n t o r ’s R o s i c t u c i a n t h e o l o g y a n d t h e p r o x i m i t y o f h i s c o u s i n
M o r i t z C a n t o r — a t t h a t t i m e a l e a d i n g e x p e r t in t h e g e o m e t r y o f E g y p t i a n a r t '
( C a n t o r 1 8 8 0 ) — it m a y b e t h a t G e o r g C a n t o r s a w t h e a n c i e n t E g y p t i a n r e p r e ­
s e n t a t i o n o f t h e l o t u s c r e a t i o n m y t h (fig. 1 3 . 1 b ) , a n d d e r i v e d i n s p i r a t i o n f r o m
t hi s A f r i c a n f r act al for t h e C a n t o r set. W e m a y n e v e r k n o w for c e r t a i n , b u t t h e
g e o m e t r i c r e s e m b l a n c e is q u i t e s t r o n g .
A s n o t e d in c h a p t e r 1, C a n t o r ’s m a t h e m a t i c s was c o n s i d e r e d u t t e r l y i m p r a c ­
ti cal; it w a s n o t u n t i l B e n o i t M a n d e l b r o t t h a t f r a c t a l g e o m e t r y b e c a m e usef ul t o
science. M a n d e l b r o t reports t h a t his in s p ira tio n c a m e from a study o f l o n g - te rm
r i ve r f l u c t u a t i o n s by B r i t i s h c i v i l s e r v a n t H . E. H u r s t . H u r s t e x a m i n e d t h e fl ood
v a r i a t i o n s o v e r s ev e r al c e n t u r i e s a n d c o n c l u d e d t h a t it c o u l d b e c h a r a c t e r i z e d in
t e r m s o f a s c a l i n g e x p o n e n t . La t e r , M a n d e l b r o t real ized t h a t t h i s w a s t h e s a m e
s cal i ng m a t h e m a t i c s t h a t t h e “r e m a r k a b l e c u r v e s " o f C a n t o r a n d o t h e r s d esc r i be d,
But w h e r e d i d H u r s t find a rel iabl e s o u r c e for sever al c e n t u r i e s o f flood d a t a ? H u r s t
l i ved in Eg y p t for 6 2 years a n d wa s a b l e t o d e m o n s t r a t e l o n g - t e r m s c a l i n g in N i l e
flood r ecor ds b ec a u s e o f t h e a c c u r a t e “ n i l o m e t e r ” r e a d i ng s g o i n g b a c k fifteen c e n -
F r a c t a ls in t h e E u r o p e a n h is t o r y o f m a t h e m a t ic s
curies. A t t e m p t s t o find p a t t e r n s in t h e flo od s a re q u i t e a n c i e n t in t h e N i l e v a l ­
ley; in s o m e w ays, H u r s t a n d M a n d e l b r o t w e r e s i m p l y t h e l a t e s t — a n d m o s t s u c ­
cessfu l— p a r t i c i p a n t s in c h a t s e a r c h .
R e c u rs io n a n d sex: a c r o s s - c u ltu r a l c o m p a riso n
T h r o u g h o u t t h e e x p l o r a t i o n o f A f r i c a n fra ctals, we failed to find a n y o n e c u l t u r a l
f e a t u r e t h a t w a s p e r s i s t e n t l y a s s o c i a t e d w i t h t h e s e fo r m s . T h e y r a n g e d from
p r a c t i c a l c o n s t r u c t i o n t e c h n i q u e s t o a b s t r a c t t h e o l o g i c a l ic o n s , f r o m w i n d ­
screen s to k in s h ip structures, f ro m e s t h e ti c p a tt e r n s to d i v i n a t i o n te c h n iq u e s. T h e r e
is n o s i n g u la r “r e a s o n ” w h y A f r i c a n s use f r a c ta l s , a n y m o r e t h a n a s i n g u l a r r e a ­
s o n w h y A m e r i c a n s like r o c k m u s ic . S u c h e n o r m o u s c u l t u r a l p r a c t i c e s j u s t c o v e r
t o o m u c h s o c ia l t e r r a i n . A t b e s t w e c a n m a k e a l o w e r - d i m e n s i o n a l p r o j e c t i o n o f
such h ig h - d im e n s io n a l d y n am ics, th e silh o u ecte th a t ap p ears g iv en o n e p a r tic ­
u la r a x is o f i l l u m i n a t i o n . T h i s s e c t i o n w ill fo c u s o n t h e r e l a t i o n b e t w e e n r e c u r ­
s i o n in m a t h e m a t i c s a n d s e x u a l i t y i n c u l t u r e . S e x is c o n v e n i e n t in t h a t o t h e r
research ers h a v e d e v e lo p e d A fric a n -E u ro p e a n co m p a riso n s , a n d th a t sexual
r e p r o d u c t i o n is o f t e n c o n n e c t e d t o r e c u r s i v e c o n c e p t s .
T a y lo r ( 1 9 9 0 ) d e s c r ib e s s e x u a l i t y in R w a n d a as b a s e d o n t h e c o n c e p t o f a
“f r a c ta l p e r s o n ” in w h i c h s o c i e t y is p e r c e i v e d “n o t in t e r m s o f m o n a d i c i n d i v i d ­
u a ls b u t i n t e r m s o f . . . s t r u c t u r e s o f m e a n i n g w h o s e p a t t e r n s r e p e a t t h e m s e l v e s
in s lig h tly v a r y i n g fo rm s like t h e c o n t o u r s o f a fra c ta l t o p o g r a p h y ” (p . 1 0 2 5 ) . H is
analysis o n e x p re s s io n s o f th is s o c i a li ty in te r m s o f t h e c i r c u l a t i o n o f fluids is used
to e x a m i n e th e failure o f p ro g ram s to e n c o u ra g e c o n d o m use. C a r o l y n M a r t i n S h a w
( 1 9 8 9 , 1 9 9 5 ) a n a ly z e s K ik u y u s e x u a l i t y in r e l a t e d w ays a n d p ro v i d e s a n i l l u m i ­
n a ti n g c o n tr a s t to E u ro p e a n sexuality. U s in g F o u c a u lt’s c ri ti q u e o f h u m a n i s m , S haw ,
c h a l l e n g e s t h e u su al p o r t r a i t o f E u r o p e a n s e x u a l r e p r e s s i o n a n d A f r i c a n s e x u a l
lic e n se . S h e d e m o n s t r a t e s t h a t in b o t h cases, t h e so c ia l s y s te m c o n t r o l s s e x u a l
b e h a v io r , b u t w h ile t h e E u r o p e a n lo cu s o f c o n t r o l is in t h e p r i v a t i z a t i o n o f p l e a ­
s u r e , t h e K i k u y u s ’s s e x u a l r e g u l a t i o n o c c u r s t h r o u g h p u b l i c e x p r e s s io n s o f p l e a ­
su re a n d “s o c i a li ty o f i n d i v i d u a l c o n s c i e n c e . ” F o r e x a m p l e , s h e h i g h l i g h t s t h e
p r a c t ic e o f ngwe/co, in w h i c h te e n a g e r s w r a p t h e m s e l v e s w i t h a few l e a t h e r strips,
oil t h e i r b o d ie s , a n d e n g a g e in a p u b l i c d is p la y o f s e x u a l b e h a v i o r . F r o m a E u r o ­
p e a n p o i n t o f v ie w t h i s s o u n d s lik e a n u n r e g u l a t e d orgy, b u t S h a w fo u n d t h a t t h e
practic e was a m e t h o d o f p r e v e n t in g te e n a g e p re g n a n c ie s a n d c h a n n e l i n g t h e t e e n s ’
s e x u a l d e s i r e i n t o so c ia lly a p p r o v e d f o r m s .4
W h e n w e l o o k a t m a n y A f r i c a n f r a c t a l s w e c a n see a n e m p h a s i s o n s e x u ­
a lity in t e r m s o f r e p r o d u c t i o n . T h e s e lf- s im i la r it y o f t h e B a m a n a c h i w a r a a n t e ­
lo p e h e a d d r e s s a n d m e r u n k u n f e r t i l i t y p u p p e t , t h e s e l f - g e n e r a t i n g D o g o n
I
210
I m p lic a tio n s
c o s m o lo g y , t h e c y c l i c k i n s h i p i c o n o g r a p h y o f t h e M i t s o g h o , F a n g , a n d B a lu b a ,
t h e i t e r a t i o n s o f b i r t h i n g in t h e N a n k a n i a r c h i t e c t u r e , a n d m a n y o t h e r c ases o f
r e c u r s i o n a r e c lo s e ly t i e d t o s e x u a l r e p r o d u c t i o n . T h u s o n e c o n t r i b u t i n g fa c to r
t o t h e A f r i c a n m a t h e m a t i c a l e m p h a s i s o n r e c u r s i o n c o u l d b e tills A f r i c a n c o n str u c tio n o f sexuality th r o u g h p o sitiv e pu b lic d o m a in ex p ressio n s.
T h e E u r o p e a n c o u n t e r p a r t o f S h a w ’s th e o r y w o u ld p r e d i c t t h e o p p o s i te , a n d
i n d e e d w e find t h a t t h e b a n i s h i n g o f i n f in i te reg ress in t h e P l a t o n i c r e f o r m was
c lo s e ly t i e d t o a k i n d o f s e x u a l p r o h i b i t i o n . In P l a t o ’s S y m p o s iu m , S o c r a t e s
e x p l a i n s t h a t t h e r e is a h i e r a r c h y o f r e p r o d u c t i o n . L o v e b e t w e e n a m a n a n d a
w o m a n w ill o n l y re su lt in a flesh c h i l d , a c r e a t u r e o f flux w h o w ill e v e n t u a l l y die,
a t b e s t p r o d u c i n g m o r e flux. L o v e b e t w e e n a m a n a n d a bo y ( l o v e r a n d b e l o v e d )
is h i g h e r , b e c a u s e it c a n re s u lt in r a i s i n g t h e b o y t o a h i g h e r p l a n e — t h a t o f a
p h il o s o p h e r. A n d a p h i l o s o p h e r c a n h a v e a “b r a i n c h il d ,” a p e r f e c t id e a t h a t n e v e r
c h a n g e s o r dies. T h e P l a t o n i c ideal o f s t a t i c , e t e r n a l p e r f e c t i o n c o n f l ic t s w i t h t h e
e v e r - c h a n g i n g d y n a m i c o f s e x u a l r e p r o d u c t i o n . T h e G r e e k p r e f e r e n c e for t h e s t a ­
t i c s h a p e o f t h e A r c h i m e d e a n s p ira l s u g g e s ts t h i s P l a t o n i c i d e a l , ju s t as t h e
g r o w i n g s h a p e o f t h e l o g a r i t h m i c s p ira l su g g es ts t h e A f r i c a n e m p h a s i s o n f e r t i l ­
ity a n d r e p r o d u c t i o n . O f c o u r s e , t h i s is a g ro ss g e n e r a l i z a t i o n ; t h e r e a re , for
e x a m p l e , p l e n t y ' o f A r c h i m e d e a n spirals in A f r i c a n design s. C o n v e r s e l y , E u r o p e a n
m a t h e m a t i c i a n J a c o b o B e r n o u l l i w a s u t t e r l y d e d i c a t e d t o t h e l o g a r i t h m i c s p ira l
a n d sp e c ifie d t h a t o n e w o u l d b e e n g r a v e d o n h i s t o m b s t o n e . B u t t h e s t o n e c u t ­
te r d id n o t go a g a in s t t h e g ra i n o f h is c u lt u r e ; B e r n o u ll i’s g ra v e is still m a r k e d w ith
a n A r c h m e d e a n s p i r a l (fig. 13-2).
I t w o u ld be d a n g e r o u s to su g g e s t t h a t t h e r e is a n e t h i c a l d i f f e r e n c e a t
s t a k e h e r e , as s o m a n y o r g a n i c r o m a n t i c i s t s h a v e m a i n t a i n e d . A g a i n , t h e r e is n o
h i s t o r i c a l e v i d e n c e for a c o n s i s t e n t r e l a t i o n s h i p b e t w e e n m a t h e m a t i c a l d i s t i n c -
FIGURE 13 .2
B e r n o u lli’s to m b s to n e
Although Bernoulli asked f o r a logarithmic spiral to be
inscribed on his tombstone, the engraver was apparently
only familiar with che linear spiral.
(From Mnor 1987, courtesy Birk/iriuser Verfag AG, Basel,
Sifiiterland.)
F r a c t a l s in th e E u r o p e a n h is t o r y o f m a t h e m a t ic s
d o n s a n d th e e th ic s o f th e i r users. S o m e strictly linear, logical th in k e rs like B e rtran d
R u s s e l l a n d N o a m C h o m s k y h a v e b e e n f a m o u s for t h e i r p r o g r e s s i v e e t h i c a l
s t a n d p o i n t s , j u s t as s o m e h o l i s t i c o r g a n i c i s t s h a v e b e e n p r o n e t o fascism . A n d
o f course vice versa. W h a t d o e s c o u n t for e th ic s is h o w peo ple a re able to use m a t h e ­
m atics' in t h e p a r t i c u l a r e v e n t s a n d ideaS t h a t s u r r o u n d e d t h e i r life. W i t h t h a t
in m i n d , l e t ’s lo o k a t t h r e e o f t h e i n n o v a t o r s w h o b r o u g h t r e c u r s i o n i n t o E u r o ­
p ean co m p u tatio n al m athem atics.
T h e story o f A d a L o v e la c e is w e ll k n o w n in c o m p u t i n g s c i e n c e history. H e r
f a m e s t e m s fr o m h e r w r i t i n g s in 1 8 4 3 o n fh e m a t h e m a t i c a l p o s s i b i l i t i e s o f
C h a r l e s B a b b a g e ’s p r o p o s e d “a n a l y t i c a l e n g i n e ”— a p l a n fo r a m e c h a n i c a l d i g i ­
tal c o m p u t e r . L o v e la c e is o f t e n p r o m o t e d as a r e c o v e r e d fe m in i s t a n c e s t o r , a p o s i­
ti o n t h a t t e n d s to o v e r e s t i m a t e h e r a c h i e v e m e n t s a n d o b sc u r e h e r o w n t h i n k i n g .
A g a in s t th e s e red u ctiv e p o rtra its, S t e i n ( 1 9 8 5 ) h as w rit te n a d etailed, critical e x a m ­
i n a t i o n o f L o v e la c e t h a t re v e a ls a m u c h m o r e i n t e r e s t i n g a n d c o m p l e x sto ry t h a n
th e p o p u larizations h a v e allo w ed .
L o v e l a c e ’s m o t h e r w as a lw a y s w o r r i e d c h a t s h e m i g h t h a v e i n h e r i t e d t h e
n o t o r i o u s s e x u a l p r o c l i v i t i e s o f h e r f a t h e r , L o r d B y ro n . H e r c h i l d h o o d r e v o l v e d
a r o u n d s t r i c t l y p r e s c r i b e d e d u c a t i o n a l a c t i v i t i e s , a n d a t ti m e s s h e w as f o r c e d to
lie perfectly still, w ith bags o v e r h e r h a n d s to w ard o ff any “w ildness.’’.T his repressed
u p b rin g in g e v e n tu a lly in sp ired re b e llio n in th e fo rm of a n a tte m p te d e lo p e ­
m e n t , b u t t h e fa iled affair left h e r h u m i l i a t e d a n d r e p e n t a n t . S h e w r o te to a f a m ­
ily fr ie n d , W i l l i a m K in g , r e q u e s t i n g m a t h e m a t i c a l i n s t r u c t i o n as a c u r e for h e r
sin fu l im p u lse s. K in g a g re e d , s e n d i n g h e r b o t h m a t h e m a t i c a l a n d relig io u s tex ts.
B u t d e s p i t e h e r d e c l a r a t i o n s to a p p l y h e r m a t h e m a t i c a l i m a g i n a t i o n “t o t h e
g r e a t e r glory o f G o d , ” s h e t u r n e d a w a y f r o m t h e m o r a l i z i n g o f K in g t o t h e m o r e
g la m o r o u s s o c ia l c o m p a n y o f B a b b a g e a n d h i s f a m o u s “ t h i n k i n g m a c h i n e s . ”
B a b b a g e ’s m o t i v a t i o n s w e r e far r e m o v e d fro m K i n g ’s relig io u s i n t e l l e c t u a ltsm . H e w as p r i m a r i ly c o n c e r n e d w i t h e c o n o m i c a n d s c i e n ti fic p ro gress. T h i s
s w i t c h fro m K in g to B a b b a g e w as a n a c t o f i n d e p e n d e n c e , a n d L o v e l a c e b e g a n
t o tu r n h e r i m a g in a t io n lo ose. W h i l e p u rs u in g a m u c h m o r e i n t e n s e area o f m a t h e ­
m a t i c a l stud y, h e r re lig io u s t h i n k i n g a ls o t o o k a n e x p a n d e d tu r n . S h e b e g a n to
de sc rib e h e rs e lf a n d h e r w o rk in te r m s o f m a g ic a l im agery: t h e m e c h a n is m s o f s y m ­
bol m a n i p u l a t i o n w e re “m a t h e m a t i c a l s p r i t e s , ” a n d s h e a d v is e d B a b b a g e to a llo w
h i m s e l f t o be “u n r e s i s t i n g l y b e w i t c h e d ” by “t h e H i g h P rie s te s s o f B a g g a g e ’s
E n g in e.”
S t e i n a ls o n o t e s t h a t it was a c t u a l l y B a b b a g e w h o first d r e w u p t h e " t a b l e
o f s t e p s ” c o n s t i t u t i n g t h e first c o m p u t e r p r o g r a m s . B a b b a g e w as h a v i n g d i f f i ­
c u lty o b t a i n i n g f u n d i n g for his w o r k , h o w e v e r , a n d re aliz ed t h a t L o v e l a c e ’s so cial
p o s i t i o n a n d n o t o r i e t y — b o t h as t h e d a u g h t e r o f B y r o n as w ell as a “L a d y o f
212
Im p l ic a tio n s
M a t h e m a t i c s ”— c o u l d b e p u t to h i s a d v a n t a g e . T h e r e p u t a t i o n o f L o v e l a c e as
t h e o r i g i n a t o r o f p r o g r a m m i n g s t e m s f r o m t h i s p u b l i c r e l a t i o n s p lo y o f B a b b a g e .
T h e r e w as, h o w e v e r , o n e t a b l e f o r w h i c h A d a w a s w h o l l y r e s p o n s i b l e : t h e
r e c u r s i v e g e n e r a t i o n o f a s e q u e n c e k n o w n as t h e B e r n o u l l b n u m b e r s . M o o r e
( L977) sta te s t h a t th is ta b le used recu rsiv e p ro g ra m m in g . H u sk e y a n d H u sk ey
( 1 9 8 4 ) , a p p a r e n t l y r e f e r r i n g t o t h i s c l a i m , s u g g e s t t h a t t h i s is a c o n f u s i o n w i t h
L o v e l a c e ’s d e s c r i p t i o n o f m a t h e m a t i c a l “ r e c u r r e n c e g r o u p s ” a n d n o t e t h a t t h e
t e r m “ r e c u r s i v e p r o g r a m m i n g " g e n e r a l l y r e f e r s t o a p r o c e d u r e t h a t c a ll s i t s e l f '
(i.e ., s e l f - r e f e r e n c e ) — im p o s s ib le fo r L o v e l a c e s i n c e h e r c o d e h a d n o p ro c e d u r e s .
B u t t h e y a ls o n o t e t h a t L o v e l a c e i n t r o d u c e d a n e w c o d e n o t a t i o n t o d e s c r i b e
w h a t s h e r e f e r r e d t o as “a c y c l e o f a c y c l e , ” w h i c h w o u l d b e e q u i v a l e n t t o t h e
r e c u r s i v e s t r u c t u r e o f n e s t e d i t e r a t i o n i n use to d a y .
S ig n i f i c a n tl y , t h i s i t e r a t i v e r e c u r s i o n w a s t h e o n e p r o g r a m fo r w h i c h B a b ­
b ag e c la i m e d c re d it: “W e d is cu ssed v ario u s illu s tr a t io n s t h a t m i g h t b e in t r o d u c e d :
I s u g g e s te d s e v e r a l, b u t t h e s e l e c t i o n w a s e n t i r e l y h e r o w n . S o a ls o w as t h e a l g e ­
braic w o rk in g o u t o f th e d ifferen t pro b lem s, e x c e p t, in d e e d , t h a t re la tin g to th e
n u m b e r s o f B e r n o u i l l i [sic], w h i c h I h a d o ff e re d to d o t o s a v e L a d y L o v e l a c e t h e
t r o u b l e ” ( q u o t e d in S t e i n 1 9 8 5 , 8 9 ) .
T h e a p p r o p r i a t i o n m ay h a v e b e e n a n t i c i p a t e d b y L o v e la c e : S t e i n n o t e s t h a t
i n t h e l e t t e r s c o n c e r n i n g t h i s p r o g r a m , L o v e l a c e is a t y p i c a l l y v a g u e — s h e h a d
a lw a y s b e e n o v e r d e p e n d e n t o n B a b b a g e fo r m a t h e m a t i c a l s p e c i f i c s — a n d s p e c ­
u l a t e s t h a t t h e v a g u e n e s s w as a d e l i b e r a t e m o v e t o p r o t e c t h e r i t e r a t i v e i n n o ­
v a t i o n . M a n y f e m i n i s t s h a v e w r i t t e n a b o u t m a l e e n v y o f w o m e n ’s r e p r o d u c t i v e
c a p a c i t y , a n d t h e r e m i g h t w e ll b e a p a r a l l e l in B a b b a g e ’s a p p r o p r i a t i o n o f
L o v e l a c e ’s r e c u r s i v e a c h i e v e m e n t . B u t t h e o r g a n i c i s t v e r s i o n s o f s u c h a n a ly s e s
p o rtra y t h e c o n f l i c t in te r m s o f w o m e n b e in g m o r e n a t u r a l o r e m b o d i e d , a n d m e n
b e i n g m o r e a r t i f i c i a l o r a b s t r a c t . In t h i s s t o r y o f m a l e w o m b e n v y a n d t h e p r o ­
t e c t i v e m o t h e r , it is t h e d i g i t a l a b s t r a c t i o n o f r e c u r s i o n , n o t c o n c r e t e e m b o d i ­
m e n t , o v e r w h i c h t h e s t r u g g le is fo u g h t . T h e b i r t h i n g m e t a p h o r w a s m e n t i o n e d
by L o v e l a c e h e r s e lf ; t h e f i n i s h e d p r o g r a m m i n g s t u d y w a s “h e r first c h i l d . " C o n ­
tra ry t o P l a t o , s e x u a l r e p r o d u c t i o n is n o t in o p p o s i t i o n t o t h e a b s t r a c t r e a l m o f
m a t h e m a t i c s ; L o v e l a c e u s e d h e r m a t h e m a t i c s t o r e b e l a g a i n s t a t t e m p t s to li m i t
h e r to a repressive fe m in in ity a n d used th is a rtific ia l s e x u a lity — a b e w itc h in g
h i g h priestess, je a l o u s l y g u a r d i n g h e r p r o g r a m m i n g p r o g e n y — t o d e v e l o p t h e first
c o m p u ta tio n a l recu rsio n .
I n t h e d i s c u s s i o n o f t h e m a t h e m a t i c a l t h e o r y o f c o m p u t a b i l i t y in c h a p ­
t e r 10 , we n o t e d t h a t t h e s e t o f “p r i m i t i v e r e c u r s i v e f u n c t i o n s , " d e v e l o p e d by
R o zsa P e t e r , h a d t h e g r e a t e s t c o m p u t i n g p o w e r s h o r t o f a T u r i n g m a c h i n e .
U n l i k e L o v e l a c e , P e t e r ’s c a p a b i l i t y as a m a t h e m a t i c i a n is u n c o n t e s t e d ; in fac t,
F r a c t a l s in th e E u r o p e a n h is t o r y o f m a t h e m a t ic s
s h e is w id e ly r e g a r d e d as “che m o t h e r o f r e c u r s i v e f u n c t i o n t h e o r y ” ( M o r r i s a n d
H a rk le r o a d 1990). B u t sh e, too, im p lie d t h a t p arallels ex isted b e tw e e n h er
g e n d e r i d e n t i t y a n d m a t h e m a t i c s ; m a i n t a i n i n g t h a t w o m e n c o u ld p r o v i d e a s p e ­
c ia l i n s i g h t t h a t m e n c o u l d n o t ( A n d r e k j a 1 9 7 4 , 1 7 3 ) . S i n c e w e k n o w t h a t , as
a m a t h e m a t i c i a n , s h e w o u ld n o t b e t h i n k i n g o f t h i s s p ec ial in s ig h t as b e i n g m o re
c o n c r e t e o r less lo gical, it m a y b e t h a t P e t e r also m a d e c o n n e c t i o n s b e t w e e n s e x ­
ual r e p r o d u c t i o n a n d r e c u r s i o n .
F o llo w in g P e t e r ’s class o f p rim itiv e recu rsiv e fu n c tio n s , o n e reach es th e upper
lim it o f re c u rsiv e p o w e r in t h e T u r i n g m a c h i n e . A l a n T u r i n g ’s c o n t r i b u t i o n s w ere
n o t o n l y in t h e m a t h e m a t i c a l a b s t r a c t i o n s o f c o m p u t i n g , b u t in its a p p l i c a t i o n
t o artific ia l i n t e l l i g e n c e as w ell. In h is c la ssic p a p e r t i t l e d “C o m p u t i n g M a c h i n ­
ery a n d I n t e l l i g e n c e , ” h e p r o p o s e d w h a t is n o w c a l l e d t h e T u r i n g te s t. A t first,
m o s t d e f i n i t i o n s o f m a c h i n e i n t e l l i g e n c e w e r e b a s e d o n a p a r t i c u l a r ta s k or
b e h a v i o r (e.g., c h e s s p la y i n g ) . B u t as t h e field o f a r t ifi c ia l i n t e l l i g e n c e ( A I ) has
d e v e l o p e d , t h e s e h a v e s h o w n t o b e i n c r e a s i n g l y i n a d e q u a t e , a n d t h e T u r i n g test
is w id e ly r e g a r d e d as t h e m o s t r e l ia b l e d e f i n i t i o n fo r A l ( i n fact, y early T u r in g
tes ts a re n o w h e l d , w i t h n o m a c h i n e w i n n e r s t h u s far).
T u r i n g b e g i n s by d e s c r i b i n g a g a m e in w h i c h a m a n a n d a w o m a n are
b e h i n d a d o o r a n d a n s w e r q u e s t i o n s fr o m a n i n t e r r o g a t o r by w r i t t e n replies. T h e
i n t e r r o g a t o r m u s t d e t e r m i n e w h o is t h e m a n a n d w h o is t h e w o m a n ; b o t h m u s t
try to d e c e i v e h i m in t h e i r a n s w e rs . T u r i n g t h e n s u g g e s te d r e p l a c i n g o n e p e r s o n
w i t h a n A I m a c h i n e ; t h e T u r i n g te s t h o l d s t h a t if t h e i n t e r r o g a t o r c a n n o t d i s ­
t i n g u i s h p e r s o n f r o m m a c h i n e , t h e n o n e h a s c r e a t e d t r u e m a c h i n e i n t e ll ig e n c e .
T u r i n g ’s b i o g r a p h e r , A n d r e w H o d g e s , su g g es ts t h a t t h i s “i m i t a t i o n g a m e ” was
i n s p ir e d by T u r i n g ’s o w n life: s t r u g g l i n g to d e f i n e h is i d e n t i t y as a h o m o s e x u a l
in a h o m o p h o b i c so ciety . B o t h t h e T u r i n g m a c h i n e ’s a b il it y to i m i t a t e o t h e r
m a c h i n e s a n d t h i s g a m e o f c o g n i t i v e i m i t a t i o n e c h o t h e so c ia l e x p e r i e n c e o f gays
w h o liv e in a c o m m u n i t y w h e r e t h e y m u s t p r e t e n d t o b e s o m e o n e t h e y a re n o t .
A n d to s o m e e x t e n t , t h e e n d le s s s e l f - r e f e r e n c e o f m e t a m a t h e m a t i c s was T u r i n g ’s
h i d i n g p l a c e fro m t h e a n t i g a y w o r l d s u r r o u n d i n g h i m . B u t t h e s e x u a l g u e ss in g
g am e o n w h ic h th e T u rin g te st was b a se d w o rk e d a g a in s t su c h n o rm a tiv e
g e n d e r r e s t r i c t i o n s : it s u g g e s te d g e n d e r as s o m e t h i n g m o r e flu id , less fi x e d — a
featu re w h ic h th e v irtu al c o m m u n itie s o n th e I n te r n e t h a v e sta rte d to d e m o n ­
s t ra te (cf. S t o n e 1995; T u rk le 1 99 5). D o u g la s H o f s t a d t e r ( 1 9 8 5 , 1 3 6 -3 6 7 ) , a m o d ­
ern m a s te r o f recu rsio n , h as also w r i t t e n a b o u t t h e p o te n t ia l for a m o r e fluid g e n d e r
i d e n t i t y in d ig i ta l d y n a m i c s .
M a t h e m a t i c s h a d a d o u b l e m e a n i n g for T u r i n g . It w as b o t h a n e m o t i o n a l
sh ie ld , a c lo s e d w o rld o f e n d le s s in t e r i o r s e lf-r e fe re n c e , as well as a n o p e n i n g in to
c o n s c i o u s n e s s a n d c o m m u n i t y . I n t h e e n d , t h i s d e s i r e fo r o p e n i n g k i l l e d T u r in g :
im p lic a t i o n s
d u rin g a robbery in v e s tig a tio n h e a d m itte d h is h o m o s e x u a lity to p o lic e d e te c ­
ti v e s a n d w as a r r e s t e d a n d f o r c e d t o s u b m i t t o h o r m o n e t r e a t m e n t s . T h i s e v e n ­
t u a l l y d r o v e h i m to s u i c id e . I t w as a tr a g ic f a i r y - t a le e n d i n g : h e k i l l e d ‘‘h i m s e l f by
e a t i n g a n a p p l e d i p p e d in p o i s o n . H o d g e s w r it e s a b o u t t h i s d e a t h in t e r m s o f t h e
d o u b l e m e a n i n g t h a t m a t h e m a t i c s h a d in h is life. “L o n e l y c o n s c i o u s n e s s o f selfc o n s c i o u s n e s s w as a t t h e c e n t e r o f h i s id eas. B u t t h a t s e l f - c o n s c i o u s n e s s w e n t
b e y o n d G o d e l i a n self-referenc e, a b s t r a c t m i n d t u r n i n g u p o n its a b s t r a c t self. T h e r e
w as in h i s life a m a t h e m a t i c a l ' s e r p e n t , b i t i n g its o w n ta il forever,-■•hut th e r e - w a s •
a n o t h e r o n e t h a t h a d b i d h i m e a t fro m t h e t r e e o f k n o w l e d g e . ”
I n A f r i c a t h e s e tw o s e r p e n t s are o n e ; s e x u a l r e p r o d u c t i o n ex is ts in t h e s a m e
p u b l i c r e a l m as s o c i a l i n t e r c o u r s e . T h a t is o n e p o s s ib le r e a s o n w h y w e see r e c u r ­
s i o n — t h e s n a k e t h a t b i t e s its o w n ta i l— so p r o m i n e n t l y e m p h a s i z e d in A f r i c a n
fractals, a n d a p o ss ib le e x p l a n a t i o n for w h y t h e s e p i o n e e r s o f r e c u r s i o n in E u r o p e
h a p p e n e d to b e p e o p l e w h o t o o k is sue w i t h s e x u a l r e p r e s s i o n . T h a t ' s n o t t o say
t h e r e is a d e t e r m i n i s t i c l i n k b e t w e e n t h e tw o . I n a n a l o g f e e d b a c k th e o r y , for
e x a m p l e , w e see b o t h a n t i - a u t h o r i t a r i a n f e m in i s ts , li k e N o r b e r t W i e n e r ( H e i m s
1 9 8 4 ) , as w ell as a u t h o r i t a r i a n p r u d e s lik e H o w a r d O d u m ( T a y lo r 1 9 8 8 ). M a t h e ­
m a t i c s is n o t a m e r e r e f l e c t i o n o f p e r s o n a l in t e r e s t s , n o r is it a n a b s t r a c t i o n t h a t
is e n t i r e l y d i v o r c e d fr o m o u r lives. W e m a k e m e a n i n g for o u r s e l v e s o u t o f w h a t ­
e v e r m e t a p h o r s — t e c h n i c a l o r o t h e r w i s e — w e fin d u sefu l; c o n v e r s e l y , p e r s o n a l
m e a n i n g s c a n o f t e n in s p ir e n e w t e c h n i c a l ideas.
W h i l e r e c u r s io n is p r o m i n e n t in A f r i c a n fractals, it h a s b e e n less so in E u r o ­
p e a n f r a c t a l g e o m e t r y . 6 I n t h e h i s t o r i c a l a p p e n d i x t o T h e F r a c ta l G e o m e tr y o f
N a tu r e , M a n d e l b r o t p ro v id e s an e r u d i te h is to r y o f m a t h e m a t i c a l d e v e l o p m e n t s
t h a t l e d t o h i s w o r k ; r e c u r s i o n is n e v e r m e n t i o n e d . E v e n w h e n r e c u r s io n , d o e s c o m e u p in t h e f r a c t a l g e o m e t r y l i t e r a t u r e , t h e t r e a t m e n t is ty p i c a l l y i n f o r m a l
o r cursory. For e x a m p le , S a u p e (1 9 8 8 , 7 2) m erely n o te s t h a t "in so m e cases th e
p r o c e d u r e c a n b e f o r m u l a t e d as a r e c u r s i o n . ” ^ S i m i l a r l y , t h e f r a c t a l t i m e se r ie s
p r o d u c e d by d e t e r m i n i s t i c c h a o s is ra r e ly r e g a r d e d as t h e p r o d u c t o f f e e d b a c k
lo o p s , a n d in o n e o f t h e few s t u d i e s t h a t is f o c u s e d o n t h i s r e l a t i o n s h i p , M e e s
( 1 9 8 4 , 1 0 1 ) m e r e ly s t a t e s t h a t “c h a o s is c e r t a i n l y p o s s ib le in f e e d b a c k s y s te m s .”
O n t h e c o n t r a r y , it is n o t t h a t c h a o s is p o s s ib le w i t h fe e d b a c k ,- b u t t h a t c h a o s
is i m p o s s i b l e w i t h o u t it.
It w o u ld b e i n a c c u r a t e t o say t h a t E u r o p e a n m a t h e m a t i c s h a s d is r e g a r d e d
r e c u r s i o n in g e n e r a l , a n d p e r h a p s r h e o b s e r v a t i o n I a m m a k i n g is s i m p l y d u e to
d is c ip l in a r y s p e c i a li z a ti o n ; t h e r e is n o r e a s o n w h y s o m e o n e s t u d y i n g a p p l i c a t i o n s
o f g r a p h i c s to a n a ly s is a n d m e n s u r a l t h e o r y s h o u l d n e c e s s a r il y b e t h i n k i n g a b o u t
T u r i n g m a c h i n e s o r re c u r s iv e f u n c t i o n s . B u t it is p r e c i s e l y t h i s l a c k o f n e c e s s ity
in m a t h e m a t i c s t h a t is so e a sily f o r g o t t e n in a d i s c i p l i n e w h e r e c e r t a i n t y goes
F r a c t a l s in th e E u r o p e a n f u s t o r ) o f m a th e m a t ic s
b e y o n d t h a t o f a n y e m p i r i c a l s c i e n c e im a g i n a b l e . M a t h e m a t i c s is b o t h a n i n v e i v
t i o n a n d a d is co v ery . W e d i s c o v e r t h e c o n s t r a i n t s i n h e r e n t in t h e fa b ric o f sp a c e
a n d t i m e , c o n s t r a i n t s t h a t a r e t h e s t u f f o f w h i c h o u r u n i v e r s e is c o m p o s e d . B u t
m a t h e m a t i c s d o e s n o t sto p th e r e . T h e c o n s t r a i n t s a re n o t ju s t n e g a tio n s , b u t r a t h e r
t h e b u i l d i n g b lo c k s w i t h w h i c h f u r t h e r m a t h e m a t i c s is c o n s t r u c t e d . A n d lik e an y
c o n s t r u c t i o n , t h e r e a re c h o i c e s t o b e m a d e , d e c i s i o n s a b o u t h o w t h e s e b u i l d i n g
b l o c k s a r e to b e c o n n e c t e d , i n t e r r o g a t e d , a n d d e p l o y e d in f u r t h e r d iscov ery .
T h i s is w h e r e t h e h u m a n side o f m a t h e m a t i c s e n t e r s t h e p i c t u r e , e s p e c ia lly t h a t
m o s t h u m a n o f e n d e a v o r s , c u l t u r e . C o n v e r s e l y , c u l t u r e is n o t m e r e w h i m , a
p u r e l y s u b j e c t i v e m a t t e r o f c h o o s i n g f a v o r e d s o c ia l p r a c t ic e s . T h i s is w h e r e th e
m a t h e m a t i c a l sid e o f h u m a n i t y e n t e r s t h e p i c t u r e , fo r we a re o n l y free t o c o n ­
s t r u c t c u l t u r e w i t h i n t h e c o n s t r a i n t s o f t h e u n i v e r s e in w h i c h w e live. N e i t h e r
m a t h e m a t i c s n o r c u l t u r e s h o u l d b e v ie w e d as firmly fixed o n t h e u n i v e r s a l/ lo c a l
d i v i d e ; t h e r e a re d iv i s i o n s w i t h i n d i v i s i o n s n e v e r e n d i n g .
CHAPTER
-Futures— for—
A frican
-fractals-
M o s t a n t h r o p o l o g i s t s h a v e l o n g a b a n d o n e d t h e t e n d e n c y t o c r e a t e a fr o z e n
“a n c i e n t t r a d i t i o n ” i n d e f i n i n g i n d i g e n o u s s o c i e ty ; c h a n g e a n d s y n t h e s i s a r e
n o w i nt eg r a l p a r t s o f t h e c u l t u r a l p o r t r a i t . S o , t oo, w i t h A f r i c a n fr actal s; t h e y ar e
n e c e s s a r i l y as m u c h o f t h e f u t u r e as t h e y a r e o f t h e p a s t .
F r a c t a l s i n A f r i c a n c o n te m p o r a r y a rts
T h e r e are m a n y works of m o d e r n A fri c a n professional art w h i c h in c o r p o ra t e
a s p e c t s o f fr actal s, s p a n n i n g a w i d e r a n g e o f c u l t u r a l v i e w p o i n t s . A t t h e N a t i o n a l
M u s e u m i n Y a o u n d e , C a m e r o o n , o n e c a n see o r g a n i c r o m a n t i c i s m i n N y a m e ’s
p a i n t i n g s o f l o g a r i t h m i c sp i r a l s m o r p h i n g i n t o p e o p l e . T h e d o u b l e - s i d e d p o s t ­
m o d e r n m e t a l s c u l p t u r e s o f L e g b a i n B e n i n , b y a r t i s t s s u c h as K o u a s s , s h o w a
c y b o r g 1 trickster w h ose tr a d itio n a l bifurcating abilities are ready for th e
b in a r y c o d e s of n e w te c h n o l o g i e s . In East Africa, p a i n t e r G e b r e Kristos D e s ta
p r o d u c e s n o n l i n e a r s c a l i n g c i r c l e s h e d e s c r i b e s as p u r e a b s t r a c t i o n i s m ( M o u n t
1 97 3 , 1 1 8 ) . A f r i c a n f r ac t a l s c o n t i n u e t o e v o l v e . B e s i d e s b e i n g p r e s e n t i n p r o ­
f e s s i o n a l s t u d i o art , f r ac t a l s h a v e a l s o a p p e a r e d in l a r g e - s c a l e p u b l i c a r t wo r k s ,
s u c h as o n t h e f a c a d e o f t h e U n i v e r s i t y o f D a k a r l i b ra r y (fig. 1 4 . 1) . T h i s s c a l i n g
d e s i g n , in w h i c h t h e a l t e r n a t i o n o f p a i n t e d r e c t a n g l e s at t h e sm all scale
216
F u t u r e s f o r A f r i c a n fr a c t a ls
217
FIGU RE 14.1
T h e lib r a ry o f th e U n i v e r s i t y o f D a k a r
This design makes use of both self-similarity (the vertical alternation of painted rectangles looks
like the alternation of buildings) and nonlinear scaling (the rectangle width decreases rapidly as
you go toward the center).
m a t c h e s t h e a l t e r n a t i o n o f t h e b u i l d i n g w alls a t t h e large scale, is r e m i n i s c e n t
o f c e r t a i n A f r i c a n fabrics.
O n e o f t h e m o s t a c t i v e a r e a s o f t o d a y ’s A f r i c a n a r t c o m e s n o t fr o m p r o ­
f e s s i o n a l s t u d i o s , b u t r a t h e r f r o m t h e u n d i s t i n g u i s h e d s e l le rs o f t o u r i s t a rt .
T o u r i s t a r t was fo r m a lly d i s r e g a r d e d in t h e p r o f e s s i o n a l a r t w o r ld , b u t c u l t u r a l
s t u d i e s h a v e i n c r e a s i n g l y s h o w n c h a t t h i s is a d u b i o u s p o s i t i o n . F irst, n e i t h e r
t h e “ t r a d i t i o n a l a r t i s t ” c r e a t i n g ro y a l w o r k s fo r a k i n g , n o r a r t s t u d e n t s t r y i n g
t o p l e a s e t h e i r i n s t r u c t o r s , n o r e v e n p r o f e s s i o n a l s t u d i o a r t is t s w h o m u s t a ls o
b e c o n c e r n e d w i t h sales a r e c o m p l e t e l y fr e e t o c r e a t e w h a t e v e r t h e y w i s h , so
t h e r e is n o r e a s o n t o s i n g le o u t t h e c r e a t o r s o f t o u r i s t a r t for b e i n g c o n s t r a i n e d .
S e c o n d , o p p o r t u n i t i e s f o r p r o f e s s i o n a l s t u d i o a r t i s t s a r e few, a n d t h e t o u r i s t
m a r k e t c r e a t e s a large n u m b e r o f e c o n o m i c o p p o r t u n i t i e s ; it s e e m s s u s p i c i o u s
t o d i s r e g a r d t h i s v i b r a n t a c t i v i t y in f a v o r o f a t i n y e li te . A n d fin ally , as C u l l e r s
( 1 9 8 1 ) n o t e s , to u r i s m is n o t t h e o p p o s i t e o f a u t h e n t i c c u l t u r e , r a t h e r t o u r i s m
c rea tes a u t h e n t i c i t y .
C u l l e r s ’s o b s e r v a t i o n was r e p e a t e d to m e by M a x ( h e d i d n o t w a n t h is last
n a m e to be u se d ), a S e n e g a le s e a rt is t in D a k a r w h o sells t o t h e to u r is t tr a d e . M a x
c o m p l a i n e d t h a t h is m o s t c r e a t i v e w o r k — t h e d e s i g n s w h i c h c a m e t o h i m in
d r e a m s — was diffic ult t o sell b e c a u s e o f t h e t o u r i s t c o n c e p t i o n o f t r a d i t i o n a n d
a u t h e n t i c i t y . L ik e m a n y c r e a t o r s o f t o u r i s t arc in D a k a r, h e p r o d u c e s i m i t a t i o n s
o f t h e k o r a , t h e S e n e g a l e s e s t r i n g e d i n s t r u m e n t t h a t f e a t u r e s a s in g le f r e t r u n ­
n i n g d o w n t h e c e n t e r a n d a h a n d g rip o n b o t h sides. F igu re 14.2 s h o w s t h e u su al
k o r a m o d e l , a l o n g w i t h M a x ’s i n n o v a t i o n , t h e re c u r s iv e k o r a . T h e re c u r s iv e
k o r a m a k e s use o f e a c h h a n d g rip a s t h e f r e t o f tw o s m a ll e r k o ras . 1 a s k e d M a x
•
2 l8
/mplicali'ons
if h e h a d e v e r c o n s i d e r e d c o n t i n u i n g to s m a l l e r sc a le s, a n d h e s a i d t h a t h e h a d
o n c e clone so, b u t t h a t it was im p o s s ib le t o sell s u c h i n n o v a t i v e w o rk ; to u r is t s
d id n o t w a n t a n y t h i n g t h a t s m a c k e d o f o rig in a l it y .
F r a c ta ls in A f r i c a n c o n te m p o r a r y a r c h i t e c t u r e
M a n y in d ig e n o u s A fric a n designs h a v e b e e n in c o r p o ra te d i n t o m o d e r n a rc h i. t e c t u r a ! p r o j e c t s in A .frica, a n d s o m e o f t h e s e h a v e b e e n f r a c t a l s . F o r e x a m p l e ,
t h e S i e r p i n s k t - l i k e i t e r a t i v e t r i a n g l e s f r o m M a u r i t a n i a w e r e u s e d in a n i n s t i t u ­
t i o n a l b u i l d i n g in S e n e g a l , a n d t h e c ir c le s o f c ir c le s in t h e a r c h i t e c t u r e o f W e s t
A f r i c a n v il la g e s b e c a m e t h e b a sis o f a d e s i g n fo r a b u i l d i n g c o m p l e x in d o w n ­
t o w n B a m a k o , t h e c a p i t a l o f M a l i (fig. 1 4 .3 ).
O n e o f t h e m o s t p o t e n t v i s io n s o f a n A f r i c a n f r a c ta l f u t u r e h a s c o m e fro m
t h e a r c h i t e c t u r a l s t u d i e s o f D r. D a v i d H u g h e s a t K e n t S t a t e U n i v e r s i t y in
O h i o . W o r k i n g as a F u lb r ig h t s c h o l a r in sev eral A f r i c a n c o u n t r i e s , H u g h e s ( 1 9 9 4 )
p u t t o g e t h e r a p o r t r a i t o f w h a t h e t e r m e d “A f r o c e n t r i c a r c h i t e c t u r e , " w h i c h
e m b o d i e s s e v e r a l a s p e c t s o f t h e f r a c t a l m o d e l . F irst, H u g h e s c o m b i n e d a c h a r ­
a c te riz a tio n of th e self-o rg an izin g p ro p e rtie s o f A fric a n b u ild in g d esig n (a n
" o r g a n i c a r c h i t e c t u r e " w h i c h “g r o w s fr o m its s i t e ” ) w i t h its s e l f - s i m i l a r p r o p ­
e r t i e s ( w h a t h e t e r m e d “t h e o u t s i d e / i n s i d e r e l a t i o n s h i p , ” a m u t u a l s h a p i n g o f
u n its, clu sters o f u n its, a n d c o m m u n a l spaces fo rm ed by th e s u r r o u n d in g c lu s ­
te r s). S e c o n d , h e e x p l i c i t l y r e j e c t e d p r i m i t i v i s t o r n a t u r a l i z i n g p o r t r a i t s . W h i l e
F i u i t r e s f o r A f r i c a n f r a c t a ls
239
n o t i n g its e n v i r o n m e n t a l h a r m o n y , H u g h e s a ls o e m p h a s i z e d t h a t A f r i c a n
a r c h i t e c t u r e is a lw a y s a n i n t e n t i o n a l a c t o f d e s i g n a n d s e m i o t i c s , n o t m e r e l y
a n u n c o n s c i o u s a d a p t a t i o n t o t h e e c o s y s t e m . I n h is f r a m e w o r k , '‘t r a d i t i o n ”
i n c l u d e s t h e t r a d i t i o n o f i n n o v a t i o n , o r as G a t e ' 5- ( i p 8 8 ) p u t s it, t h e A f r i c a n
t h e m e ot “r e p e t i t i o n w i t h r e v i s i o n . ”
FIGURE I 4 . 3
I n d i g e n o u s fra cta ls in m o d e rn a rch itectu re
(a) Here a traditional Mauritanian fractal design is used in a modern building in the Casamance,
Senegal, ( b ) The DPC building in Burkina Faso, using traditional scaling cylinders with contem ­
porary construction techniques. Architects such as Issiaka Isaac Drabo have made many large-scale
buildings based on this syncretic approach.
220
Jmpiications
G i v e n t hi s c o m b i n a t i o n o f s e l f- o r g a n i z e d s t r u c t u r e a n d i n t e n t i o n a l d e s i gn ,
it is n o t sur prising t h a t H u g h e s ’s wo r k led h i m t o a be a ut i f ul e x a m p l e o f t h e p o t e n ­
ti al f r ac t a l f ut ur e . Fi g u r e 14.4 s h o w s a d e s i g n by A l e x N y a n g u l a , o n e o f U y g h e s ’s
s t u d e n t s at t h e C o p p e r b e l t U n i v e r s i t y in Z a m b i a ( H u g h e s 1 9 9 4 , ‘’1 6 5 - 1 6 6 ) . T h i s
ar c h i t e c t u r e provi des a powerful syncr eti c fusion of in d i g e n o u s a n d m o d e r n
for ms. T h e figure t r a c e d b y t h e w a l k w a y s h o w n i n t h e g r o u n d p l a n is a classic
f ig u r e
14.4
D e s i g n for K itiv e
C o m m u n ity C linic
(a) Kitwe C o m m u n i t y Clinic
in Zambi a; design by David
Hu g h e s and Alex Nyangula.
(b) Kitwe C o m m u n i t y Clinic
ground plan.
CPhotos courtesy David Hughes.)
FIGURE 1 4 . 5
F r a c ta l.ite r a tio n s o f N y a n g a l a ’s c o m m u n i t y c l i n i c d e s i g n
Fractal based on Nyangula’s architectural design. T he ‘‘active lines” of the generation process have
been removed, as have any self-intersecting hexagons.
222
I m p l ic a tio n s
e x a m p l e o f ch e f r a c t a l b r a n c h i n g p a t t e r n r e f e r r e d t o a s a C a y l e y t r e e (s e e
S c h r o e d e r 1 9 9 1 , 8 7 - 8 8 ; P e i t g e n e t al. 1 9 9 1 , 1 9 - 2 0 ) , a n d c a n b e e x t e n d e d fro m
t h e tw o i t e r a t i o n s g i v e n by N y a n g u l a t o in finity. A d d i n g t h e hexag on sT C a s y n ­
c r e t i s m b e t w e e n t h e c y l i n d e r o f Z a m b i a n in d i g e n o u s a r c h i t e c t u r e a n d t h e r e c ­
ti li n e a r form s o f m o d e r n m a t e ri a ls ) v io la te s t h e C a y le y r e q u i r e m e n t t h a t t h e g ra p h
is s e l f - a v o i d i n g ( t h a t is, t h a t t h e b r a n c h e s d o n o t i n t e r s e c t ) . S i n c e I w a s i n t e r ­
e ste d in e x p l o r i n g th e fra c ta l s t r u c t u r e by t a k i n g N y a n g u l a ’s d e s i g n t o h i g h e r i t e r ­
a tio n s , I m a d e tw o a d ju s tm e n ts for th is p ro b l e m . O n e is su gg ested by t h e a p p r o a c h
e l e v a t i o n s k e t c h ( H u g h e s 1 9 9 4 , 1 6 7 ) , w h e r e it is c l e a r t h a t t h e c e n t r a l xtnit is
sligh tly larger t h a n t h e o th e r s . T h i s m e a n s t h a t s e l f - i n te r s e c ti o n will be fo restalled
to h i g h e r i t e r a t i o n s . 2 T h e o t h e r is sim p ly t h e e l i m i n a t i o n o f u n i t s w h e n e v e r th e y
o v e r l a p . W i t h th e s e tw o q u a li f i c a ti o n s , N y a n g u l a ’s d e s i g n m a k e s for a n in f in i te l y
e x p a n d a b l e ( y e t b o u n d e d ) a r c h i t e c t u r e , a s s h o w n i n figure 1 4 .5 . S u c h f l e x ib i l­
ity c o u l d c o n t r i b u t e to t h e e ffo rts to e n c o u r a g e a m o r e p a r t i c i p a t o r y a p p r o a c h
to A fric a n a rc h ite c tu ra l d e sig n (F a th y 1973; O z k a n 1997).
If w e t a k e a n a e r i a l v i e w o f t h e m o d e r n E u r o p e a n s e t t l e m e n t o f P a ris ,
F r a n c e , w e w o u l d see l i n e a r c o n c e n t r i c c i r c l e s s u r r o u n d i n g i t s c e n t e r o f s o c i a l
p o w e r. T h e d i f f e r e n c e b e t w e e n t h i s l i n e a r , r a d i a ll y s y m m e t r i c s e r ie s o f c i r c l e s
a n d A f r i c a ’s n o n l i n e a r , d e c e n t r a l i z e d a r c h i t e c t u r e is p e r h a p s s u b t l e , b u t i m p o r ­
t a n t . T h e t e r m " A f r o c e n t r i c " is m i s l e a d i n g in t h a t “c e n t r i c " is m u c h m o r e t h e
g e o m e try o f P aris th a n of L o g o n e -B irn i, M o k o u le k , L ab b ezan g a, a n d th e o th e r
A f r i c a n a r c h i t e c t u r e s w e h a v e e x p l o r e d . H u g h e s ’s c a l l f o r a “m u l t i d i m e n s i o n a l
A f r o c e n t r i s m ” is b o t h a n a f f i r m a t i o n o f “ A f r o ” a n d a c h a l l e n g e t o “ c e n t r i s m ” ;
it is a c a ll for c u l t u r a l p o r t r a i t s t h a t d o n o t r e d u c e to a s i n g l e o n e - d i m e n s i o n a l
c e n te r b u t ra th e r co m b in e th e b o u n d a rie s of trad itio n w ith th e in fin ite e x p a n ­
sion o f in n o v a tio n .
A f r i c a n fr a c ta ls in m a t h e d u c a t i o n
S e v e ra l re se a rc h e rs h a v e i n d e p e n d e n tly e x p lo re d fractal a s p e c ts o f A f r ic a n
m a th e m a tic s . C h o n a t G e tz of th e U n iv e rsity of th e W itw a te rs ra n d h a s c reated
I t e r a t e d F u n c t i o n S y s t e m s i m u l a t i o n s o f Z u lu b a s k e t w e a v in g . J o h n S im s, m a t h e ­
m a t i c i a n a n d a r t i s t a t t h e R i n g l i n g S c h o o l o f D e s ig n in F lo r id a , h a s d e v e l o p e d
fra c ta l p a t t e r n s based o n B aku ba .rafia c l o t h ( a n d in s p ired by his A f r i c a n h e r i ta g e ) .
I n c h a p t e r 5 w e e n c o u n t e r e d t h e l u s o n a a n a ly s is o f P a u l u s G e r d s , a p r o f e s s o r at
U n t v e r s i d a d e P e d a g o g ic a o f M a p u t o , M o z a m b i q u e , w h o s e p ro lific w r i t i n g s h a v e
r e c e n t l y r a n g e d fr o m t h e e t h n o m a t h e m a t i c s o f w o m e n 's a r t in s o u t h e r n A f r i c a
( G e r d e s 1 9 9 8 a ) to t h e use o f M o z a m b i q u e b a s k e t w e a v i n g g e o m e t r y in m o d e l ­
in g f u l l e r e n e m o l e c u l e s ( G e r d e s 1 9 9 8 b ).
F u t u r e s f o r A f r i c a n fr a c t a ls
W h i l e t h e r e a re c l e a r l y b e n e f i t s t o u t i l i z i n g i n d i g e n o u s k n o w l e d g e for
d e v e l o p m e n t a n d e d u c a t i o n in A f r i c a , A f r i c a n frac tals m i g h t also b e o f use in th e
U n i t e d S t a t e s . D e s p it e t h e low m a t h e m a t i c s p a r t i c i p a t i o n o f A f r i c a n A m e r i c a n
s t u d e n t s as a n e t h n i c g r o u p , it h a s b e e n d e m o n s t r a t e d t h a t c h a n g e s in t h e l e a r n ­
in g e n v i r o n m e n t c a n i m p r o v e t h e i r m a t h e m a t i c s p ro f ic ie n c y to le v e ls e q u a l to
t h e m a jo rity p o p u l a t i o n . E v i d e n c e suggests t h a t a l t h o u g h d ir e c t i n s t i t u t i o n a l b a r ­
riers in e c o n o m i c a l l y d i s a d v a n t a g e d s c h o o l s , s u c h as t h e e m p h a s i s o f v o c a t i o n a l
o v e r a c a d e m i c s u b j e c ts ( D a v i s 1 9 8 6 ) a n d l a c k o f c o m p u t e r a c c e s s ( A n d e r s o n ,
W e l c h , a n d H a rri s , 1 9 8 4 ) c a n a c c o u n t for s o m e o f th i s d if f e r e n c e , m o r e s u b t le
c u rr ic u la r c h a n g e s c a n play a key ro le in r e t e n t i o n a n d a c h i e v e m e n t . For e x a m p le ,
B a ra tz e t a l. ( 1 9 8 5 ) f o u n d t h a t A f r i c a n A m e r i c a n s t u d e n t s a re m o r e li k e ly to
use c o m p u t e r s f o r r o u t i n e d rill; h e n c e , t h e p r o b l e m is n o t s i m p l y t h e a v a i l a b i l ­
ity o f c o m p u t e r s , b u t a ls o t h e i r s t y le o f u t i l i z a t i o n . T h e N a t i o n a l A s s e s s m e n t o f
E d u c a t i o n a l P ro g r e s s ( 1 9 8 3 ) s t u d y o f m a t h p e r f o r m a n c e in s e v e n t e e n - y e a r - o l d
A fric a n A m e ric a n s re p o rte d th e g re a te st d e fic ie n c ie s a t t h e a p p lic a tio n s level,
a n d s e v e r a l r e s e a r c h e r s ( U s i s k i n 198 5; D a v i s 1 9 8 9 ; M a l c o m 1 9 8 3 ) h a v e r e c o m ­
m e n d e d r e v i s i o n o f c o u r s e s to e m p h a s i z e m o r e i n t e r d i s c i p l i n a r y a n d “re a lw o r l d ” m a t h e m a t i c s i n s t r u c t i o n as w e ll as “ a c t i o n - o r i e n t e d ” p ed a g o g y .
C o m p u t e r - b a s e d l e a r n i n g h a s d e m o n s t r a t e d t h e c a p a b i l i t y fo r b o t h i n t e r a c t i v e
a n d i n t e r d i s c i p l i n a r y m a t h e m a t i c s i n s t r u c t i o n (K e it e l a n d R u r h v e n 1 9 9 3 ) , a n d
S t i f f e t al. ( 1 9 9 3 ) s p e c i f ic a lly p o i n t t o c o m p u t e r - b a s e d le a r n in g , as a p r o m i s i n g
fo r u m fo r b r i n g i n g t h e s e c h a n g e s t o A f r i c a n A m e r i c a n s t u d e n t s . T h e s e n e e d s
c o u l d be d i r e c t l y a d d r e s s e d by a p p l y i n g A f r i c a n f r a c ta ls t o t h e c la s s r o o m .
I n a d d i t i o n to c h a n g e s in s t r u c t u r a l a s p e c t s o f m a t h e m a t i c s t e a c h i n g ,
s e v e r a l r e s e a r c h e r s a n d i n s t r u c t o r s h a v e i n i t i a t e d c u l t u r a l l y e n r i c h e d c u r r ic u l a .
T h e r a t i o n a l e for t h i s a p p r o a c h c o m e s f r o m a v a r i e t y o f p e r s p e c t i v e s (e.g.,
V y g o tsk ian le a rn in g th e o ry ). P o w ell (1 9 9 0 ) n o te s th a t p e rv a s iv e m a in s tre a m
stereo ty p es o f scien tists a n d m a t h e m a tic ia n s co n flic t w ith A fric a n A m e ric a n
c u ltu ral o rie n ta tio n . S im ila r c o n flic ts b e tw e e n A fric a n A m e ric a n id e n tity and
m a t h e m a t i c s e d u c a t i o n in t e r m s o f s e l f - p e r c e p t i o n , c o u r s e s e l e c t i o n , a n d
c a r e e r g u i d a n c e h a v e b e e n n o t e d (cf. H a l l a n d P o s t m a n - K a m m e r 1 9 8 7 ; B o y e r
1 9 8 3 ). B u t we s h o u l d n o t a ss u m e t h a t th is c o n s t i t u t e s a p r o b l e m o f “se lf-e s te e m .”
T h e r e l a t i o n b e t w e e n c u l t u r a l i d e n t i t y a n d l e a r n i n g is q u i t e c o m p l e x ; it w o u ld
be n a i v e to su g g e s t t h a t t o d a y ’s A f r i c a n A m e r i c a n s t u d e n t s h a v e t h e s a m e r e l a ­
t i o n to i d e a s a b o u t t h e i r a n c e s t r y as d i d s t u d e n t s in p r e v i o u s d e c a d e s , a n d in
n o case h as th e re ever b e e n a sim p le “m im ic ry ” of A fric a n cu ltu re. R ath er,
e th n o g r a p h ic research (H e b d ig e . 1987; M e r c e r 1988; R ose 1994) show s th a t
A f r i c a n A m e r i c a n y o u t h a c ti v e ly c o n s t r u c t i d e n t i t y u sin g a w id e v a r i e ty o f c u l ­
t u r a l sig n ifie rs .
224
] m j) lic a tio n s
F o r th is re as o n , a p p li c a ti o n s o f A f r i c a n fractals will h a v e to .stress d esig n tools
a n d g u id e d dis c o v ery , a n d a v o id p assiv e p r e s e n t a t i o n . W h i l e “ i n t e r a c t i v e ” h a s
b e c o m e a c a t c h w o r d in m u l ti m e d i a , m a n y o f th e s e system s m e r e ly use t h e c o m p u t e r
like a slide p ro je c to r, w i t h s t u d e n ts pressing d if fe re n t b u t t o n s t o see v a r i o u s im ages.
M u l t i m e d i a in t h i s fo r m h a s a d i s t i n c tl y “c a n n e d ” feel t o it. T h e d e s i g n a p p r o a c h ,
in c o n t r a s t , offers s t u d e n t s to o ls for c o n s t r u c t i n g p a t t e r n s o f t h e i r o w n c r e a t i o n .
T h a n k s t o m a n y p a r t i c i p a n t s — in p a rtic u la r, p r o g r a m m e r s T Q B erg a n d J a r o n
S a m p s o n , a n d m i n o r i ty m a t h - e d u c a t i o n sp ecialist G l o r i a G i l m e r — we h a v e s t a rte d
d e v e l o p m e n t o f a n A f r i c a n fra c ta ls s o f tw a r e m a t h lab. T h e la b b e g in s w i t h s i m u ­
la t io n s o f t r a d i t i o n a l A f r i c a n p a t t e r n s a n d s h o w s s t u d e n t s h o w t h e m a t h e m a t i c a l
s t r u c t u r e b e h i n d t h e s e d e s i g n s offers t h e m to o ls t o c r e a t e t h e i r o w n .
A g a i n , i t is i m p o r t a n t to stress t h a t A f r i c a n A m e r i c a n s t u d e n t s a r e n o t
e x p e c t e d t o b e i n t e r e s t e d in t h e m a t e r i a l o u t o f a s i m p l e i d e n t i t y r e f l e c t i o n ,
a n y m o r e t h a n t h e y w o u ld n e c e s s a rily be i n t e r e s t e d in w e a r i n g D n s h ik i s h i r t s a n d
A fro s . R a t h e r , it is t h e o p p o r t u n i t y t o c r e a t e n e w c o n f i g u r a t i o n s a n d s y n t h e s e s
t h a t C o m b in e tr a d it io n a n d i n n o v a t i o n t h a t are significant. A t t h e J u n e 199 6 m e e t ­
in g o f t h e C o l u m b u s U r b a n Y o u t h C o n f e r e n c e , w e e x p l o r e d t h e s e c o n n e c t i o n s
w i t h a class o f e i g h t e e n 1 2 - y e a r - o ld A f r i c a n A m e r i c a n s t u d e n t s . T h e first class
m e e t i n g i n t r o d u c i n g t r a d i t i o n a l a r c h i t e c t u r e w as a n e a r d is a s te r ; d e s p i t e m u l t i m e d i a a n d m a n i p u l a t i v e s , it a p p e a r e d t h a t t h e p r i m i e i v i s t a s s o c i a t i o n s w ith
“ m u d h u t s ” w e r e a s t r o n g d e t e r r e n t . T h e f o l l o w i n g s e s s io n , u s i n g t h e G h a n a i a n
log s p i r a l— c e l l u l a r a u t o m a t a — o w a ri r e l a t i o n s , w as q u i t e su c c e s s fu l, p r o b a b l y
b e c a u s e t h e c o m b i n a t i o n o f t r a d i t i o n a l re lig io u s k n o w l e d g e a n d m a t h e m a t i c a l
g r a p h i c s s e n t a m o r e c l e a r a n t i p r i m i t i v i s t m e s s a g e d B u t in a d e s i g n e x e r c i s e
w h e re th e s tu d e n ts b eg an w ith c o m p u te r g ra p h ic s sim u la tio n s o f th e G h a n a ia n
l o g a r i t h m i c s p i r a l p a t t e r n s , t h e y s h o w e d l i t t l e i n t e r e s t in p r o d u c i n g f u r t h e r
im itations o f t h e A f r i c a n d e s i g n s . R a t h e r , t h e s t u d e n t s q u i c k l y c a u g h t o n to
v is u a l c o r r e l a t e s o f t h e e q u a t i o n p a r a m e t e r s a n d b e g a n a fr e e -fo r- a ll c o m p e t i t i o n
t o see w h o c o u l d m a k e t h e m o s t b izarre p a t t e r n s . T h e i r i n t e r e s t a p p e a r e d to be
s p a r k e d by t h e A f r i c a n c o n n e c t i o n s , h u t q u i c k l y w e n t b e y o n d t h e m .
P erh ap s m ore im p o rta n t th a n m itig a tin g a d ire c t co n flic t b e tw e e n e th n ic
id e n t it y a n d m a t h e m a t i c s , u sing A f r i c a n fractals in t h e c la s s ro o m m i g h t h e l p guard
a g a in st a n o v e re m p h a sis o n b io lo g ical d e te r m in is m , w h ic h h a s b e e n fou n d
ad versely to affect m a t h e m a t i c s le a r n in g . G e a r y ( 1 9 9 4 ) rev ie w s cro s s-c u ltu ra l s t u d ­
ies t h a t i n d i c a t e t h a t w h i l e c h i l d r e n , t e a c h e r s a n d p a r e n t s in C h i n a a n d j a p a n
t e n d t o v ie w d if fic u lty w i t h m a t h e m a t i c s as a p r o b l e m o f t i m e a n d e f f o r t, t h e i r
A m e r i c a n c o u n t e r p a r t s a t t r i b u t e d i f f e r e n c e s in m a t h e m a t i c s p e r f o r m a n c e to
i n n a t e a b il it y ( w h i c h c a n t h e n b e c o m e a self-fu lfillin g p r o p h e c y ) . F o r A f r i c a n
A m e r i c a n s , b i o l o g i c a l d e t e r m i n i s m h a s b e e n c l o s e ly a s s o c i a t e d w i t h m y t h i c
F utures fo r A fr ic a n fractals
s t e r e o t y p e s a b o u t “p r i m i t i v e p e o p l e ” (e.g., t h e f a b l e t h a t A f r i c a n s c o u n t “o n e ,
t wo, t h r e e , m a n y ”). By s h o w i n g t h e p r e s e n c e o f c o m p l e x m a t h e m a t i c a l c o n c e p t s
in A f r i c a n c u l t u r e , w e c a n m e n d s o m e o f t h a t d a m a g e . S i n c e r e d u c t i v e m y t h s o f
b i o l o g i c a l d e t e r m i n i s m a r e d e t r i m e n t a l t o m a t h e m a t i c s l e a r n i n g for s t u d e n t s o f
all e t h n i c b a c k g r o u n d s , all s t u d e n t s c o u l d p o t e n t i a l l y b e n e f i t f r o m t h i s m a t e r i a l .
Finally, w e s h o u l d n o t e t h a t t h e i n c r e a s i n g use o f m u l t i c u l t u r a l c u r r i c u l u m
ma t e r i a l s in t h e arts a n d h u m a n i t i e s h a v e n o t b e e n m a t c h e d in t h e sci ences. T h i s
c o u l d s e n d a m e s s a g e t o m i n o r i t y s t u d e n t s t h a t t h e i r h e r i t a g e is o n l y p e r t i n e n t
t o t h e a r t s a n d h u m a n i t i e s , a n d t h a t t h e s c i e n c e s a r e r e a l l y for p e o p l e f r o m
o t h e r e t h n i c groups. In a d d i t i o n , s o m e text s s u c h as Multicultural Mathematics ( N e l ­
s o n 1 9 9 3 ) h a v e e m p h a s i z e d o n l y C h i n e s e , H i n d u , a n d M u s l i m e x a m p l e s , so
t h a t e v e n in c ase s w h e r e m u l t i c u l t u r a l i s m is used, A f r i c a n m a t h ma y be left o u t
( s ee K at z 1 992 for a s i mi l a r c r i t i q u e ) . A n d o f t h e f e w t e x t s t h a t d o use A f r i c a n
m a t h , a lm o s t all e x a mp l es are re s t ri ct ed t o p r i ma r y sc h o o l level. A g a i n , thi s restri c­
t i o n m i g h t u n i n t e n t i o n a l l y i m p l y p r i m i t i v i s m (i.e., t h a t m a t h e m a t i c a l c o n c e p t s
f r o m A f r i c a n c u l t u r e a r e o n l y c h i l d l i k e ) . F o r thi s r e a s o n , o u r l a b ’s i n c l u s i o n of
a d v a n c e d t o p i c s s u c h as f r ac t a l g e o m e t r y , c e l l u l a r a u t o m a t a , a n d c q m p l e x i t y
a r e w o r t h t h e e x tr a effort t o t i e i n t o a s e c o n d a r y s c h o o l c u r r i c u l u m ( w i t h o u t o v e r ­
l o o k i n g t h e use o f s t a n d a r d t o p i c s s u c h as l o g a r i t h m i c s ca l i n g , g e o m e t r i c c o n ­
struction, and trigonometry),
W h i l e t h e mu l t i m e d i a l a b ’s mo s t significant p o t e n t i a l for i m p r o v i ng e d u c a t i o n
is in m a t h e m a t i c s , we s h o u l d n o t i gnor e A f r i c a n St u d i es . A f r i c a n art, for e x a mp l e ,
is i ncr easi ngly used in s e c o n d a r y s c h o o l s across t h e n a t i o n , a n d use o f our lab c o u l d
g r eat l y e n h a n c e s u c h courses. First, as n o t e d a b o v e , it p r o v i d e s a n a l t e r n a t i v e to
d e t r i m e n t a l m i s r e p r e s e n t a t i o n s o f A f r i c a n s as “p r i m i t i v e " p e o p l e . In art h i s t o r y
l es s ons , for i n s t a n c e , s t u d e n t s o f t e n l e a r n a b o u t t h e g e o m e t r i c basis for G r e e k
a r c h i t e c t u r e o r R e n a i s s a n c e p a i n t i n g , w h i l e c o m m e n t a r y o n A f r i c a n w o r k s is
o f t e n r es t r i ct e d t o d i s c u s s i on o f " n a t u r a l n e s s ” o r “e m o t i o n a l e x p r e s s i o n . ” S e c o n d ,
t h e l ab aids in i n t e g r a t i v e c u r r i c u l a d e v e l o p m e n t (see R o t h 1 9 9 4 o n difficul ti es
in t h i s ar ea) . It w o u l d a l l o w m a t h t e a c h e r s w h o w o u l d li ke t o i n c l u d e e t h n o m a t h e m a t i c s c o m p o n e n t s in t h e i r t e a c h i n g t o refer t o e x a m p l e s in w h i c h s t u d e n t s
are al r eady e ng a g e d , a n d w o u l d p r o v i d e art t e a c h e r s w i t h n e w t ool s for d e s i g n a n d
analysis. S i m i l a r a d v a n t a g e s c o u l d b e o b t a i n e d i n o t h e r A f r i c a n S t u d i e s areas.
I n f o r m a t i o n tec h n o lo g ie s a n d s u s t a i n a b l e d e v e l o p m e n t
T h e use o f i n d i g e n o u s k n o w l e d g e s y s t e m s i n d e v e l o p m e n t g o e s b a c k t o c o l o ­
n i a l a p p r o p r i a t i o n s , b u t in t h e p o s t c o l o n i a l c o n t e x t t h e s e s y s t e m s h a v e t a k e n
o n n e w m e a n i n g as a s i g n o f e i t h e r e p i s t e m o l o g i c a l i n d e p e n d e n c e , o r a t l e a s t
226
implicatfoni
a m o r e e g a l i t a r i a n v i e w o f k n o w l e d g e s y s te m s . In c h a p t e r 10 , f o r e x a m p l e , w e
sa w t h e s c a l i n g s p i r a l s o f J o l a s e t t l e m e n t a r c h i t e c t u r e t h a t a r o s e f r o m t h e i r
c ir c u la r b u ild in g s; th e F re n c h re s e a rc h o rg a n iz a tio n E N D A h asT b u ilt a n
i m p l u v i u m c r e a t e d by t h e c o m b i n a t i o n o f m o d e r n m a t e r i a l s a n d t h i s t r a d i ­
t i o n a l J o l a d e s i g n . A n o t h e r o f E N D A ’s ru r a l d e v e l o p m e n t p r o j e c t s t h a t i n c o r ­
p o r a t e b o t h t r a d i t i o n a l f r a c t a l a r c h i t e c t u r e a n d m o d e r n t e c h n i q u e s is s h o w n
i n f i g u r e 14 .6.
I n c h a p t e r 6 w e saw h o w t h e s c a l i n g p a t t e r n s o f k c n t e c l o t h w e re c r e a t e d
to m a t c h t h e sc a lin g o f s a c c a d i c eye m o v e m e n t s as t h e y s c a n fro m t h e face to t h e
body. T h e G h a n a i a n B r o a d c a s t i n g C o r p o r a t i o n , G h a n a ’s n a t i o n a l t e l e v i s i o n
c h a n n e l , h a s c o n t i n u e d t h i s p r a c t i c e in t h e c o n t e x t o f m o d e r n i n f o r m a t i o n t e c h ­
n olo gie s, utilizing t h e scalin g p a t t e r n o f k e n t e c l o t h in t h e i r test p a t t e r n (fig. 14.7).
W h e r e a s t h e t r a d i t i o n a l s c a l in g w as a p p l i e d t o t h e h u m a n v is u a l s c a n , t h i s t e c h nologized versio n m a kes use o f th e sa m e p a t t e r n for t e s ti n g t h e v id e o sca n . A sim ple
a p p l i c a t i o n , b u t it s h o w s t h a t A f r i c a n f r a c t a l s a re n o t j u s t r e s t r i c t e d t o l o w - t e c h
a d a p t a t i o n s ; t h e y c a n a ls o p r o v i d e s o m e useful b ri d g e s b e t w e e n t r a d i t i o n a l a n d
h i g h - t e c h w orld s.
I n c h a p t e r 10 w e sa w t h a t t h e r e w e r e ties b e t w e e n t h e t r a d i t i o n a l k n o w l ­
e d g e s y s te m s s u p p o r t e d b y A f r i c a n f r a c ta l s a n d t h e p r o d u c t i v e m a i n t e n a n c e o f
th e s e s o c ietie s in w h a t P e r B ak w o u ld call a s t a t e o f se lf-o rg a n iz e d c ritica lity . T h i s
su g g es ts t h a t m o s t o f t h e in d i g e n o u s A f r i c a n s o c i e t i e s w e r e n e i t h e r u t t e r l y a n a r ­
c h i c , n o r frozen in s t a t i c o r d e r ; r a t h e r , t h e y u ti li z e d a n a d a p t i v e f le x ib ility t h a t
c o u ld b e a p p l i e d t o m o d e r n d e v e l o p m e n t . B u t d e c a d e s o f r e s e a r c h h a v e s h o w n
figure
1 4 .6
M o d e rn ise d
f r a c t a l v illag e
This ENDA project in Burkina
Faso combined the traditional
fractal structure with modern
construction techniques.
Futures fo r A fr ic a n fracials
227
FIGURE I 4 . 7
K e n t e c lo th in th e
Q h c in a ia n B r o a d c a s tin g
C o r p o ra tio n t e s t p a t t e r n
Kente cloth pattern is used in
the upper right-hand quadrant
of the large circle.
t h a t a t o p - d o w n a p p r o a c h t o d e v e l o p m e n t , e v e n t h a t m a k i n g use o f i n d i g e n o u s
k n o w l e d g e , is o f t e n less e f f e c t i v e t h a n a b o t t o m - u p , “grass r o o t s ” a p p r o a c h .
A d o p t i n g in f o rm a tio n tec h n o lo g y t o nir al areas could prov id e t h e o p p o rtu n it y
for p u t t i n g A f r i c a n f r ac t a l s t o w o r k i n s u s t a i n a b l e d e v e l o p m e n t .
I n a d d i t i o n t o t h e n e e d for b o t t o m - u p a u t h o r i t y , r e s e a r c h e r s h a v e d e m o n ­
s t r a t e d t h e c r i t i c a l r o l e o f w o m e n i n A f r i c a n d e v e l o p m e n t ( e. g . , B o s e r u p
1 9 70 ; N e l s o n 1 9 8 1 ; A d e p o j u a n d O p p o n g 1 9 9 4 ) ; p a r t i c u l a r l y in t e r m s o f t h e
g e n d e r e d d i v i s i o n o f l a b o r in r u r a l s o c i e t i e s ( B e n e r i a 1 9 8 2 ) . W h i l e m u c h o f
t hi s a n a l y s i s h a s f o c u s e d o n t h e v u l n e r a b i l i t y o f w o m e n i n b e a r i n g t h e b r u n t
o f e c o n o m i c c h a n g e , it h a s a l s o s t n r t e d h n e w a p p r e c i a t i o n f o r t h e e x t e n s i v e
k n o w l e d g e s y s t e m s t h a t e x i s t e d i n p r e c o l o n i a l w o m e n ’s a c t i v i t i e s . S i n c e m a n y
o f t h e s e p r a c t i c e s c o n t i n u e t o d a y ( a l b e i t i n m o d i f i e d f o r m ) , w o m e n ’s i n d i g e ­
n o u s k n o w l e d g e sy s t ems h a v e b e c o m e a n i m p o r t a n t r e s o u r c e in n e w a p p r o a c h e s
to d e v e lo p m e n t.
S o m e obvious challenges in cl ude e n v i r o n m e n t a l d am ag e (increasing salin­
ization, def o r es t a t i o n , a n d d e s er t i f i c a t i o n ) , e x t e r n a l e c o n o m i c pressures ( t h e m o v e
to c a s h - c ro p p in g , tou ri sm , a n d m i g r a t i o n to cities; abus e of p o w e r by pr i va te
c o r p o r a t i o n s ) , incr eas ed disease ( A I D S a n d o t h e r viruses), poli ti cal u nr es t
( e t h n i c conflict, u n c o n t r o l l e d m ili ta r y force, abuse of a u th o r it y ) , a n d dam a g e
t o t h e s o c i o c u l t u r a l s y s t e m ( d i s r u p t i o n s o f w o m e n ’s t r a d i t i o n a l a u t h o r i t y , loss
of t r a d i t i o n a l k n o w l e d g e s y s t e m s ) . W h i l e al! o f t h e s e a r e far t o o l a r g e t o be
a d d r e s s e d by a n y o n e a p p r o a c h , n o n e o f t h e m c a n be v i e w e d i n i s o l a t i o n f r o m
t h e o t h e r s . I n Ni g e r i a, for e x a m p l e , t h e S h e l l P e t r o l e u m D e v e l o p m e n t C o m p a n y
I m p l ic a tio n s
228
b e g a n o p e r a t i o n s in O g o n i l a n d t h a t e v e n t u a l l y le d t o w i d e s p r e a d e n v i r o n ­
m e n t a l d a m a g e ; a t t e m p t s t o p r o t e s t t h r o u g h t h e p re s s a n d o t h e r c o m m u n i c a ­
t i o n e v e n t u a l l y led t o t h e e x e c u t i o n o f O g o n i w r i t e r K e n S a r o - W i w a ( S o y n i k a
1 9 9 4 ) . F r e e d o m o f t h e p re s s is n o t a s e p a 'r a t e issu e from,, p r o t e c t i o n o f t h e
e n v iro n m e n t.
It is righ t to decry abuse o f a u th o rity , b u t r e p la c in g o n e a u t h o r i t y w ith a n o t h e r
is n o t n e c e s s a r i l y g o i n g t o p r o v i d e a l o n g - t e r m s o l u t i o n . A f r i c a n f r a c ta l s s u g g e s t
t w o a l t e r n a t i v e a p p r o a c h e s . F ir s t, w h a t is n e e d e d is n o t E. F. S c h u m a k e r ’s c a ll
fo r " s m a ll is b e a u t i f u l , ” b u t r a t h e r , a s e l f - o r g a n i z e d a p p r o a c h t o c h a n g e s in t h e
re la tio n s b e tw e e n scale a n d th e s o c i o e n v ir o n m c n ta l sy stem s— n o t just a p p r o ­
p ria te tec h n o lo g y , b u t a p p ro p r ia te scalin g . S e c o n d , 'm o r e c ritic a l a tt e n ti o n
n e e d s t o be p a i d to t h e a r t i f i c i a l / n a t u r a l d i c h o t o m y , w h i c h t e n d s t o b e t r a p p e d
in e i t h e r t h e o r g a n i c i s t s ’ d e s i r e fo r u n t o u c h e d n a t u r e ( e .g ., H u g h e s 1 9 9 6 ) , o r t h e
t e c h n o - o p t i m i s t ’s d e s i r e fo r r e s o u r c e e x t r a c t i o n .
A n a l t e r n a t i v e t o t h e s e d a m a g i n g e x t r e m e s c a n b e f o u n d in C a l e s t o u s
j u m a ’s 1 9 8 9 classic, T h e G e n e Fh m ters. R a t h e r t h a n a p r e s e r v a t i o n i s t p e r s p e c ti v e ,
i n w h i c h i n d i g e n o u s s o c i e t y w o u l d b e p o r t r a y e d as n a t u r a l e l e m e n t s o f a n
u n c h a n g i n g e c o s y s te m , o r a t e c h n o c r a t i c p r o f ite e r in g p e r s p e c t i v e , . i n w h i c h a g ri­
c u l t u r a l d e v e l o p m e n t is m e r e ly a q u e s t i o n o f m a x i m i z i n g y ie l d s w i t h i m p o r t e d
s tra in s , J u m a p ro v i d e s e v i d e n c e for in d i g e n o u s a g r i c u l t u r a l a c t i v i t y as s u s t a i n a b l e
b i o t e c h n o l o g y . H is s t u d i e s s h o w a l o n g - s t a n d i n g A f r i c a n t r a d i t i o n o f n e w s e e d
v a riety d e v e lo p m e n t th a t c o m b in e d eco lo g ical su s ta in a b lilty w ith in n o v a tio n
a n d e x p e r i m e n t a t i o n . T h e s e p r a c t ic e s h a v e b e e n t h r e a t e n e d hy c o r p o r a t e m o n o c r o p p i n g , w h i c h c a n c a u s e soil d e p l e t i o n , o v e r - d e p e n d a n c e ora i n s e c t i c i d e s , loss
o f g e n e t i c v a r i a t i o n , a n d o t h e r s o c ia l a n d e c o l o g i c a l crises, as w e ll as t h e a p p r o ­
p r i a t i o n o f t h e s e g e n e t i c re s o u r c e s by a b i o t e c h n o l o g y i n d u s t r y w i t h li t t l e i n t e r ­
est in i n d i g e n o u s le g a l rig h ts . J u i n a n o t e s t h a t t h e c h a l l e n g e n o w f a c i n g A f r i c a n
a g r i c u lt u r a li s t s is n o t j u s t p r e s e r v a t i o n o f b io d iv e rsity , b u t a ls o a c c e s s to t h e legal,
t e c h n i c a l , a n d f i n a n c i a l a p p a r a t u s t h a t w o u ld a ll o w t h e m t o r e a p t h e p ro fit t h a t
c o u l d s u s t a i n s u c h e c o l o g i c a l l y s o u n d efforts.
F r o m t h e v i e w p o i n t o f c o m p l e x i t y th e o r y , J u m n ’s c r i t i q u e s u g g e s ts t h a t w e
a re t r a p p e d b e t w e e n t h e p e r i o d i c sta sis o f t h e p r e s e r v a t i o n i s t s ’ li m i t c y c le , a n d
t h e w h i t e n o i s e o f t h e p r o f i t e e r i n g p o s i t i v e f e e d b a c k l o o p . A s w e saw in t h e s e
m a t h e m a t i c a l m o d e l s , b o t h a r e l a c k i n g in f l e x i b l e i n t e r a c t i o n s w i t h m e m o r y ;
t h e l i m i t c y c le b e i n g t o o t i e d t o it, a n d t h e w h i t e n o i s e b e i n g t o o free f r o m it.
I n f o r m a t io n te c h n o l o g ie s h a v e t h e p o t e n t i a l to p r o v i d e th is m e m o ry , d o c u m e n t i n g
i n d i g e n o u s k n o w l e d g e fr o m s e e d v a r i e t i e s a n d soil t y p e s t o g e n e s e q u e n c e s to
e c o t o p e s . By p r o v i d i n g i n f o r m e d ru ra l a c c e s s t o i n f o r m a t i o n t e c h n o l o g i e s ,
A f r i c a n a g r i c u l t u r a l i s t s c a n ta k e a s t e p t o w a r d p r o t e c t i n g t h e i r g e n e t i c re s o u rc e s
Futures fo r A fr ic a n fractals
f r o m a p p r o p r i a t i o n a n d m o v e t o w a r d J u m a ’s a p p r o a c h , w h i c h we m i g h t call
“ b i o t e c h - d i v e r s i t y ” (cf. H a r a w a y 1 9 9 7 ; S h i v a 1 9 9 7 ) .
T o v i e w i n d i g e n o u s k n o w l e d g e as a s e l f - o r g a n iz i n g s y s t e m is o n e t h i n g , b u t
c r e a t in g t h e same b o t t o m - u p a p p r o a c h for a synth esis o f ecological sustainabli li ty a n d t e c h n o l o g i c a l ' d e v e l o p m e n t is a m u c h gr e a t e r c h a l l e n g e . For e x a m p l e ,
Rus s e l Bar s h n o t e s : “T h e r e is a n i m p l i c i t a s s u m p t i o n in t h e r e s e a r c h m e t h o d o l ­
ogy used t o e l i c i t t r a d i t i o n a l p h a r m a c o l o g i c a l k n o w l e d g e t h a t t hi s i n f o r m a t i o n
is r e c o r d e d a n d t r a n s m i t t e d d i g i t a l l y ( n u m b e r s a n d / o r w o r d s ) . . . [ r a t h e r t h a n ]
internalizing an an a lo g m o d e l ” (19 9 7 , 3 3 - 3 4 ) .
N a t i v e Seeds, a b o t a n i c a l o rg an i z at i o n d ed ic a te d to t h e c o n t i n u a t i o n of
indi genous p l a n t st ock, h a s b e e n c r e a t i n g a “ c ult ural m e m o r y b a n k ” t h a t will record
b o t h a n a l o g a n d d i g i t a l i n f o r m a t i o n o n N a t i v e A m e r i c a n a g ri c u lt u r e . T h e c o n ­
cept, originating from P hilippine e th n o b o t a n is t Virginia N azarea-Sandoval
( 1 9 9 6 ) , d o c u m e n t s t h e c o m b i n a t i o n o f c u l t u r a l a n d b i o l og i c a l i n f o r m a t i o n a b o u t
t h e crops , seeds, f a r m i n g , a n d u t i l i z a t i o n m e t h o d s . T h e i n f o r m a t i o n , i n c l u d i n g
v i d e o i n t e r v i e w s , is s t o r e d o n C D - R O M , w i t h a cc e ss c o n t r o l l e d e n t i r e l y by t h e
i n d i ge no u s farmers. In t h e U . S . c o n t e x t , w h i c h is o v e r l o a d e d w i t h e l e c t r o n i c t e c h ­
nology a n d e t h n o c i d e , this a p p r o a c h ma k es sense, b u t th e A fr i c a n c o n te x t ,
w i t h its e n o r m o u s i n d i g e n o u s p o p u l a t i o n a n d s p a r se e l e c t r o n i c t e c h n o l o g y , will
call for t e c h n i q u e s t h a t c a n h a v e a w i d e r i m p a c t , o n e t h a t i n c l ud e s d e v e l o p m e n t
o f a t e c h n o l o g i c a l i n f r a s t r u c t u r e as well.
If t h e r e is t o be soci al t r a n s f o r m a t i o n t h r o u g h grass-roots t e c h n o l o g i c a l i n n o ­
v a t i o n , it will r e q u i r e m u c h m o r e p a r t i c i p a t i o n t h a n a g r i c u l t u r a l s y s t e m s a l o n e .
O t h e r k i n d s o f i n f o r m a t i o n t e c h n o l o g y d e v e l o p m e n t c o u l d i n c l u d e f l exibl e e c o ­
n o m i c n e t w o r k s , w h i c h a l l o w s ma l l -scale business''- > • c o l l a b o r a t e in t h e m a n u ­
f a c t u r e ot p r o d u c t s a n d se r vi c e * u . c y c o u l d n o t p r o d u c e i n d e p e n d e n t l y . T h e s e
n e t w o r k s h a v e c r e a t e d s t r o n g r e v i t a l i z a t i o n i n c e r t a i n r ur al a r e a s o f E u r o p e
( Sabel a n d Piore 1990), a n d h a v e s h o w n p r o mi se in pil ot studies in t h e rural U n i t e d
S t a r e s as wel t (e.g., A C E n e t in s o u t h e r n O h i o ) . T h e use o f c o m p u t e r s t o o r g a ­
nize p r o d u c t i o n a n d v e n d i n g a n d p r o v i d e d y n a m i c s e a r c h e s for t h e a p p r o p r i a t e
m a r k e t n i c h e — o n e w h i c h w o u l d b e e n v i r o n m e n t a l l y a n d soci all y s u s t a i n a b l e as
wel l as pr of i t a b l e— c o u l d s p r e a d t h e b e n e f i t s o f n e w i n f o r m a t i o n t e c h n o l o g i e s to
r he mi crobusi ness level w i t h o u t h a v i n g t o p ut a lap t o p in every pushcart, anti microf i n a n c i n g p r o g r ams h a v e al r ea d y p r o v e d successful in m a n y T h i r d W o r l d c o u n t r i e s
(Sera geld in 1997).
A f r i c a n t r a d i t i o n s o f d e c e n t r a l i z e d d e c i s i o n m a k i n g c o u l d a l s o be c o m ­
b i n e d w i t h n e w i n f o r m a t i o n t e c h n o l o g i e s , c r e a t i n g n e w f or ms t h a t c o m b i n e
d e m o c r a t i c r u l e w i t h c o l l e c t i v e i n f o r m a t i o n s h a r i n g . T h e i d e a o f “e l e c t r o n i c
d e m o c r a c y " h a s s l owl y b e e n d e v e l o p i n g o v e r t h e I n t e r n e t ; b u t t h e ef f o r t s h a v e
229
230
I m p l ic a tio n s
b e e n h a m p e r e d by t h e t e n d e n c y t o a s s u m e t h a t v i r t u a l v o t i n g m u s t b e t h e s a m e
as o r d i n a r y v o ti n g . P e r h a p s t h e n e u r a l n e t sty le o f A f r i c a n d e c i s i o n m a k i n g c o u ld
b e u t i l i z e d i n t h e W e s t as w e ll, w i t h v o t e r s i n d i c a t i n g p r o p o r t i o n a b s t r e n g t h s
fo r v a r i o u s o p t i o n s . C o n v e r s e l y , p e r h a p s t h e r e a r e w a y s to” a p p l y c o m p u t e r
m e d i a t o e n h a n c e A f r i c a n d e c i s i o n m a k i n g . O n e a p p r o a c h w o u ld b e t h e d e v e l o p ­
m e n t o f c o m m u n i t y n e t w o r k s t h r o u g h p u b l i c - a c c e s s t e r m i n a l s ( S c h u l e r 1 9 9 5 ).
A n d t h e e n o r m o u s d e v e l o p m e n t in e l e c t r o n i c s e c u r i t y m e a s u r e s , c r e a t i n g sy s­
te m s t h a t sty m ie e v e n t h e m o s t s o p h i s t i c a t e d h a c k e r s ( e n c r y p t i o n c o d e s , fin g er­
p r i n t s c a n n e r s , e t c . ) , m i g h t f i n d u s e s in p r e v e n t i n g v o t e r f r a u d t h a t is so
c o m m o n in u n s t a b l e p o l i t i c a l r e g i m e s .
N ig e ria n A m e r ic a n c o m p u te r e n g in e e r E g o n d u O n y e je k w e h a s started
e ff o rts t o a p p ly i n f o r m a t i o n t e c h n o l o g y n e t w o r k i n g in A f r i c a n d e v e l o p m e n t a l
p r o j e c t s u s i n g c o m p l e x i t y t h e o r y a s a g u i d i n g p r i n c i p l e . O n e a r e a s h e c it e s is th e
p r o b l e m o f la n d o w n e r s h i p (fo r e x a m p l e , see C h a r n l e y 1 9 9 6 ) . S h e n o t e s t h a t t h e
c o n t i n u a l d i v i s i o n o f l a n d p r o m o t e d by t h e c o l o n i a l le g a c y o f t e n r e s u l t s in
u n p r o d u c t i v e e c o n o m i e s o f s c a le , b u t t h a t g o v e r n m e n t o w n e r s h i p t e n d s t o m a k e
c o n d i t i o n s w o rse hy a d d i n g m o r e h ie r a r c h y . “R e s o l v i n g t h e l a n d p r o b l e m re q u ires
a n o n - h i e r a r c h i c a l m e t h o d o f o r g a n i z a t i o n , a sy s te m in w h i c h c o o p e r a t i v e b e h a v ­
i o r is r e w a r d e d a t t h e s a m e t i m e t h a t i n d i v i d u a l i n n o v a t i o n c a n fl o u ris h ; a c o m ­
b i n a t i o n o f c o o p e r a t i o n a n d c o m p e t i t i o n like we see in c e ll u la r a u t o m a t a a n d o t h e r
c o m p u t a t i o n a l m o d e l s o f s e l f - o r g a n iz i n g sy s tem s . W h a t b e t t e r w ay to e n c o u r a g e
t h i s t h a n t h r o u g h c o m p u t i n g a n d i n f o r m a t i o n n e t w o r k s ? ” **
N e i t h e r t h e A f r i c a n f r a c ta l s f r a m e w o r k n o r d i s s e m i n a t i o n o f i n f o r m a t i o n
te c h n o l o g ie s offers p a n a c e a s . M y p o i n t is, rath er, t h a t t h e sh ift in p e rs p e c tiv e often
c a l l e d for in d e v e l o p m e n t n e e c t n o t b e e i t h e r c o n s e r v a t i v e r e t u r n t o -the p a s t, n p r
t h e e p i s t e m o l o g i c a l e q u i v a l e n t o f a n a l i e n i n v a s i o n . A f r i c a n f r a c t a l s o ff e r a
f r a m e w o r k t h a t is b o t h r o o t e d i n i n d i g e n o u s c u l t u r e s a n d c r o s s - p o l l i n a t e s w i t h
n ew hybrids.
APPENDIX
-Measuring--------------------------------------------- th e fractal---------------------------------------------dimension---------------------------------------------o f African---------------------------------------------settlem ent--------------------------------------------architecture--------------------------------------------
T h e r e a re s e v e r a l d i f f e r e n t w ays co e s t i m a t e t h e f r a c t a l d i m e n s i o n o f a s p a t ia l
p a t t e r n . In t h e c a s e o f M o k o u l e k (fig. 2 . 4 o f c h a p t e r 2 ) we h a v e a b l a c k - a n d w h i t e a r c h i t e c t u r a l d i a g r a m , w h i c h a l l o w s us to d o a t w o - d i m e n s i o n a l v e r s i o n
o f che r u l e r size v e r s u s l e n g t h p l o t s w e sa w i n c h a p t e r 1. By p l a c i n g t h e a r c h i ­
t e c t u r a l d i a g r a m o f M o k o u l e k u n d e r g rid s o f i n c r e a s i n g r e s o l u t i o n , a n d c o u n t ­
in g tire n u m b e r o f g rid c e lls t h a t c o n r a i n s o m e p a r t o f t h e d i a g r a m , we c a n p l o t
t h e in c r e a s e o f a r e a w i t h d e c r e a s i n g c e ll size ( j u s t as we o b t a i n e d a p l o t o f t h e
i n c r e a s i n g l e n g t h w i t h d e c r e a s i n g r u l e r size). F ig u r e
a
.i
s h o w s t h e re s u lts , i n d i ­
c a t i n g a f r a c t a l d i m e n s i o n o f 1.67— n o t t o o far f r o m t h e i .53 f r a c t a l d i m e n s i o n
t h a t is o b t a i n e d a n a l y t i c a l l y fr o m t h e c o m p u t e r s i m u l a t i o n .
F o r t h e a e r i a l p h o t o o f L a b b a z a n g a (fig . 2 .5 o f c h a p t e r 2 ) w e h a v e a n
im ag e in s h a d e s o f gray, a n d t h e s i m p l e g r i d - c o u n t i n g m e t h o d c a n n o t be app lie d .
It is p o s s i b l e t o r e d u c e t h e g ra y s c a l e t o b l a c k a n d w h i t e , b u t a n a l t e r n a t i v e
m e t h o d a ll o w s us to m a k e a m o r e d i r e c t m e a s u r e o f t h e s c a l in g p r o p e r t i e s . F i g ­
ure
a
. 2a
s h o w s _ t h e m e t h o d for f i n d i n g t h e s c a l i n g s l o p e o f 1 / F n o is e in a o n e ­
d i m e n s i o n a l t i m e s e r ie s hy a p p l y i n g a F o u r i e r t r a n s f o r m . In fig u re
a
.2b
w e see
h o w t h i s c a n b e a p p l i e d to a t w o - d i m e n s i o n a l s p a t i a l d i s t r i b u t i o n by s w e e p ­
in g t h e s a m e s p e c t r a l d e n s i t y m e a s u r e a r o u n d in p o l a r c o o r d i n a t e s . R a t h e r t h a n
tire l i n e o f o n e - d i m e n s i o n a l 1 / F n o i s e , a t w o - d i m e n s i o n a l d i s t r i b u t i o n is
A p p en d ix
232
c h a r a c t e r i z e d b y a c o n e . It is d i f f i c u l t t o s h o w t h e e n t i r e c o n e , b u t w e c a n t a k e
h o r i z o n t a l s l i c e s (fig.
a
.2b),
w hich show
b a z a n g a a n d its f r a c t a l s i m u l a t i o n (fig.
a
sim ilar
c h a r a c t e r i s t i c s f o r b o t h L ab -
.3)
log (cell size)
FIGURE A.J
M e a s u r i n g t h e f r a c ta l d i m e n s i o n o f M o k o u l e k
frequency
O n e - d i m e n s i o n a l t i m e s e r i e s f o r 1/ F n o i s e .
1/ F n o i s e s p e c t r a l d e n s i t y
from i - D Fo u rier transform .
low fre q u e n c ie s a t h ig h p o w e r
j-D F o u rier tra n sfo rm , w ith fre q u e n c y in p olar
coo rd in ates: w id e r c ir c le = h i g h e r fre q u e n c y .
T h e l i n e o f 1/ F n o i s e is r o t a t e d t o b e c o m e a c o n e .
h ig h fre q u e n c ie s at low p o w e r
F I G U R E A . 2 ..
U s i n g a 2 ' D F o u r i e r t r a n s f o r m to d e t e c t f r a c ta l s p a tia l d is tr ib u tio n s
low fre q u e n c ie s a t h ig h p o w e r
h ig h fre q u e n c ie s a t low po w er
a
h ig h f re q u e n c ie s a t lo w p o w e r
lo w fre q u e n c ie s a t h ig h p o w e r
FIGURE A .3
R e s u l t s o f a 2 - V F o u r ie r t r a n s f o r m a p p l i e d to a e r ia l p h o t o o f L a b b a z a n g a
(a) Spectra for aerial photo of Labhazunga (fig. 2.50 from chapter 2). (b) Spectra for fractal image
(Jig. 2.5b from chapter 2). Note that the axes of symmetry in the fractal can he seen in this spectral
density distribution, while none exist for that of Labbazanga.
Notes'
c h a p te r
i
i n t r o d u c t i o n to fr a cta l g e o m e t r y
j. For a hexagon example, see Washburn and Crowe (1988, 237). Numerical examples
can be found in Crump (1990, 39-40, 50-54, 105-106, 128-133).
2 . The number 10 was not only a basis for counting, but it also appeared in Chinese nat­
ural philosophy. In acupuncture, for example, the number 10 is created by the combi­
nation of the "five elements" (wu-yiin) and the binary yin/yang.
3. Michael Polanyi (1966) referred to this as "tacit knowledge.”
c h a p te r
2
Fractals
in A f n c c . n
settlement architecture
1. O n triangular churches, see Norberg-Schulz (1965, 172); for che Pantheon, see
ibid., 124.
2. Another passage, “path of the serpent,” is used only by royalty. It alternates left and
right as it approaches the center of the palace, and thus creates a scaling zigzag pattern.
The implication seems to be that even royalty must negotiate the fractal ranking, but
they can traverse it in a more direct route.
3. American readers are probably most familiar with nuclear families, but in Africa the
family structure typically extends to much larger networks. The English term "cousins,”
for example, emphasizes the nuclear family by .lumping all these relatives together, while
many African kinship systems have distinct terms for paternal parallel cousins, mater­
nal parallel cousins, paternal cross cousins, etc.
4. T he status difference between front and back is also expressed in the Ba-ila term for
slave: “one who grows up at the doorway” (Smith and Dale 1968 [ 1920I vol. 1, 304).
5. This is another meaning for the term "participant simulation.” In the first meaning, briefly
mentioned in the introduction, I defined it as an effort in cooperative modeling and
analysts, a rechnologized version of recent attempts in collaborative ethnography by
some anthropologists and their informants. In that sense it supports che humanist goals
236
N otes
of self-governing autonomy. But in the Mokoulek case I am also using it in the post­
modernist sense, a participant in a virtual world. The contrasting meanings and their
consequences are discussed in detail in chapter 10, where the two are brought together.
6. T h e results were published in Eglash and Broadwetl (1989), and are reproduced in
the appendix.
""
ch ap ter 3
F ra c ta ls in c ro ss -c u ltu ra l co m p ariso n
1. in general, anthropologists divide nonstate societies between “band” organization,
which is entirely decentralized and based mainly on consensus, and “tribal” organiza­
tion, in which there is an official leader but otherwise little political hierarchy. The term
"tribe” is controversial, however, since colonialists often used it to deny the existence
of indigenous state societies, so it is important ro separate the technical designation
front its colloquial use2. This is a complex designation in cultural studies, since the label of “traditional’'— or
worse yet, "authentic”— was used by colonial authorities to exercise control over
indigenous populations, and still o c c u r s in the neocolonial context to valorize the ‘‘van­
ishing native” while appropriating their cultural resources. See Minh-ha (1986),
Anzaldua (1987), Clifford (1988), and Bhaltba (1990) for discussion of some of these
issues.
3. Crowe and Nagy (1992), for example, have done extensive analysis of Fiji decoration,
and found 1 2 o f the 17 mathematically possible two-color strip symmetries, but none
of the designs they show are fractal.
4. Of course, nothing is absolutely certain when it comes to ancient history. Several
researchers have suggested that the Coptic designs from Egypt were an important
influence 011 the Celtic interlace patterns, and some Italian floor tiles were created by
North African artisans (Argiro 1968, 22). But one could just as easily argue the influ­
ence in reverse. Given the history of trade routes and travel, we should not attempt
to reduce designs to a singular origin; the goal is to see how any one society has built
up its particular repertoire of designs— from whatever sources— as part of a dynamic
yet culturally cp. <ific practice.
chapter
4
Intention and invention in design
1. This spatial metaphor of “underlying”— truth beneath the surface— can be a delusion
if we assume that there is never more than one true "essence” to be found. On the other
hand, claiming that no model is more accurate a generalization than any other is equally
misguided.
2. T h e postwar era marked a significant change in the role of nature as a potential model
for scientific discovery, as seen in the emerging disciplines of cybernetics and bionics
(Gray 1995).
chapter
7
Numeric systems
1. It is unfortunate that an otherwise excellent paper comparing African and Australian
ethnomathematics (Watson-Verran and Turnbull 1994) fails to make this distinction
between the iterative generation of linear and nonlinear number series.
2. Readers who recall the definition of nonlinear functions as involving, at minimum, some­
thing tike
may be puzzled by the idea of a nonlinear additive series. That is because
most of us were first exposed to the definition of “nonlinear” in the context of continuous
functions (e.g., differential equations). But discrete iteration (what is often called a “dif­
ference equation") can produce nonlinear steps with simple addition.
3. After giving a lecture on Bamana divination in the United States, I was approached
by a mathematics faculty member who was quite taken by this phrase. "That’s just like
us,” he exclaimed. "We get the power of mathematics only at the cost of our social defor­
mity as nerds.”
N u ic s
T h e s e r i e s w a s first i n t r o d u c e d a s a n e x a m p l e o f a r e c u r s i v e l y c o m p u t a b l e a p e r i o d i c s t r i n g
b y A x e l T h u e ( 1 8 6 3 - 1 9 2 2 ) , u s in g rh e r e p l a c e m e n t rules
tial
0.
0
-4
0 1 , 1 —» 10,
w ilh an in i­
M o r s e d i s c o v e r e d it s a p p l i c a t i o n t o d e t e r m i n i s t i c c h a o s , i n w h i c h it m o d e l s i h c
fra ctal tim e series p r o d u c e d by c e r ta in n o n l in e a r e q u a tio n s. S e e S c h r o c d e r ( t y y i ,
2 6 4 - 2 6 8 ) o n these aspects o f the seq u en ce.
5 . O n e - d im e n s io n a l version s c a n s h o w all th e d y n a m ic s o f tw o d im e n sio n s, an d ca n
e v e n b e used as a k i n d o f p a r a l l e l c o m p u t e r . C o n s i d e r , fo r e x a m p l e , a ru le t h a t in e a c h
i t e r a t i o n t h e n u m b e r o f c o u n t e r s i n a c u p is r e p l a c e d b y t h e s u m o f i t s e l f a n d its l e f t
neigh bor. S tartin g w ith on e:
0100000
—4
0110000
—4
0 1 2 1000
-4
0133 1 0 0
—4 0 1 4 6 4 1 0 .
T h i s f o u r t h i t e r a t i o n g i v e s us c h e b i n o m i a l c o e f f i c i e n t s f o r e x p a n s i o n o f ( a + b ) “*,
w h ic h e q u a ls a4 +
CHAPTER 3
I.
4 a 3b
+ 6 a 2b2 +
4 ab^
+ iA
Recursion
T h e s t a n d a r d t e r m i n o l o g y is s o m e w h a t a m b i g u o u s , s i n c e “ r e c u r s i o n ” is s o m e t i m e s
u s e d t o r e f e r s p e c i f i c a l l y t o w h a t w e w i l l c a l l “ s e l f - r e f e r e n c e , " a n d a t o t h e r t i m e s it is
u s e d i n t h e m o r e g e n e r a ! s e n s e a p p l i e d h e r e . “ I t e r a t i o n ” is u s e d in it s n o r m a l d e f i n i t i o n ,
a n d fo r t h e l e a s t p o w e r f u l w e w i l l u s e t h e t e r m “ c a s c a d e . ” T e c h n i c a l l y , t h e s e t h r e e t y p e s
o f recu rsio n ro u g h ly co r r e sp o n d to T u r i n g m a c h in e s , p u s h - d o w n a u t o m a ta , an d finites ta te a u t o m a t a , b u t th e s e m o d e l s are a lit tle t o o a b s tr a c t to b e d ir e c t ly useful in h e l p ­
in g readers d e v e lo p a sen se o f t h e d is t i n c t io n s t h a t are o f in terest h ere.
2.
S a g a y ( 1 9 8 3 ) e x p lic it ly m e n tio n s s ta rtin g w i t h th e sm all sh a p e in th e ce n te r, w h erea s
t h e I p a k o E le d e row s lo o k lik e t h e y m ig h t b e b e t t e r d e s c r ib e d as a p r e e s ta b lis h e d
lin ear s e q u e n ce (a lth o u g h S a g a y d o es n o t g iv e details h e re ).
3.
A c t u a l l y , it is n o t w a x t h a t is u s e d i n m u c h o f A f r i c a , b u t r a t h e r a l a t e x c r e a t e d b y b o i l ­
i n g t h e s a p o f c h e E u p h o r b i a p l a n t . W i l l i a m s n o t e s t h a t it c a n p r o d u c e l o n g , d e l i c a t e
th rea d s th a t are im p o ssib le for w a x .
4 . P e lto n (19 8 0 , 230) co n trasts rh e sin g u la r r a n d o m e v e n ts o f th e N a tiv e A m e r i c a n
t r i c k s t e r m y t h s w i t h “ t h e le s s e p i s o d i c , m o r e n a r r a t i v e m y t h s o f L e g b a a n d O g o - Y u r u g a
[ i n A f r i c a ] . ” T h e r e a s o n f o r c h e d i f f e r e n c e is p a r t l y m a t h e m a t i c a l . T h e N a t i v e A m e r i ­
c a n c o n c e p t o f u n p r e d i c t a b i l i t y is b a s e d m o r e o n c h a n c e ( s e e A s c h e r 1 9 9 1 , 8 7 - 9 4 ) ,
w h i l e t h e A f r i c a n c o n c e p t t e n d s co b e c l o s e r t o d e t e r m i n i s t i c c h a o s , as w e s a w in B a m a n a
sand d iv in atio n .
5.
C u r t i n ( 1 9 7 1 ) s h o w s t h a t c h e s l a v e t r a d e f r o m w h a t is n o w n o r t h e r n S e n e g a l d i m i n ­
i s h e d a f t e r 1 7 0 0 , a n d t h a t t h e N i g e r i a n a r e a d i d n o t b e g i n m a j o r a c t i v i t y u n t i l aft er. 1 7 3 0 .
T h i s still le a ve s th e p ossib ility chat F uller c a m e fro m th e area o f p re s e n t-d a y B e n i n an d
^ G h a n a " w h i c h w o u l d 'i Y iT t 0 6 ' f a r s o u t h ' t o h a v e ' d i r e c t l y s h a r e d i n f l u e n c e s w i t h t h e B a s s a r i , b u t H o l l o w a y ( 1 9 9 0 , 1 0 ) n o t e s t h a t V i r g i n i a n s s h o w e d s o m e p r e f e r e n c e for
A f r ic a n s from the S e n e g a in b ia n regio n .
1.
1 q u a lif ie d th is as “ s t a n d a r d ” b e c a u s e t h e r e h a s b e e n a g r o w i n g c o n c e r n t h a t a n t h r o ­
pologists m ay h a v e o ve re m p h a s ize d th e im p o rta n c e o f age-grade an d kin sh ip by p ro ­
j e c t i n g t h e i r o w n d e s i r e s a s w e l l a s c h e i n t e r e s t s o f t h e i r i n f o r m a n t s . S h a w ( 1 9 9 5 ) , fo r
e x a m p le , s h o w s h o w L o u is L ea ky's d e s c r ip tio n o f th e e x tr e m e o b e d ie n c e o f th e K ik u y u
to th e ir ag e -g ra d e syste m w a s c o l o r e d b o t h b y L e a k y's d e sire for che o rd er o f a “sm a ll
E n g lish v illa g e ” th a t h e n e v e r e x p e r ie n c e d ( h a v in g g r o w n u p w ith m ission a ry p aren ts)
a n d t h e K i k u y u e l d e r s ’ o w n i n t e r e s t s in r e c e i v i n g t h e i n i t i a t i o n p a y m e n t s t h a t w e r e o v e r ­
d u e to th em .
’ . In a d d i t i o n t o che a s s o c i a t i o n o f t h e v e r t i c a l w i t h t h e s p ir itu a l, F e r n a n d e z su g g e s ts th a t
t h e s p a t i a l d i s t i n c t i o n d e r i v e s f r o m t h e F a n g ’s p e r i o d i c c l a n f i s s i o n / r e l o c a t i o n . T h e f r a g ­
m e n t a t i o n o f a s o c i a l g r o u p c o m e s w i t h h o r i z o n t a l m o v e m e n t a n d is s e e n a s t h e r e s u l t
o f s t a g n a t i o n o r s t r i f e , w h i l e t h e e s t a b l i s h m e n t o f t h e g r o u p i n a n e w l o c a t i o n is s e e n
as p o s it iv e r e g e n e r a t i o n , b u i l d i n g f r o m t h e g r o u n d up.
L M a u r e r a n d R o b e r t s ( 1 9 8 5 , 2 5 ) d e s c r i b e che T a b w a b e lt, a le a th e r strip w i t h b a n d s o f
b e a d s o r w i r e a s r e p r e s e n t a t i o n s o f a s i n g l e d e s c e n t l i n e . S i n c e t h e T a b w a use c h e m p a n d e
Notes
238
disk to represent the expansion of all kinship groups from a singular origin, it is not
unreasonable to think of the irmkaba belt as a lower-dimensional projection of the
tnpancle disk. If one is willing to speculate so wildly that even [ would hesitate to do
so, the aardvark's winding tunnel could be viewed^as a three-dimensional spiral pro­
jected onto the two-dimensional mpande disk, just as the belt is a'bne-dimensional
projection of the mpande spiral. A similar practice, the “Poincar£ slice,” is used in non­
linear dynamics (see Abraham and Shaw .982).
9. It is important to understand that the problem is not one of “authenticity.” 1 agree with
the critiques of modernist anthropology’s tendency to make one individual represen­
tative of an entire society and to focus on a false homogeneous past. In ethnomathematics we are interested in the invention of mathematical concepts; so it doesn't
matter whether the source is an- entire society or a single creative individual. What does
matter is the precision and accuracy of che math, and it is here that the interpretive
flexibility offered by narratives presents problems.
jo. Note that I wrote "has trouble with” rather than “cannot do"— in fact, a programmer
could write a kind of “metaloop” of iteration that would figure out how many nestings
are needed. But in doing so, the program has to be able to refer to a part of itself (its
loops), so this is already a partial or limited self-tefetence. Of course we could then play
the same trick, demanding that we can’t tell ahead of time how many metaloops wili
be needed, and our sfnarty-pants programmer could again make a meta-metaloop, and
soon. It is only when we generalize the trick itself that full self-reference-will be required.
And even then, it too will meet up with undoable tasks— because that very property
of not bounding the process ahead of time leaves it vulnerable to other problems. As
Alan Turing proved for computing, and as Kurt Godel showed for all mathematics in
general, any system that is sufficiently powerful to fully utilize self-reference will have
to be incomplete in its ability to resolve all the theorems it can ask (see Hofstadter 1980).
11. The most specific connection made by Taylor is the possibility that the material attrib­
uted to Hermes-Thoth was derived from some of che Egyptian priesthood writings men­
tioned by Clement of Alexandria.
12. Stephan ides (1922, 192) suggests a more direcc sub-Saharan origin of alchemy, enter­
ing Egypt around 7 1 8 b . c . e ., following the invasions o f Ethiopia.
13. T h a t’s not to say that the Legba drum beats were random; but the drumming did
indeed have an unexpected change of pacechapter
10
Complexity
J, The analog/digital dichotomy in computing is often confused with other dualisms. The
same terms are used by engineers to describe the continuous/discrete dichotomy, and
by cognitive scientists to discuss “reasoning by analogy” versus inductive analysis, but
these distinctions are irrelevant to the sense in which it is used here. Musical nores,
for example, are excellent examples of analog communication, but they are entirely dis­
crete. See Eglash (1993) for details.
2. Blum et al. show that an analog Turing machine would be susceptible to tire halting
problem. See Eglash (1992, 1998c) for more details on this recent history of cybernetics.
We can think of the wave/particle duality in physics as another indication that the
analog/digital distinction is fundamentally egalitarian.
3. We can also look at this in terms of psychopathology. A neurotic will often repeat
the same phrase over and over, while a psychotic tends to be talking “word salad,”
a jumble of nonsense. In both cases, their mental relation to memory is pathologi­
cally simplified: the neurotic slavishly follows memory, whilethe psychotic completely
ignores it. Complex information processing requires a dynamic interaction with
memory, a nontrivial recursive loop.
4. For example, say there are choices A, B, and C. A wins, bur B and C voters say, "If only
I trad known A was going to win, I would have been willing to vote the other way.”
N otes
Tank and Hopfield (1987, 106) contrast this one-shot majority rule voting with the
collective-decision-making process in neural nets: “In a collective-dectsion commit­
tee the members vore together and can express a range of opinions; the members
know all about the other votes and can change their opinions. The committee gener­
ates . . . whac might be called a sense of the meeting."--5. Recall that we scaled down P to a number between 0 and 1 . That means that (1 - Pn )
will always he a fraction, which reduces P,i— in face, the larger Pn, the smaller the
fraction.
6. The reason it never lands back on exactly the same spot is not because of external noise;
it is rather for the same reason that the number P never repeats. Gottfried Mayer-Kress
suggested that a good way to understand this is to note that the drunken driver never
scops missceering, even while che sober one is overpowering him. I suspect that this com­
bination of negative feedback and positive feedback is at the heart of every case of deter­
ministic chaos, although I have yet to prove it. in Eglash (1992) I reported that che
Lorenz attractor consisted of only positive feedback, but this turns our to be incorrect.
In terms of dynamical systems theory (Abraham and Shaw 1982; Devaney J986),
positive feedback is the counterpart to spreading in phase space, and negative feedback
corresponds to folding in phase space. The phase-space combination of local spread­
ing and global folding is a common definition for chaos; the conjecture simply trans­
lates the phase-space definition into a control theory formulation.
7. I’ve oversimplified the relations here. For example, a finer distinction can be made about
‘'disorder” if we consider white-noise versus brown-noise distribution on a surface
(Gardner 1978; Voss 1990). In Brownian motion, a particle moves in a random, con­
tinuous trajectory; given an infinite amount of time, such “brown noise” will approach
a two-dimensional curve. In white noise, single points on che surface are selected at
random, so an infinite amount of time will still only leave us with disconnected points,
which is a zero-dimensional curve. Between zero and one dimension, we have objects
like the Cantor set, and between one and two dimensions we have objects like the Koch
curve. This is slightly different when we chink about noise as a single time-varying sig­
nal (as in acoustic noise) because the single points of the white distribution will also
be connected into a continuous (but nondifferentiable) curve, now of dimension one,
while brown noise as a time series will still be at dimension two.
8. Achebe himself prevents such a reading by highlighting a precoloniaf corastrophe that
befalls his main character, Okonkwo. At the same time, the contrast between Okonkwo’s
misery due to indigenous accident and his suicide as a result of che colonial encounter
makes it clear rhaf these are entirely different orders of chaos.
y. There is also a good illustration of collective fractal generation in the arts: che Mbutt
bark-cloih design shown in chapter 3 is actually llie producr of multiple artists.
c h a p te r
) 1
T h e o r e tic a l /rm n e iu o r k s in c u ltu r a l stu d ie s o f k n o w le d g e
1. Popper might object to the characterization of “fractal geometry minus dimensional mea­
sures,” since it sounds like an ad hoc adjustment, but the important thing is that the
four attributes (scaling, recursion, infinity, and dimension) were tested in a more or less
falsifiable manner. Whether or not one can still call it fractal geometry if one of the
four is missing is an important question; but we need to address the possibility of a weak
characterization of recursion in European fractals before making that judgment.
2. This should not necessarily he assumed to mean "closer to nature,” since it could also
refer to an indigenous knowledge system chat promotes good ecological practices; but
the ambiguity is problematic.
3. In fact I’m not— my master’s degree is in systems engineering, anti although I rook a
few graduate seminars in mathematics for my interdisciplinary Ph.D. (thanks to die flex­
ibility of the History of Consciousness board ar the University of California at Santa
Cruz), 1 wouldn’t dare call myself a mathematician in professional company, 1 have always
239
N otes
240
tried to introduce myself as an crhnomnthematician during field work, but sometimes
translation problems took time to get that across.
4. Worth it not just in ethical and methodological terms; it often came to my aid in dire
circumstances. On a hot road neat the Lake Chad region, I was stopped by military police
who were clearly looking for a bribe. I was released only when I began to launch into
a lengthy explanation of fractal geometry. Knowing the Baka counting system saved
my skin when a group of teenagers in a village in southern Cameroon took me for a
disrespectful tourist; unlike the gendarmes, they were delighted to find mathematics
in their midst.
5. O n the role of neologisms in the work ofCesaire, see Clifford (1988). On the construction
of negritude as a set of binary oppositions, see Mud ini be ( jy88).
6. For example, the octopus arose, millions of years before vertebrates but has a nervous
system more sophisticated than that of some reptiles (see Eglash 1984, 161). This is a
dangerous analogy, of course, because people often confuse biological and cultural evo­
lution. Here are two crucial differences. First, cultural evolution is Lamarckian— we
can pass our acquired knowledge to the next generation— while biological evolution
is Darwinian, with the rare lucky mutant having an advantage that is then passed on.
Second, the timescales are of different orders of magnitude. Significant biological
evolution requires on the order of a million years, while dramatic cultural evolution
requires no more than a few thousand years. This is why human beings have such a tiny
amount of genetic variation: the first modern humans, from their singular origin in Africa,
quickly spread across the earth over a few thousand years. Our nearly identical genetic
composition is a result of speedy Lamarckian cultural evolution adapting us into these
new environments.
c h a tte r
12 The
politics o/ A f r i c a n f r a c t a ls
1. Derrida's promotion of arbitrary signifiers and artificiality was not the sole voice for this
position. Black activists like James Boggs (1968) have also been champions of artifice.
Wittig’s (1973) L e s b ia n B o d y takes a topic that was often created as the unassailable ground
of feminist meaning, the authentic physical self, and dismantles this construction
through textual erotics. Like Derrida, she shows that a system of arbitrary symbols is
just as capable of carrying the kind of human essence often attributed to the Real or
Natural.
2. Angela Davis has pointed out Ellison’s denaturalizing tropes in lectures at UCSC; her
recent work continues to tease out these threads of self-assembly in black cultural iden­
tity and community.
3. My favorite illustration of analog artifice in black intellectual works occurs in chap­
ter 11 of Audre Lourd's Zflmi. Like Witfig (1973), she describes the self-assembly of a
lesbian body, hut her techniques for this artificial reconstruction come through the ana­
log media of scent, vibration, and form. See Eglash (1995) for other examples.
4. Consider, for example, the mojo hnnd/dataglove comparision in Dery (1994, 210), or
the following passage from Williams (1974, 40): “ ‘Simply anything can become a God,'
a Yoruba informant once remarked. ‘This button (pointing to the dashboard of the car
in which we were), ‘it only needs to he built up hy prayer’ (hy invocation).”
5. Similar views can be found in several other intellectual works of the time; e.g.,
Joreen’s (1972) critique of the women's movement, "Tyranny of Structurelessness." There
ate, of course, many centralist critiques of decentralization, but Joreen's text cook a
more complex angle of analysis. See Ehrlich (1979) for a critical view. Invocations
of African royalty in black cultural representations are typically viewed as commen­
tary on self-esteem. While that may he true, in most cases there are hints that it. also
serves to question the humanist control enacted in a political democracy that can sup­
port such deep economic subservience (see Queen Lntifa's "Queen of Royal Badness”
in Smith J990).
N otes
6. In face, this was how 1 got started on African fractals. It occurred to me that aerial
photos might show the difference between these architectural designs as fractal ver­
sus Euclidean. Pat Caplan generously provided me with aerial photos of the area in which
she worked, and the indigenous housing did indeed appear to be less Euclidean.
7. Recursive architectural structure is linguistically indicated by the Yoruba term for
homestead: ot k a 0 1, or "“house within the house.”
8. The 1993 Supreme Court ruling in Sfiau/ v. Reno used the terms “bizarre” and “snakelike,” the latter echoing historian John Fiske’s 1812 characterization of a "dragon Iike”
contour, a phrase changed to "salamander” and finally to “gerrymander" (after Mass­
achusetts governor Elbridge Gerry) by political cartoonist Gilbert Stuart.
9. The insistence that stochastic variation implies free will and deterministic variation
implies domination is made by several authors besides Porush (e.g., Hakim Bey). I think
that individuals or groups can indeed create such associations, just as they can create
the opposite (e.g., that a simple bounded system can still have the liberty of infinite
variation, as we will see argued by Gilroy, Van Wyk, and Heaver). The error is in assum­
ing universal meaning to what has to be local semiotics. A closer examination of the
social meanings for statistics (Porter 1986) reveals that its political associations are often
dependent on modernist concepts of humanist individualism, which is strongly critiqued
in the Foucaultian and other postmodernist analyses championed by Porush, Hayles,
Sobchack, and others.
10. Just as important is the reverse influence, e.g., Jewish jazz musician' Mezz Mezzrow pass­
ing for black while in prison so that he could play in the band.
11. Gilroy’s work in this area should be seen as part of a larger community of researchers
and cultural workers (e.g., artists) who have developed a postmodern emphasis on hybrid-,
ity, creolization, and other impure identities (cf. Minh-ha 1 9 8 6 ; Anzaldua 1 9 8 7 ;
Bhabba 1 9 9 0 ; Sandoval 1 9 9 5 ; Haraway 1 9 9 6 ) .
12. Digital and analog are also confusing terms because digital technology is now commonly
used to generate the analog waveforms of music. But it is necessary to see how these
representations are layered. The electronic “on-off” .code pulses are actually noisy
waveforms that must be processed with analog control circuits at the lowest level of
the silicon chip; eventually they are decoded in binary form, then converted to an elec­
trical waveform that will modulate the speaker. T h e resulting acoustic waveform can
be analog, digital, or— especially in the case of rap music— somewhere in between. See
Eglash (1993) for details.
chapter
13
Frncwls in t/ie Europemi
h i s t o r y o f m a t h e m a t ic s
1. According to ancient accounts, the discovery of irrationals was in the middle of the
fifth century b.c. e. Modern scholars generally agree that the proof for the incommen­
surability of the square of a diagonal with respect to its side, first mentioned explicitly
in Plato's dialog T/ienems, is too abstract to have been used at this time. Von Fritz.(1944)
provides a resolution for this conflict in his speculative reconstruction of Hippasus’ analy­
sis of the pentagon. See Knorr (1975) and Fowler (1987) for discussion of the origi­
nal texts relevant to this area.
2. Plato was not the only influence at the time, nor were irrationals only granted one per­
spective. Fowler (1987), for example, maintains rhat the significance of irrationals has
been misunderstood and suggests that even Plato presented their proof as “a source of
interesting and fruitful problems” rather than as a disturbing paradox. Nevertheless,
it was the homogeneous representations of Platonic thought deployed centuries later,
not irs contemporary diversity, which would matter for the intuition and practice of
modern mathematicians.
3. "We add to the first number the second one, i.e., 1 and 2 , the second to the third; the
third to the fourth; the fourth to the fifth . . . and it is possible to do this order for an
infinite number of months” (trans. Maxey Brooke).
241
242
N o te s
4. Similar analysis was provided hy Henry Louis Gates ( 1990) and others in the censor­
ship trial of rap group 2 Live Crew, maintaining that the explicit sexual lyrics were not
acultura! profanity but rather modern variations of a long-standing black tradition of
public sexual commentary.
*
v',-«
5. Tuana (1989), for example, notes that the male homunculus theory,‘which locates the
active principle of birth in sperm only, dominated European medical thinking from Aris­
totle to van Leeuwenhoek (and in some senses even to the present; see Hnrrouni
1997). Again, the African version is in strong contrast; recall from chapter 8 that the
Fang believe that the homunculus or active principle is contained in the female blood
(the division is more egalitarian than the European model, however, since the male Fang
are said to provide a complementary protective, skeletal principle).
6. That is, prior to complexity theory, at which point advances in the application of frac­
tal geometry were made precisely because of the growing recognition of a relationship
between computational recursion and self-organizing phenomena. Complexity theory
is a marker distinguishing the transitional postmodernism of the 1970s from the stable
postmodernism of the 1980s (Eglash 1998c).
7. The qualification is not inaccurate; the problem is that sometimes the authors of this
text { T h e Science o f F r a c t a l Images) use the term “recursion" to mean iteration, and some­
times (as in this case) it means self-referential programming. This level of ambiguity
would not be tolerated for any other mathematical terminology used in the text.
chapter
14
Futures f o r
A f r i c a n f r a c t a ls
1. For more on cyborgs, see Haraway (1996) and Gray (1995).
In fact, if I had used a large enough size difference, self-intersection could have been
avoided altogether, but 1 think that would not do justice to the African tradition of
putting similar-sized houses together— a tradition that has its roots in egalitarian
socioeconomic structure, and one to which Nyangula was no doubt sensitive.
3. But there was more to it than that. Perhaps in port because it implied a Platonic view,
it made sense to the students chat religious symbolism would be mathematical, while
.something as concrete as 0 mud wall was too hard to reimage. There was also the visual
effect of seeing computer simulations of the African log spirals; for a generation
brought up on video games and MTV, this placed it in a contemporary framework. Finally,
there was something about the religious subject matter itself— the very concept of a
"life force" expressed as a self-organizing system— that may have created a resonance
for these students.
4. Onyejekwe’s African Women Global Network is available from http://www.osu.edu/
org/awognet.
2.
References
Abraham, Ralph, and Shaw, Christopher. Dynamics: The Geometry o f B e h a v i o r . Santa Cruz,
Calif.: Aerial Press, 1982.
Abrahams, R. D. A f r i c a n Folktales. New York: Pantheon, 1983.
Achebe, Chinua. T h i n g s F a ll Apart. New York: Astor-Honor, 1959.
Adams, L. ‘'North-West Indian art and its early Chinese parallels.” Man 3 6 , no. 3 (1936).
Adepoju, Aderanci, and Oppong, Christine, eds. Gender, Work and Population in SubSaharan A/rica. Geneva: International Labor Office, 1994.
Ayudoawu, Kofi. Rides f u r Playing Orvare. Kumpsi, Ghana: KofiTall, 1991.
Akpabot, Samuel. "Random music of the Birom.” A f r i c a n Arts 8, no. 2 ( 1975): 46-47.
Anderson, R. E.; Welch, W. W.; and Harris, L. J. “Inequities in opportunities for computer
literacy." Computing Teacher u (1984): 10-12.
Andreka, Hajnal. "Rozsa Peter.” In Lynn Osen, ed., Women in Mathematics, 171-174. Cam­
bridge, Mass.: MIT Press, 1974.
Anzaldiia, G. Borderlands, L a F r o n t e r a . San Francisco: Spinsters, 1987.
Appadurai, Arjun. “The production of locality.” In Richard Fafdon, ed., C o i m t e n v o r k s : Man­
aging the Diversity o f K n o w l e d g e . New York: Routledge, 1995.
Appiah, A. In My Father’s House. New York: Oxford, 1992.
Argiro, Larry. Mosaic An Today. Scranton, Penn.: International Textbook Co., 1968.
Argyros, A. J. A B l e s s e d R a g e f o r O r d e r ; D e c o n s t r u c t i o n , E v o l u t i o n a n d C h a o s . Ann Arbor:
University of Michigan Press, 1991.
Arnoldi, Mary Jo. Bamana and Bo^o Puppetry o f t h e Segou Region Youth S o c i e t i e s . West
Lafayette, Ind.: Purdue University Press, 1977.
Asante, M. K. and Asante, K. W. A f r i c a n Culture; T h e Rhythms o f U n i t y . Westport, Conn.:
Greenwood Press, 1985.
Ascher, M. Ethnomathematics. New York; Chapman and Hall, 1991.
243
""nt.uywuF
244
R eferences
Atinore, Anthony; Stacey, Gillian; mu! Forman, Werner. Black Kingdoms, Black Peoples: The
West A f r i c a n Heritage. New York: G. P. Putnam's Sons, 1979.
Auerbach, E. Mimesis. Princeton^N.J.: Princeton University Press, 1953.
Aveni, A. Skywatchers o f Ancient Mexico. Austin: University of Texas I?ress, 1980.
B a d a w y , A l e x a n d e r . A H is t o r y o f E g y p t i a n A r c h i t e c t u r e . G iz a : S t u d i o M i s r , 1 9 5 4 Badawy, Alexander. Ancient Egyptian Architectural Design. Berkeley: University of Califor­
nia Press, 1965.
B a k , P e r, a n d C h e n , K a n . “S e l f - o r g a n i z e d c r it ic a lit y . ” S c i e n t i f i c A m e r i c a n , J a n . 1 9 9 1 , 4 6 - 5 3 .
Baratz, J. C-; Ficklen, M. S.; King, B.; and Rosenbaum, P. Who Is Going to Medical School?
Princeton, N.J.: Educational Testing Service, (985.
Barbier-Mueller Collection. A f r i c a n A r t : From the Barbier-Mueller Collection. Geneva: PrestelVeriag, 1988.
Barsh, Russel. “The epistemology of traditional healing systems." Human Organisation 56,
no. 1 (1997)-.
Bascom, William Russell. A f r i c a n A r t in Cultural Perspective. New York: Norton, J973.
Bastin, Maiie-Louise. "The Mwnnangana Chokwe chief anil art.” In Erna Beumers and HansJoachim Koloss, eds., Kings o f A f r i c a . Berlin: Museum fur Volkerkunde, 1992.
Batty, Michael, and Longley, Paul. Fractal Cities: A Geometry o f Form and Function. San Diego:
Academic Press, c. 1994.
Beauvilain, A., et al. Atlas Aerien du Cameroun. Universite de Yaounde, 1983.
Beguin, Jean-Pierre. L'Habttat au Cameroun. Paris: Office de la Recherche Scientifique OutreMer, 1952.
Bell, Eric T. Men 0 / Mathematics. New York: Simon and Schuster, 1939.
Benena, Lourdes, ed. Women and Development: The Sexual Division o f L a b o r i n R u r a l S o c i ­
e t i e s . New York: Praeger, 1982.
Berjonneau, Gerald; Sonnery, jean-Louis; and Fondation Dapper, eds. Rediscovered Master­
pieces o f A f r i c a n Art. Boulogne, France: Art 135, 1987.
Bhablvi, H. N a t i o n a n d N a r r a t i o n . London: Routledge, 1990.
Biebuyck, Daniel. Lega Culture. Los Angeles: University of California Press, 1973.
Black Elk. Black Elk Speaks:.Being the L i f e S t o r y o f a H o l y Man o f t h e O g a l a l a S i o u x , as told
to John G. Neihardt (Flaming Rainbow); illustrated by Standing Bear. New York:
W. Morrow and Company, 1932.
Blier, Suzanne. The Anatomy 0 / Architecture: Ontology and Metaphor in B a t a m m a li b a A r c h i - _
l e c u i r a l Expression. New York: Cambridge University Press, 1987.
Blier, Suzanne. African Vodun: Art, Psychology, and Power. Chicago: University of Chicago
Press, 1995'
Blum, L.; Shub, M.; and Smale, S. “O n a theory of computation and complexity over the
real numbers.” Bull. AMS 2 t, no. 1 (July 1989): j-46.
Boggs, J. “The Negro in Cybernation.” In T h e E v o l v i n g S o c i e t y : T h e Proceedings o f t h e First
Annual Conference on th e Cybercultural Revolution —Cybernetics and Automation, ed. Alice
Mary Hilton. New York: Institute for Cyberculturul Research, 1966.
Boserup, Ester. W o m a n ’s R o l e in E c o n o m i c D e v e l o p m e n t . London: Allen and Unwin, 1970.
Bourdier, J., and Trinh Minh-ha. A f r i c a n Spaces. New York: Africana, 1985.
Boyer, C. A History of Mathematics. New York: John Wiley and Sons, 1991.
Boyer, E. E. High School: A R e p o r t on S e c o n d a r y Education in America. New York: Harper and
Row.
Burland, Cottie. North American Indian Mythology. New York: Hamlyn Publishing Group, 1965.
Cantor, M. Vorle.sungen i ib e r Geschichte der Mathematik. Leipzig: Teubner, 1880.
Caplan, Pat. “Development policies in Tanzania." In Nici Nelson, ed., African Women in the
Development Process. London: Frank Cass, 1981.
Carnegie Institute. T h e A r t o f B l a c k A f r i c a : Collection o f J a y C Leff. Pittsburgh: Museum of
Art, Carnegie Institute, 1970.
Chanda, Jacqueline, A f r i c a n Arts and C u l t u r e s . Worcester, Mass.: Davis Publications, 1993-
R eferences
C h a p p e l , T . J . H . D e c o r a te d G o u r d s in N o r t h - e a s t e r n N i g e r i a . L o n d o n : E t h n o g r a p h i e s , 1 9 7 7 .
C h a r n l e y , S u s a n . " P a s t o r a l i s m a n d t h e d e m i s e o f c o m m u n a l p r o p e r t y in T a n z a n i a . ” C u l t u r a l
Su rvival Q u a rterly , S p r in g 19 9 6 , 4 1 - 4 4 .
C l i f f o r d , J. “ O n e t h n o g r a p h i c a u t h o r i t y . ” R e p r e s e n t a t i o n s - . u . . ( i 9 8 3 ) : 1 1 8 - 1 4 6 .
C l i f f o r d , J. " A p o l i t i c s o f n e o l o g i s m : A i m e C e s £ i } r e . ” I n T h e P r e d i c a m e n t o f C u l t u r e . C a m ­
b rid g e , M ass.: H a r v a r d U n i v e r s i t y Press, 1988.
C I o s s , M . P., e d . N a t i v e A m e r i c a n M a t h e m a t i c s . A u s t i n : U n i v e r s i t y o f T e x a s P r e s s , 1 9 8 6 .
C o n n a h , G . “ T h e q u e s t i o n o f e c o n o m i c b a s e s: G r e a t Z i m b a b w e a n d r e l a t e d s ite s ." In
A f r ic a n C i v i l i s a t i o n s . N e w Y o r k : C a m b r i d g e U n i v e r s i t y P r e s s ; 1 9 8 7 .
C o o k , T h e o d o r e A . T h e C u r v e s 0/ L ife . L o n d o n : C o n s t a b l e a n d C o m p a n y , 1 9 1 4 .
C r o w e , D . , a n d N a g y , D . “ C a k a u d r o v e - s t y l e M a s i K e s a o f F i j i .” A r s T ex tr t n a 1 8 ( 1 9 9 2 ) : 1 1 9 - 1 5 5 .
C r u m p , T . T h e A n t h r o p o l o g y o f'N u m b e r s . C a m b r i d g e , U . K . : C a m b r i d g e U n i v e r s i t y Pre ss , 1 9 9 0 .
C u l l e r s , J o n a t h a n . " S e m i o t i c s o f t o u r i s m . ” A m e r ic a n J o u r n a l o f S e m i o t i c s 1 , n o s . 1 - 2 ( 1 9 8 1 ) :
127-145.
C u r t i n , P h i l i p D . T h e A l a n t i c S l a v e T r a d e : A C e n s u s . M a d i s o n : U n i v e r s i t y o f W i s c o n s i n P re s s ,
1971.
D a u b e n , J. W . G e o r g C a n t o r . C a m b r i d g e , M a s s . : H a r v a r d U n i v e r s i t y P r e s s , 1 9 7 9 .
D a v i d s o n , B a s i l . A f r ic a n K i n g d o m s . N e w Y o r k : T i m e - L i f e , 1 9 7 1 .
D a v i s , j . D . T h e E ffe c t o f M a th e m a tic s C o u r s e E n r o l l m e n t o n R a c i a l / E t h n i c D iffe r e n c e s in S e c ­
o n d a r y S c h o o l M a t h e m a t i c s A c h i e v e m e n t . N A E P R e p o r t . P r i n c e t o n , N .J .: E d u c a t i o n a l T e s t ­
in g S e r v ic e , 1986.
D a v i s , J. D . “ T h e m a t h e m a t i c s e d u c a t i o n o f B l a c k h i g h s c h o o l s t u d e n t s . ” I n P e a r s o n a n d
B e c h t e l 1989.
D e l a v a l , B e r n a r d . " U r b a n c o m m u n i t i e s o f t h e A l g e r i a n S a h a r a . ” Elcistics 2 2 7 ( O c t o b e r
I 9 7 4 )D e l e u z e , G i l l e s , a n d G u a t t a r i , F e l i x . A T h o u s a n d P l a t e a u s : C a p i t a l i s m a n d S c h iz o p h r e n ia . T r a n s .
B r a in M a s s u m i . 'M i n n e a p o l i s : U n i v e r s i t y o f M i n n e s o t a Press, 1 9 8 7 .
D e r r i d a , J. O f G r a m m a to lo g y . T r a n s . G . C . S p i v a k . B a l t i m o r e : J o h n s H o p k i n s U n i v e r s i t y Press,
1974D e n y e r , S . A f r ic a n T r a d itio n a l A r c h ite c tu r e . N e w Y o r k : A f r i c a n a P u b l i s h i n g , 1 9 7 8 .
D e r y , M a r k . “ B l a c k t o t h e f u t u r e . ” In M a r k D e ry , e d ., F la m e W a r s , 1 7 9 - 2 2 2 . D u r h a m , N . C . :
D u k e U n i v e r s i t y Press, 1 9 9 4 .
D e t r o it I n s tit u te o f A r t s . D etroit C o lle c ts A fr ic a n A r t . D e tr o it : W a y n e S t a t e U n i v e r s i t y
Press, 1 9 7 7 .
D e v a n y , R o b e r t . A n I n tr o d u c t io n to C h a o t i c D y n a m i c a l S y s t e m s . M e n l o P a r k , C a l i f . : B e n i a m i n
C u m m in g s P u b lish in g, 1986.
D e v i s s e , J. “ U r b a n h i s t o r y a n d t r a d i t i o n i n t h e S a h e l . " I n R e a d i n g the C o n t e m p o i w y A f r i c a n
C i t y . A g a K h a n A w a r d for A r c h i t e c t u r e . S i n g a p o r e : C o n c e p t M e d i a , 1 9 8 3 .
D e w d n e y , A . K . " A n a l o g g a d g e t s .” Scientific A m e r i c a n , J u n e 1 8 - 2 9 , * 9 8 5 .
D iaz, R o g e l i o P it r a c h o . “T h e m a t h e m a t ic s o f n atu re ; t h e c a n a m a y t e q u a d r i v e r t e x .” J S G E M
N e u ' s d e tt e r 1 1 , n o . 1 ( D e c e m b e r 1 9 9 5 ) : 7 - 1 2 .
D r a k e , S t . C l a i r . B la c k F olk H e r e an d T h e r e . L o s A n g e l e s : U C L A C e n t e r fo r A f r o - A m e r i c a n
Studies, 1987.
D uly, C o l i n . T h e H ou ses o f M a n k in d . L o n d o n : B l a c k e r C a l m a n n C o o p e r L td ., 1 97 9 .
E h r l ic h , C . “ S o c i a li s m , a n a r c h is m , a n d f e m in is m ." h i Rein ven tin g A n a r c h y . L o n d o n : R o u tled ge a n d K e g a n P aul, 197 9 .
E g la s h , R . " T h e c y b e r n e t i c s o f c e t a c e a . ” Investigations o n C e ta c e a 1 6 ( 1 9 8 4 ) : 1 5 0 - 1 9 8 .
E g la s h , R . “ A c y b e r n e t i c s o f c h a o s . " P h . D . diss., B o a r d o f S t u d i e s in H i s t o r y o f C o n s c i o u s ­
ness, U n iv e r sity o f C a lifo r n ia , S a n ta C ru z, 1992.
E g l a s h , R . “ I n f e r r i n g r e p r e s e n t a t i o n t y p e f r o m s p e c t r a l e s t i m a t e s o f f r a c t a l d i m e n s i o n in c o m ­
m u n ic a t io n w a v e fo r m s ." jo u r n a l o f Social and E v o lu tion a ry Structures 1 6, n o . 4 ( 1 9 9 3 ) .
E g l a s h , R . “ A f r i c a n i n f l u e n c e s in c y b e r n e t i c s . ” I n T h e C y b o r g H a n d b o o k , e d . C h r i s G r a y . N e w
York: R o u tle d g e , 1995.
245
246
R eferences
“When math worlds collide: Intention and invention in ethnomathematics.” S c i­
ence, Technology and Human Values 22, no. 1 (Winter 1997a): 79-97.
Eglash, R. “Bamana sand divination: Recursion in ethnomathematics.” American Anthro­
pologist 99 no. 1 (March 1997b): 112-122.
Eglash, R. “The African heritage of Benjamin Banneker." Social Studies o f S c i e n c e 27 (April
1997c): 307-315.
Eglash, R. “Geometric algorithms in Mangbetu design." Mathematics Teacher 91, no, 5
(May 1998a): 376-381.
Eglash, R. “Computation, complexity and coding in Native American knowledge systems.”
In Judith Hanks and Gerald Fast, eds., Changing the Faces o f Mathematics: North Ameri­
can /ndtgenous P e o p l e ’s Perspective. Rcston, Va.: NCTM, 1998b.
Eglash, R. “Cybernetics and American youth subculture.” Cultural Studies 12 (1998c): 3.
Eglash, R., and Broadwell, P. “Fractal geometry in traditional African architecture.” Dynam­
ics Newsletter, June 1989, 1-10.
Eglash, R.; Diatta, C.; and Badiane, N. "Fractal structure in Jola material culture." E k i s t i c s
61 no. 368/369 (September-December 1994): 367-371.
Fagg, Will tarn. “T he study of African art." Bulletin o f the Allen Memorial Art Museum,
Winter 1955, 44-61.
Fagg, William. The Art 0 / Western A f r i c a : Sculpture and Tribal Masks. New York: New
American Library, by arrangement with UNESCO, 1967.
Faris, J. C- “From form to content in the structural study of aesthetic systems." In D. Wash­
burn, ed., Structure and Cognition in Art, 90-11 2. Cambridge, U.K.: Cambridge U ni­
versity Press, 1983.
Fathy, Hassan. Architecture f o r the Poor. Chicago: University of Chicago Press, 1973Fauvei, John, and Gerdes, Paulus. “African slave and calculating prodigy: Bicentenary of
the death of Thomas Fuller." Historin Mathematica 17 (1990): 141-151.
Fernandez, James W. Bwili: An Ethnography o f the Religious Imagination in A f r i c a . Princeton,
N.J.: Princeton University Press, c. 1982.
Fourier, Joseph. Description de I’Egypte. Paris: Commission des Sciences et Arts d'Egypte, 1821.
Foucault, M. D i s c i p l i n e and Punish. New York: Pantheon, 1979.
Fraser, D. Village Planning in the Primitive World. New York: George Braziller, 1968.
Freeman, M., and Carbyn, L. Traditional Knowledge and Renewable Resource Management.
Alberta, Canada: Boreal Institute for North American Studies, 1988.
Freeman, W. “Physiological basis of mental images.” Biological'Psychiatry 18 (1981):
1107-1125.
Garcia, Linda. The Fractal Explorer. Santa Cruz: Dynamic Press, 1991.
Gardi, Rene. Indigenous A f r i c a n Architecture. New York: Van Nostrand Reinhold, 1973.
Gardner, Martin. "Mathematical recreations: The dragon curve.” Scientific American, April
1967, 11 7-120.
Gardner, Martin. Aha.1 New York: Scientific American, 1978.
Gates, Henry Louis. The Signi/ying Monkey. Oxford: Oxford University Press, 1988.
Gates, Henry Louis. “Rap music: Don’t knock it if you’re not onto its ‘ lies. ’ " New York Her­
ald Tribune, June 20, 1990.
Geory, D. C. Children’s Mathematical Development: Research and Practical Applications. Wash­
ington D. C-: American Psychological Association, 1994.
Geertz, Clifford. The Interpretation o f Cultures: Selected Essays. New York: Basic Books, 1973.
Gerdes, Paulus. Une tradition g e o m e t r i q u e en A/rique: les dessrns sttr le s a b l e . Paris: Harmnttan, 1995Gerdes, Paulus. W o m e n , A r c , a n d Geometry i n S o u t h e r n A f r i c a . Lawrenceville, N.J.: Africa
World Press, 1998a.
Gerdes, Paulus. “Molecular modelling of fullerenes." The Chemical Intelligencer 1 ( 1998I1).
Gies, J., and Gies, F. Leonardo o f P is a and the New Mathematics o f the Middle Ages. New York:
Thomas Orowell, 1969.
E g la s h , R .
R eferences
G i l l i n g s , R i c h a r d J. M a t h e m a t i c s in the T i m e o f the P h a r a o h s . N e w Y o r k : D o v e r , 1 9 7 2 .
G i l r o y , P. T h e B l a c k A t l a n t i c . C a m b r i d g e , M a s s . : H a r v a r d U n i v e r s i t y P r e s s , 1 9 9 3 .
G l e i c k , J. C h a o s — M a k i n g o f a N e w S c i e n c e . N e w Y o r k : V i k i n g , 1 9 8 7 .
G o i d w a t e r , R o b e r t . B a m b a ra S c u l p t u r e fro m th e W e s t e r n S u d a n . N e w Y o r k : U n i v e r s i t y P u b ­
lishers, i9 6 0 .
G r a t t a h - G u i n n e s s , 1. " T o w a ' r d s a b i o g r a p h y o f G e o r g C a n t o r . " A n n a l s o f S c ie n c e 2 7 , n o . 4
( ' 9 7 ' )'
G r a y , C h r i s . T h e C y b o rg H a n d b o o k . N e w Y o r k : R o u t l e d g e , 1 9 9 5 .
G ria u le , M a r c e l. C o n v e rsa tio n s w ith.O gotem m eli. T r a n s . R . B u tle r a n d A . R ic h a rd s. L o n d o n :
O x f o r d L l n iv e r s ity Press, 1965.
H a e c k e l , E r n s t . K u n s t / b n n e n ti e r N c it ur. L e ip z ig : V e r l a g d e s B i b l i o g r a p h i s c h e n l n s t i t u t s , 19 0 4 .
H a l l , E. R ., a n d P o s t - K a m m e r , P. “ B l a c k m a t h e m a t i c s a n d s c i e n c e m a j o r s : W h y s o fe w ? ” C a r e e r
D e v e lo p m en t Q u a r te r ly 35 ( 1 9 8 7 ) : 2 0 6 - 2 1 9 .
H a m i l t o n , A . M a o ri A r t . L o n d o n : N e w H o l l a n d P u b l i s h e r s , 1 9 7 7 .
H a n s o n , F. A . " W h e n c h e m a p is t h e t e r r i t o r y : A r t i n M a o r i c u l t u r e . ” I n D . W a s h b u r n , e d . ,
S t r u c t u r e a n d C o g n i t i o n in A r t , 7 4 - 8 9 . C a m b r i d g e , U . K . : C a m b r i d g e U n i v e r s i t y P r e s s ,
1983.
H a r a w a y , D o n n a . " A m a n i f e s t o f o r c y b o r g s . ” S oc ia lis t R e v i e w 8 0 { 1 9 8 5 ) : 6 5 - 1 0 8 .
H a r a w a y , D o n n a . “T h e p r o m i s e s o f m o n s t e r s : A r e g e n e r a t i v e p o litic s fo r i n a p p r o p r i a t e / d
o t h e r s . ” I n C u l t u r a l S tu d i e s , e d . L. G r o s b e r g , C . N e l s o n , a n d P. T r e i c h l e r , 2 9 5 - 3 3 7 . N e w
Y ork: R o u t l e d g e , 1992.
H a ra w a y , D o n n a . M odest_ W itn ess@ S eco n d _ M illen n iin n : F e m a le M a n _ M ee ts_ O n co M o u se .
N e w Y o rk : R o u t l e d g e , 1 9 9 7 .
H a r t o u n i , V a l e r i e . C u l t u r a l C o n c e p t i o n s : O n R ep r o d u c t iv e Technologies a n d the R e m a k i n g o f L ife.
M in n e a p o lis : U n iv e r s ity o f M i n n e s o t a Press, 1997.
H a y l e s , K . C h a o s B o u n d . I t h a c a , N .Y . : C o r n e l l U n i v e r s i t y P r e s s , 1 9 9 0 .
H e a v e r , H a n n a n . “H e b r e w in a n Israeli A r a b h a n d : S ix m i n i a t u r e s o n A n t o n S h a m m a s ’
A r a b e s q u e s . ” C u l t u r a l C r i t i q u e , F a ll 1 9 8 7 , 4 7 - 7 6 .
H e b d i g e , D . C u t ’n ’ M i x : C u l t u r e , I d e n t i t y , a n d C a r i b b e a n M u s i c . N e w Y o rk : M e t h u e n ,
1987.
H e i m s , S . J. J o h n v o n N e u m a n n a n d N o r b e r t W i e n e r . C a m b r i d g e , M a s s . : M I T P r e s s , 1 9 8 0 .
H i n c h c l i f f , K . H . V il la g e H o u s i n g in the T r o p ic s . W a s h i n g t o n , D . C . : H U D , 1 9 4 6 .
H e r s k o v i t s , M e l v i l l e . “ W a n in t h e n e w w o r l d . ” P a p e r r e a d a t t h e A m e r i c a n i s t C o n g r e s s , H a m ­
burg. 1930.
H o d g e s , A'. A k i n T u r i n g : T h e E n ig m a . B u r n e t t B o o k s , 1 9 8 3 .
M o f s t a d t e r , D o u g l a s . G o d e l , E s c h e r , B a c h . N e w Y o rk : B a s i c B o o k s , 1 9 8 0 .
H o i s t a d t e r , D o u g la s. “T h e p r is o n e r 's d i l e m m a : C o m p u t e r t o u r n a m e n t s a n d t h e e v o l u t i o n
o f c o o p e r a t i o n . " i n M c t a m f l g i c a ! T h e m a s , 7 1 5 - 7 3 8 . N e w Y ork : B a s i c B o o k s , 1 9 8 5 .
H o l a s , B. P o n e s S c u l p t e e s d u M u s e e d ' A b i d j t m . D a k a r : i f a n , 1 9 5 2 .
H o l l o w a y , J. E. A / r i c a n i s m s in A m e r i c a n C u l t u r e . B l o o m i n g t o n : I n d i a n a U n i v e r s i t y P r e s s , 19 9 0 .
H o l m , B ill. N o r th w e s t C o a s t In d ia n A r t . S e a t t l e : U n i v e r s i t y o f W a s h i n g t o n Press, 1965.
H o o k s , B e l l . Y e ar n in g : R a c e , G e n d e r , a n d C u l t u r a l Po li ti cs. B o s t o n : S o u t h E n d P r e s s , c. 1 9 9 0 .
H o u n t o n d j i , P. A fr ic a n P h i l o s o p h y : M y t h a n d R e a l i t y . B l o o m i n g t o n : I n d i a n a U n i v e r s i t y
Press, 1983.
H u g h e s , D a v id . A /ro c e m ric A rc h ite c tu re : A Design P rim er. C o l u m b u s , O h i o : G r e y d e n Press,
1994.
H u g h e s , D a v id M . “ W h e n p a rk s e n c r o a c h u p o n p e o p l e . ” C u ltu ra l S urvival Q u a rte rly , S p r in g
1996, 36 -4 0 .
H u l l , R . A fr ic a n C i t i e s a n d T o w n s before th e E u r o p e a n C o n q u e s t . N e w Y o r k : W . W . N o r t o n ,
1976.
H u r a u l t , M . J. H i s t o i r e d a L a m id a i P eal d e B a n y o . P a r i s ; L’A c a d e m i e d e s S c i e n c e s d ’O u t r e M er, 1975.
H u r s t , H . E. T h e N i l e B a s i n . C a i r o : G o v e r n m e n t P r e s s , 1 9 3 1 .
247
R eferences
Huskey, Velma, and Huskey, Harry. “Ada, Countess of Lovelace and her contribution to com­
puting.” Abacus i, no. 2 (1984).
Hyndman, D. “Conservation through self-determination: Promoting the interdependence
of cultural and biological diversity.” Human Organisation 53, no. 3 (1994).
Imperato, Pascal James. Dogon C l i f f D w e l l e r s : T h e Art o f Mali’s Mountain P e o p l e . New York:
L. Kahan Gallery Inc. /African Arts, 1978.
Isichet, Elizabeth Alio. A History o f N i g e r i a . New York: Longman, 1983.
Jefferson, Louise E. The Decorative Arts o f A f r i c a . New York: Viking Press, 1973.
Jones, Vaughan. "Knot theory and statistical mechanics.” Scient//ic American, November
1990,98-103.
Jorccn. “Tyranny of srnicturclcssness.” T/ic Second Wove 2, no. 1 (1972).
Juma, C. The Gene Hunters. Princeton, N.J.: Princeton University Press, 1989.
Kay, P. Explorations in Mathematical Anthropology. Cambridge, Mass.: MIT Press, 197 t .
Katz, V. J. Book review of Ethnomathematics. Historia Mathematica 19 no- 3 (August 1992):
310-314.
Keitel, C., and Ruthven, K., eds., Learning from Computers: Mathematics Education and
Technology. Berlin: Springer-Verlag, 1993.
Kent, Kate P. N a v a j o W e a v i n g : Three Centuries o f C h a n g e . Seattle: University of Washing­
ton Press, 1985.
Klein, C. “Woven heaven, tangled earth.” Annals o f t h e N e w Y o r k A c a d e m y o f S c i e n c e s 385
(May 14, 1982): 1-36.
Knorr, W. R. T h e E v o l u t i o n o f th e E u c l i d i a n Elements. Dordrecht: Reidel, 1975Koloseike, A. “Models of terraces in Ecuador." In P. A. Ballonoff, ed.. Mathematical Models
o f S o c i a l a n d C o g n i t i v e S t r u c t u r e s . Urbana: University of Illinois Press, 1974.
Krantz, S. G. "Fractal geometry.” Mathematical Intelligencer 11, no. 4 (1989).
Laude, Jean. A f r i c a n Art o f th e D o g o n . New York: Viking Press, 1973.
Lebeuf, A.M.D. Les P r i n c i p a u t e s K o t o k o . Paris: Editions du Centre National de la Recherche
Scientifique, 1969.
Lefkowitz, Mary. Not Out o f A f r i c a : How A/rocentrism Became an Excuse to Teach Myth as
History. New York: Basic Books, 1996.
Leuzinger, Elsy. T h e A r t o f B l a c k A f r i c a . New York: Rizzoli International, 1977Levey, Martin. T h e A l g e b r a o f A b u K a m i l . Madison: University of Wisconsin Press,
1966.
Lloyd, P. C- “The Yoruba: An urban people?” In A. Southall, ed., Urban Anthropology. New
York: Oxford University Press, 1973.
Lyotard, J. F. The Postmodern Condition: A R e p o r t on Knowledge. Minneapolis: University of
Minnesota Press, 1984.
McClelland, E. M. Cult o f l f a Among the Yoruba. London: Ethnographica, 1982.
Macintosh, Roderick J., and Macintosh, Susan K. “People without history.” Archaeology 42,
no. i-(January 1989): 75-80.
Malcom, S. M. An assessment of programs that facilitate increased access and achievement of females
and minorities in K-12 mathematics and science e d u c a t i o n . National Science Board Com­
mission on Precollege Education in Mathematics, Science, and Technology. Washington,
D.C.: Office of Opportunities in Science, and American Association for the Advance­
ment of Science.
Mamedov, Kh. S. “Crystallographic patterns.” Computing and Mathematics with Applications
12B, no. 3/4 (1986): 511-529.
Mandelbrot, B. The F r a c t a l G e o m e t r y o f N a t u r e . San Francisco: W. H. Freeman, j 977Mandelbrot, B. “Scalebound or scaling shapes: a useful distinction in the visual arts.”
Leonardo 14 (1981): 45-47.
Mandelbrot, B. The Fracra! Geometry o f N a t u r e . San Francisco: W. H. Freeman, 1982.
Maor, E. 7 'o Infinity and Beyond: A Cultural History of the In/in/te. Princeton, N.J.: Prince­
ton University Press, 1987.
R eferences
Markus, Mario. "Autonomous organization of a chaotic medium into spirals.” In Istvan Hargittai and Clifford Pickover, eds., S p i r a l Symmetry, 165-186. London: World Scientific,
jppj.
Martel, E. "How not to teach ancient history.” American Educator, Spring 1994, 33-37.
Maurer,. Evan M., and Roberts, Allen F. T a b w a , th e Rising o f a New Moon: A Century o f T a b w a
Art. Ann Arbor: University of Michigan Museum of Art, 1985.
May, Robert M. "Simple mathematical models with very complicated dynamics.” Nature 261
(1976): 459-467.
Mees, A. "Chaos in feedback systems.” In A. V. Holden, ed., C h a o s . Princeton, N.J.:
Princeton University Press, 1984.
Mercer, Kobena. "Black hair/style politics.” New Formations no. 5 (1988): 33-54.
Meurant, Georges. Shoowa Design: African Textiles f r o m the Kingdom o f K u b a . London:
Thames and Hudson, 1986.
Meurant, Georges, and Thompson, Robert Farris. Mbuti Design: Paintings by Pygmy Women
o f the Ituri Forest. New York: Thames and Hudson, 1996.
Miles, Charles. Indian and Eskimo A r t i f a c t s 0 / North America. Chicago: Henry Regnery, 1958.
Minh-ha, T. “She, the inappropriated Other.” Discourse 8 (1986).
Moore, C. G. "Research in Native American mathematics education.” For the Learning o f
Mathematics 14, no. 2 (June 1994): 9-14.
Moore, Doris. A d a , C o u n t e s s o f L o v e l a c e . London: J. Murray, 1977Morris, E., and Harkleroad, L. “R6zsa Peter; Recursive function theory’s founding mother.”
The Mathematical Intelligencer 12, no. 1, (1990): 59.
Mount, Marshall W. African Art: The Years since 1920. Bloomington: Indiana University Press,
>9 7 3 '
Mudimbe, V. Y. The Invention of A f r i c a . Bloomington: Indiana University Press, 1988.
Mveng, Engelbert. L'nrt d ' A f r i q u e Noire. Paris: Point Omega/Mame, 1964.
Nabokov, Peter, and Easton, Robert. Native American A r c h i t e c t u r e . Oxford: Oxford University
Press, 1989.
National Assessment of Educational Progress. Princeton, N.J., 1983.
Nazarea-Sandoval, Virginia. "Fields of memories as everyday resistance." Cultural Survival
Quarterly, Spring 1996, 61-66.
Neihardt, John. Black Elk Speaks. Lincoln: University of Nebraska Press, 1972.
Nelson, D-; Joseph, G. G.; and Williams,). Multicultural Mathematics. Oxford; Oxford U ni­
versity Press, 1993.
Nelson, Nici, ed. A f r i c a n W o m e n in t h e D e v e l o p m e n t P r o c e s s . London: Frank Cass, 1981.
Nolan, Robert. Bassari Migrations. Boulder, Colo.: Westview Press, 1986.
Nooter, N .I., and Robbins, W. M. “Bembe,” plate 1221, in African Art in American Collec­
tions. Washington D.C.: Smithsonian Institution Press, 1989.
Norberg-Schulz, Christian. Intentions in Architecture. Cambridge, Mass.: MIT Press, 1965.
Nordenfalk, Carl. Celtic and Anglo-Saxon Painting. New York: George Braziller, 1977.
Odica, Okechukwu. Traditional African Art. History of Art 505. Columbus: Ohio State
University, 1971.
Oritz de Montellano, B. “Melanin, Afrocentricity, and pseudoscience." Yearbook of Physi­
cal Anthropology 36 (1993): 33-58.
Ozkan, Suha. “Architecture to change the world I” In Ismail Serageldin, ed., T h e . A r c h i c e c tu r e o f E t n l m v e r m c n t . Lnnhmn, Md.: Academy Editions, 19157.
Parrinder, Geoffrey. A f r i c a n Mythology. London: Paul Hamlyn, 1967.
Pearson, W., and Bechtel, H. K., eds. Blacks, Science, and American Education. New
Brunswick, N.J.: Rutgers University Press, 1989.
Peirgen, H. O.; Saupe D.; Jiirgens, H.; Maletsky, E.; Perciante, T.; and Yunker, L., eds. Fractalsfor the Classroom: Strategic Activities. Vol. 1. New York; Springer-Verlag/NCTM, 1991.
Peitgen, H. O., and Saupe D., eds. The Science o f F r a c t a l Images. New York: Springer-Verlag,
1988.
249
250
R eferences
Pelton, Robert D. The Trickster in West A f r i c a : A Study o f Mythic Irony and Sacred Delight.
Berkeley: University of California Press, 1980.
Pennant, T. “Housing the urban labor force in Malawi: An historical overview, 1930-1980.”
A/rican Urban Studies, 16 (Spring 1983), 1-22.
v~'Perczel, C. F. "Ethiopian crosses at the Portland Art Museum.” A f r i c a n Arts”i 4, no. 3 (May
1981): 51-55.
Perrois, Louis. Ancestral Art o f G a b o n : F r o m t h e C o l l e c t i o n s o f t h e B a r b i e r - M u e l l e r M u s e u m .
Geneva: Barbier-Mueller Museum, 1985.
Petitto, A. L. "Practical arithmetic and transfer: A study among West African tribesmen.”
Journal 0 / Cross Cultural Psychology 13 (1982): 15-28.
Petruso, K. M. "Additive progression in prehistoric mathematics: A conjecture." Hixtoria
Mathematica 12 (1985): 101-106.
Phillips, Tom. A f r i c a : The Art o f a Continent. New York: Prestel, 1995.
Pickover, Clifford. C o m p u t e r s , P a t t e r n , C h a o s a n d B e a u t y . New York: St. Martin's P r e s s , i< y g o .
Ptcton, John, and Mack, John. A f r i c a n T e x t i le s : Looms, Weaving and Design. London: British
Musem, *979.
Polanyi, Michael. The Tacit Dimension. Garden City, N.Y.: Doubleday, 1966.
Porter, T. M. The Rise 0/ Statistical Thinking, 1820-1900. Princeton, N.J.: Princeton University Press, 1986.
Porush, D. "Fictions as dissipative structures: Prigogine’s theory and postmodernism’s road­
show."'In N. Katherine Hayles, ed., C h a o s a n d O r d e r . Chicago: University of Chicago
Press, J 9 9 1 .
Powell, L. “Factors associated with the underrepresentation of African Americans in mathe­
matics and science." Journal 0/ Negro Education 59, no. 3 (1990).
Preziosi, D. A. “Harmonic design in Minoan architecture.” Fibonacci Quarterly, December
3968,371-383.
Restivo, Sal. The Social Relations o f P h y s i c s , M y s t i c i s m , and Mathematics. Dordrecht:
D. Reidel, 1985.
Roberts, Allen F., and Maurer, Evan M. T a b w a : T h e R is in g o f a Neiu Moon. A nn Arbor: U n i­
versity of Michigan Museum of Art, 1985.
Roberts, Allen F., and Roberts, Mary Nooter, eds. Memory: Luba Art and the Making 0 / His­
tory. New York: Museum for African Art, 1996.
Roberts, Mary Nooter. “Luba memory theater.” In Allen F. Roberts and Mary Nooter
Roberts, eds., Memory: Luba Art and the Making o f History. New York: Museum for African
Art, 1996.
Rose, T. Black Noise. Hanover, N.H.: Wesleyan University Press, 1994.
Roth, K . J. “Second thoughts about interdisciplinary s t u d i e s . ' ' A m e r i c a n E d u c a t o r 44-48
(Spring I 9 9 4 )Rowe, Walter. "School daze: A critical review of the 'African-American Baseline Essays for
Science and Mathematics.’ ” Skeptical Inquirer, September 1995.
Rubel, L. A. “Digital simulation of analog computation and Church's thesis." J. Symbolic
Logic 34, no. 3 (September 1989): toi 1-101 7.
Rucker, Rudy. CALAB. San Jose: Autodesk, 1989.
Sabel, Charles, and Piore, Michael. D i a l o g o n F le x i lr f e M a n u f a c t u r i n g N e t t v o r k s . Research
Triangle Park, N.C.: Southern Technology Council, 1990.
Sagay, Esi. A f r i c a n Hairstyles. Oxford: Heinemann International, 1983.
Sandoval, Chela. “New sciences: Cyborg feminism and the methodology of the oppressed."
In Chris Gray, ed., The Cyborg Handbook, 407-422. New York: Routledge, 1995Saupe, Dietmar. “Algorithms for random fractals.” In Heinz-Otto Peitgen and Dietmar Saupe,
eds., The Science o f Fractal linages, 71-113. Berlin: Springer-Verlag, 1988.
Schildkrout, Enid, and Keim, C u r t i s A . A f r i c a n R e f l e c t i o n s : A r t f r o m N o r t h e a s t e r n Z a i r e .
Seattle: University of Washington Press, 1900.
Schroeder, M. F r a c t a l s , C h a o s , and P o w e r Lniw. New York: W. H. Freeman, 1991.
R eferences
Schuler, Douglas. New Community Networks: W i r e d f o r C h a n g e . New York: AddisonWesley, 1996.
Schwab, W. B. “Oshogbo: An urban community?” In H. Kuper, ed., Urbflni?;ation a n d M i g r a ­
t i o n in W e s t A f r i c a . Berkeley: University of California Press, 1965.
Seignobos, C. Nord C a m e r o u n : Montages et hautes t e r f e s . Roquevaire: Editions Parentheses,
.1982.
Serageldin, Ismail. “'Micro-finance; Reaching the poorest.” in Ismail Serageldin, ed., T h e A r c h i ­
t e c t u r e o f E m p o w e r m e n t . Lanham, Md.: Academy Editions, 1997.
Shaw, Carolyn Martin. "The achievement of virginity: Sexual morality among the Kikuyu
of Kenya.” National Endowment for the Humanities Summer Seminar, 1989.
Shaw, Carolyn Martin. Colonial Inscriptions. Minneapolis: University of Minnesota Press, 1995.
Shiva, Vandana. B io p ir a c y : T h e P lu n d e r o f N a t u r e a n d K n o w le d g e . Boston: South End Press, 1997.
Sieber, Roy. African Textiles and Decorative Arts. New York: Museum of Modern Art. Dis­
tributed by New York Graphic Society, Greenwich, Conn., 1972.
Skinner, S. Terrestrial Astrology: Divination b y G e o m a n c y . London: Routledge and Kegan Paul,
1980.
Smith, D. "The queen of rap." New York, December 3, 1990, 123-146.
Smith, E. W., and Dale, A. M. The lla-Speaking P e o p le s o f N o r t h e r n R h o d e s ia . New York: Uni­
versity Books, 1968 [1920].
Sobchack, Vivian. "Theories of everything: A meditation on chaos.” A r t f o r u m , October 1989.
Soppelsa, Robert T. Sculpted Wooden Doors o f I v o r y C o a s t : T h e B a u l e a n d S e n u f o . M. A.
thesis S71 2, Ohio State University, 1974.
Soyinka, Wole. “The last despot and the end of Nigerian history?” index on Censorship 23,
no. 6 (November 3994): 67-75.
Spivak, Gayatri Chakravorty. Jn Other Worlds: E s s a y s in Cultural Politics. New York:
Methuen, 1987.
Steenburg, D. “Chaos at the marriage of heaven and hell.” H a r v a r d T h e o l o g i c a l Review 84
(1991): 447-466.
Stein, Dorothy. Ada: A L i f e a n d a Legacy. Cambridge, Mass.: MIT Press, 1985.
Stephanides, Michel. "La naissance de la chimie.” Scientia 21 ( j 922): 189-196.
Stiff, L. V.; Johnson, J. L.; and Johnson, M. R. “Cognitive issues in mathematics education."
in P. S. Wilson, ed., Research I d e a s f o r th e C la s s r o o m : H ig h S c h o o l M a t h e m a t i c s . New York:
Macmillan, 1993.
Sroller, Paul. "The negotiation of Songhay space: Phenomenology in the heart of darkness.”
American Ethnologist, 1980.
Stone, Aliucqiiei e' Rosanne. The War o f D e s i r e and Technology at the Close o f th e M e c h a n i c a l
Age. Cambridge, Mass.: MIT Press, 1995.
Suvan, D. E. The Magic o f Bimdelier. Santa Fe, N.M.: Ancient City Press, 1989.
Tank, David W., and Hopheld, John J. “Collective computation in neuronlike circuits." S c i ­
e n t i f i c A m e r i c a n , December 1987, 104-J14.
Tanner, Clara L. Southwest Indian Craft Arts. Tucson: University of Arizona Press, 1968.
Taylor, C. C. “Condoms and cosmology: The “fractal’ person and sexual risk in Rwanda.”
Soc. Sci. Med. 31, no. 9 (1990): 1023-1028.
Taylor, F. Sherwood. “A survey of Greek alchemy." Journal of Hellenic Studies, 50 (1930):
109-119.
Taylor, P. J. “Technocratic optimism, H. T. Odum, and che partial transformation of eco­
logical metaphor after World War 11." Journal o f the History o f B i o lo g y 21, no. 2 (Sum­
mer 1988): 2 13-244.
Thomas, Elise, and Slockish, Marie. T h e H e r i t a g e o f K l i c k i t a t B a s k e tr y . Portland: Oregon His­
torical Society, 1982.
Thompson, D’Arcy Wentworth. On Growth and Form. Cambridge, U.K.: Cambridge Uni­
versity Press, 1917.
Thompson, J.E.S. Maya History and Religion. Norman: University of Oklahoma Press, 1970.
25 2
References
Thompson, Robert F. Flash o f th e S p i r it . New York: Vintage Books, 1983.
Trowell, Margaret. A f r i c a n D e s i g n . New York: Frederick A. Praeger, i960.
Tuana, Nancy. “The weaker seed." In Nancy Tuana, ed., Feminism and Science, 147-171. BtoOmington: Indiana University Press, 1989.
Tnrkle, Sherry. L i f e on th e Screen: Identity in t h e A g e o f th e I n t e r n e t . New York: Simon and
Schuster, 1995.
Usiskin, Z. “ We need another revolution in school mathematics." In T h e S e c o n d a r y S c h o o l
Curriculum 1983 Y e a r b o o k , ed. C. R. Hirsch and M. J. Zweng, 1-21. Reston, Va.:
National Council of Teachers of Mathematics, 1985.
Van Wyk, Gary. "Secrecy, knowledge, power and postinodernity.” In David Franke), ed.,
S e c r e c y : A f r i c a n Art dint C o n c e a l s a n d R e v e a l s . New York: Prestel-Verlag, 1993.
Vergis, A.; Sceiglitz, K.; and Dickinson, B. "The complexity of analog computation.” Tech­
nical report no. 337, Department of Electrical Engineering and Computer Science, Prince­
ton University, February 1985.
Vogel, Susan. Baule Art a s th e E x p r e s s i o n o f W o r l d V i e w . Ann Arbor, Mich.: University Micro­
films, 1977.
von Fritz., Kurt. "The discovery of incommensurability by Hippasus of Metapontum.” Annals
o f M a t h e m a t i c s 46, no- 2 (April 1945).
Voss, R. F. “Fractals in nature." In H. O. Peicgen and D. Saupe, eds., T h e S c i e n c e o f F r a c t a l
I m a g e s , 21-69. New York: Springer-Verlag, 1988.
Wahl, B. Exploring Fractals on th e M a c i n t o s h . New York: Addison-Wesley, 1995.
Washburn, D. K.( and Crowe, D. W. Symmetries o f C u l t u r e . Seattle: University of Washington
Press, 1988.
Wassing, Rene. A f r i c a n A r t : Its Background and Traditions. New York-. Harry N. Abrams, 1968.
Waters, Frank. Book o f th e Hopi. New York: Viking Press, 1963.
WatsorvVerran, H., and Turnbull, D. “Science and other indigenous knowledge systems.”
In Sheila Jasnnoff, Trevor Pinch, Gerald Markle, and James Petersen, eds., H a n d b o o k
o f S c i e n c e a n d T e c h n o lo g y S t u d i e s . Beverly Hills, Calif.: Sage Publications, 1994.
Witherspoon, Gary, and Peterson, Glen. Dynamic Symmetry and Holistic Asymmetry in
N a v a jo a n d Western Art and C o s m o lo g y . Bern and New York: Peter Lang Publishing, 1995Williams, Denis, /con and Image. New York: New York University Tress, 1974.
Wilson, Eva. N o r t h American Indian Designs. Bath: Pitman Press, 1984.
Wiredu, J. E. "How not to compare African thought with-Westem thought.” In R. Wright,
ed-, A f r i c a n P h i lo s o p h y : An Introduction. Washington, D. C.: University Press of Amer­
ica, 1979.
Wittig, M. The Lesbian Body. Boston: Beacon Press, 1973.
Wolfram, S. “Universality and complexity in cellular automata.” Physica 10D (1984): 1—35.
Wolfram, S., ed. T h e o r y a n d Application o f C e l l u l a r A u t o m a t a . Singapore: World Scientific,
1986.
Zahan, Dominique. The Bambara. Leiden: E. J. Brill, 1974.
Zaslavsky, Claudia. A f r i c a Counts. Boston: Prindle, Weber, and Schmidt, 1973.
Zaslow, Bert, and Dittert, Alfred E. Pattern Mathematics and Archaeology. Phoenix: Arizona
State University, 1977-
Index-
abbia,120,138-140,145-146
Abraham, Ralph, 193, 238118, 239116
abstraction, 17, 51, 53, 62, 78, 102, 109, 131,
133, 202, 21 2, 213, 214, 216
Achebe, Chinua, 173
addition modulo. Sett mod 2
additive scries, 186-189
aesthetics- See esthetics
affine transformation, 75
Afiocenirism, 180-181, 218, 222
age-grade, 68, 87, 121, 124, 237116
agriculture, 24, 31, 125, 227-229
Agudoawu, Kofi, 107
Akan, 77-78, 81, 104
alchemy, yy, 100, 101, J40, 141,2381112
algorithm, 38, 47, 6 1, 68, 77, 97, 113, 118,
' 3 3 . «53 —) 5 4 . 17°. ' 7 4 .206
analog, 151-154, 158-161, 164, 192-194,200,
202, 214, 229, 23811m , 2, 240113, 24ini 2
Ananse, 137
Angola, 68, 186
animism, 194
anthropology: authority in, 183—184; function­
alist, 170; mathematical, 185-187, 191;
modernist, 131,238; reflexive, 95; struc­
turalist, 18 x, 188. S ue a ls o ethnography
apartheid, 184, 200
apei iodicity, 108, 172
.Arabic culture, 98-99, 205
archaeology, 61, 87, 89
architecture: African, 4-8, 19-40, 87-89,
110—111, 124, 126-128, 131, 135,
148-149, 162-164,'^6.174, '95-199,
205, 210, 216-222, 224, 226; American,
3 - 5 . 3 9 . 4 9 - 5 0 . 55. ' 9 7 - ' 9 9 ; Chinese, 4;
European, 3, 20, 39, 48-51, 55, 89, 174,
195-196, 225; Indian, 47-48; Mative
Am. t k ; n, 39-42; South Pacific, 47
Aristotle, d, 51, 147-148, 205-206, 242
arithmetic, 86-108
arithmetic series. Sec additive series
art education, 225
artificial intelligence, 213
Ascher, Marcia, 45, 47, 186, 237114
Ashanti, 137
authenticity, 74, 184, 193-194, 217, 23602,
238119, 240m
authority, 31, 133, 183, 186, 203, 227—228
Babbage, Charles, 21 1-212
Badinne, Nfally, 162, 164
Ba-ila, 26-29, 5 5 . ' >o. 23 5 n 4
Bak, Per, 161,170, 226
Baka, 183, 240114
Baker, Houston, 194
Bakuba, 172-173, 222
Baluha, 130, 166, 210
Bambara, Toni Cade, 194
Bamilekc, 24-25
Banneker, Benjamin, 55,90, 182, 183
253
In d e x
254
B an tu, 62
C h a i t i n , G re g o ry , 153
Banyo, 34-36
c h a o s , 9 3 , 9 5 , 103, 108, 143, 159, 162, 168,
1 7 4 , 1 8 2 , 1 9 0 , 1 9 3 , 1 9 7 , 1 9 9 , 2 1 4 , 2 37iM
b asket w eaving, 4 5 -4 6 , 222
B nssari, 1 2 1 - 1 2 2 , 2 3 7 0 5
chi w ara, 1 2 4 -1 2 5 , 127, 134, 209
B a ta m m a lib a , 121, 126, 135
C h in e se m ath em atics, 4, 4 7 -4 8 , 185, 225,
23502
B atty , M ic h a e l, 4 9 - 5 0
b ead w ork, j J3 , j 19, 166, 2 3 7 0 5
C h o k w e , 6 1 , 6 8 , 6 9 , 7 0 , 8 4 , 187
B e l l , E r i c T . , 2 0 7 —2 0 8
C h o m s k y , N o a m : c o g n itiv e th e o r y of, 2 11;
h ie r a rc h y of, 1 5 6 - 1 5 8
B e m b e , j 23
B enin, 91, 1 2 4 ,1 4 1 -1 4 3 , 166, 182, 216,
C h ristia n ity , 20, 4 8 , 9 0 , 127, 1 3 5 -1 3 6 , 149
cities.
237 n 5
See
B etg, T Q , 224
c la s s , 81
a rch itectu re
B ern o u lli, Ja c o b o , 210
C lif f o rd , J a m e s , 131, 1 8 3 , 1 9 3 , 236112, 2 4 0 0 5
B e y , H a k i m , 2 4 1n 9
co astlin es, 1 5 .1 7
b i n a r y c o d e , 9 5 , 9 8 , 1 01
c o lo n ialism , 1 9 5 -1 9 7
b in o m ia l c o efficien ts, 2 3 7 n s
c o m p l e x i t y , 5, 4 5 , 6 8 , 1 4 6 , 1 5 1 - 1 7 6 , 1 8 4 , 1 8 9 ,
b io lo g ical d e te r m in is m , 187, 191, 2 2 4 - 2 2 5
biology, 3 , 3 4 , 8 4 , 1 0 2 - 1 0 5 , 1 0 7 - 1 0 8 , 1 24,
225, 2 2 8 ,2 3 0
c o m p u t e r : a n a l o g , 1 5 1 —1 5 5 , 1 5 8 - 1 6 1 , 1 6 4 - 1 6 6 ;
1 3 1 ,1 3 3 , 1 4 1 ,1 5 9 ,1 8 9 , 191, 2 2 7 -2 2 9 ,
c a l c u l a t i o n by, 7 4 , 8 9 , 9 7 , 1 5 1 ; i n d e v e l o p ­
24006
m en t, 2 2 9 -2 3 0 ; ed u catio n , 2 2 3 -2 2 5 ; h ard ­
w a r e , 9 5 , 9 8 , 1 0 1 ; p r o g r a m s , 1 1 0 —1 1 2 , 1 3 2 ,
b io tech n o lo g y , 2 2 8 -2 2 9
b irth , 34, 9 0 , 109, 127, 131, 133, 168, 170,
135, 1 3 7 -1 3 8 , 188, 2 11; s im u la tio n , 3 ,1 2 ,
21, 28, 31, 3 2 , 34, 38, 6 i ,
208, 210, 212, 24205
B lix e n , K a r e n (lsak D in e s e n ), 197
71,
77, io r- ro 4 ,
147, 172; th eo ry , 146, 1 5 6 -1 5 8 , 2 1 2 - 2 1 4
B l y d e n , E. W ., 2 0 0
C o n g o . See D e m o c ra tic R e p u b lic o f C o n g o
body, 12, 6 3 - 6 5 , 7 5 -7 6 , 1 3 1 -1 3 3 , 164, 226,
C onw ay, Jo h n H o rto n , 103 -1 0 4 , (70
c o o r d in a te sy stem s: C a r te s i a n , 3 - 5 , 4 2 , 8 5 ,
24011m , 3
196; polar, 2 3 1 -2 3 4 ; sp h erical, 83
Boggs, Ja m e s, 2 4 0 m
B ourdier, Jc n n -P a u f, 3 2 - 3 3
C o p t ic d e sig n , 2 3 6 0 4
b raid in g . See h airsty les
c o rn ro w s. See h airsty les
b rid ew ealth , 89
cosm ology, 4 3 - 4 4 , 4 8 , 1 3 1 - 1 3 5 , 2 0 4 , 210
B ro ad w ell, P eter, 3 1
c o u n t i n g : b a s e six , 122; b a s e t e n , 4 , 9 9 , 2 3 5 0 2 ;
base tw o, 8 9 - 9 1 , 100
b ro n ze s c u lp tu re , 1 3 8 -1 3 9
C ro w e , D o n a ld , 4 7. 48
B row n, Jam es, 1 9 9 -2 0 0
b ro w n noise, 2 3 9 0 7
C row ley, A lcister, 99
B u r k i n a F a s o , 3 1 - 3 3 , 182
C ru tc h fie ld , Ja m e s, 1 5 9 -1 6 0 , 174
cy b ern etics, 2 3 6 n 2 , 2 3 8 0 2
B u tler, O c t a v i a , 194
B w am i, 52, 123
•
cyborgs, 2 1 6 , 2 4 2 m
B w iti, 129
D a n , 1 4 1 - 1 4 3 , 1 6 6 . 170, 175
C airo , 3 7 - 3 8 , 2 0 1 -2 0 2
D a n g b e . See D a n
C a m e r o o n , 2 1 —2 5 , 2 9 - 3 1 , 3 4 - 3 6 , 1 1 3 ,
D a u b e n , J. W ., 2 0 8
1 1 9 - 1 2 0 , 1 3 8 - 1 3 9 , 145, 14 9 - 1 5 0 , 182,
D a v i s , A n g e l a , 240112
1 9 0 , 2 1 6 , 2 3 9 0 7 , 240114
d e S o u s a , M a r t i n e , 141
C a n t o r , G e o r g , 8 - 1 0 , 1 9 7 , 2 0 6 —2 0 8
d e S o u z a , F r a n c i s c o , 141
C a n to r, M oritz, 208
d e a th , 3 4 , 164, 170, 2 0 4 ,2 1 4
C a n t o r set, 1 2 - 1 3 , 15, 17, 9 3 , 9 9 , 1 4 7 - 1 4 8 ,
d ecen tralizatio n , 31, 39, 189, 197, 222, 229,
206-208
236111
C a p la n , P at, 195, 2 4 1 0 6
D e la n y , S a m u e l R ., 194
C a rb y , H a z e l, 194
D e m o c r a t ic R e p u b l ic o f C o n g o , 6 1 , 1 2 7 , 166
C a r v e r, G e o r g e W a s h i n g t o n , 194
D e rrid a , Ja cq u es, 1 9 2 -1 9 3
c a r v i n g , 7, 4 3 - 4 4 . 4 5 , 6 2 - 6 3 , 6 8 , 1 0 8 , 1 13,
D escartes, R e n e , 1 9 5 -1 9 6
1 1 7 , 1 2 0 ,1 3 8 , 1 4 3 ,1 6 6 , 187, 189
d e s c e n t , 8 , 1 2 4 - 1 3 1 , 1 4 9 , 2 0 6 , 237118
C a s a m a n c e , 162, 164
d esig n th em e s, 3, 4 , 6, 27, 3 9 - 4 0
c a sc a d e , 1 0 9 - 1 1 0 , 1 n - i 14, 145
D e stn , G e h r e K ristos, 2 1 6
C ay ley tree, 222
d e te r m in is tic c h a o s. Sec c h a o s
Cayuga, 186
dev elo p m en t, 2 2 5 -2 3 0
ce llu la r a u to m a ta , 1 0 2 -1 0 8 , 143, 1 5 4 -1 5 5 ,
diaspo ra, 55, 180, 199
i 58, 162, 164, 168, 170
D in rta , C h r i s t i a n S i n a , 7, 1 6 1 - 1 6 2 , 164
C e l t i c d e s i g n , 7, 4 8
D iaz, R o g c lin , 4 3 - 4 4
C e s a i r e , A i m e , 1 8 8 , 240115
differen tial e q u a tio n s , 2 3 6 0 2
Index
diffusion limited aggregation, 49
digital, 101, 104, 151-152, 156-158, 166, 190,
192-194, 200, 211-213, 229, 23811m, 2,
2411112
dimension, 12, 15, 18-19, 32, 43, 81, 83-84,
93, 104, 113, 115, 154, 170-172, 176, 209,
23808, 239007, 1
^
disease, 17, 227
disequilibrium, 170
divination, 31, 93-101, 108, 122, 124, 133,
143, 151, 183, 190, 209, 237n4
Dogon, 131-134, 138, 140, 146, 170,175
doubling. See counting: base two
Du Bois, W.E.B., 200
dynamical systems theory, 23906
East Africa, 86, 99, 216
economics, 189, 196, 211, 217, 223, 227, 229,
240115
education. See art education: mathematics
education
Egypt. 37-38, 87-89, 9 9 . ‘ 3 4 - 1 3 5 . ‘ 3 7 .
140-141,188-189,‘91.204-208,23604
Ellison, Ralph, 194
engineering, 5, 73-74, 85, 143, 230
Eno, Brian, 101
environment, 20, 39, 50-51, 133, 219,
227-229,24on6
Epimenides of Crete, 111, 137
epistemology, 180, 189, 193, 225, 230
Eshu,174,175
cssentialism, 180-182
esthetics, 7, 38, 50, 52, 53-57, 62—63, 81, 113,
209
ethics, 192, 194-195, 2 10, 24004
ethnography, 28, 31, 45, 127, 131, 181 —184,
200, 203, 223, 235115. See a lso anthropol­
ogy
ethnophilosopliy, 149, 189-190
Ethiopia, 101,1 35-136
Euclidean construction method, 65, 68-69,
113, 118
Eulerian path, 48. 68, 70, 186
evolution, 161, 187, 189-190, 240116
Fagg, Will vam, 7, 84, 139, 190
falsili.ibiiity, 6, 179, 240111
Fang, 127, 129, 149, 210, 23707
fetus. See birth
Fibonacci series, 87-89, 11o - 1 11, 156,
205-206
finite state automaton, 156-158, 237m
fluid flow, 47-48, 77-78, 97, 104, 209, 213
F o n ,190
Foucault, Michel, 189, 194-195, 209, 24109
Fourier transform, 231, 233-234
fractal dimension. See dimension
fractal geometry: definition of, 8-19; European
history of, 8-17, 203-215. See also com­
puter: simulation; dimension; infinity;
recursion; scaling; self-similarity
fractions, 204, 205, 23905
free will, 97, 199,24109
Fulani, 29, 113, 119
Fuller, Thomas, 122, 23705
functionalism. See anthropology: functionalist
funepal rituals, 164
Gabon, 127
Gambia, 121, 182, 23705
game of life. See cellular automata
game theory, 101
Garcia, Linda, 93
Garvey, Marcus, 200
Gates, Flenry Louis, 90, 190, 219, 24204
Gauss, Carl Friedrich, 206
Geertz, Clifford, 181-182
gender, 190, 212-213, 227
genetics, 124, 161, 180, 188, 228, 24006
geometry. See affine transformation; computer:
simulation; coordinate systems; dimension;
Euclidean construction method; Eulerian
path; fractal geometry; graphing; helix;
hexagon; iterated function systems; nondifferenciable curve; pentagon; Poincan§ slice;
quincunx; scaling; self-similarity; Sierpinski gasket; sinusoidal waves; spiral; tiling;
trigonometry
geomancy, 98-101
Gerdes, Paulus, 68, 122, 186, 222
Getz, Chonut, 222
Ghana, 74, 77-80, 101, 104-108, 113, 115,
124, 182, 226-227, 237n5
Gilmer, Gloria, 224
Gleick, James, 182
Gcidel, Kurt, 199, 214, 238mo
graphing, 4, 12, 14, 47, 73-74, 79, 81,83-85
graphics. See computer: simulation
Greek culture, 76, 89, 99, 141, 147—148,
203-206,210, 225
Griaule, Marcel, 131, 133
griot, 164
Guinea-Bissau, 44, 121
hairstyles, 7, 63, 81-84, 112-114
Hausdorff, Felix, 12
Hausdorff-Besicovirch measure. See dimension
Heaver, Hannan, 38, 200, 202
Heighway, John, 113
helix, 112, 114
Hermes Trismegistus, 99, 134, 141,238ml
HerskovitS, Melville, 107
hexagon, 4, 5, 121—122, 214, 222, 235m
"■hierarchy, 39, 120, 122, 156-158, 189, 197,
2 1o, 230, 236m
Hindu culture, 99, 185, 187, 225
Hofstadter, Douglas, 1io, 213, 2381110
homosexuality, 213-214
homunculus, 127, 242115
Hughes, David, 218-222
humanism, 194-195, 209
Hurst, H. £., 12, 208-209
255
In d e x
256
Hurston, 2 orn Neale, 188
hybrids, 187, 200, 230, 24in 11
Ibo, 197
lfa, 9 3 ~ 9 5
India, 7, 47-48
infinity, 8-9, 12, 13, 18, 34, 4J-42, 70, 76-77,
91,111,135,138-139,146-150, 153,
157—759, j 76, 790, 204-207, 2jo, 222,
239007,1,24inn9,3
information technology. S e e computer
initiation, 68, 87, 100, 121—123, 133, 23706
intentionalt'ty, 5-6,19,49-57,81, 113, 123,.
162, 165, 174, 184-187, 219-220, 225
inunlion, 53, 56-57, 68, 71, 113, 154, 24102
iron work, 61, 89-90, 141, 143
irrational numbers, 97, 204, 24innt, 2
lshango bone, 89, 91
Islam, 29, 3j , 38, 93, 162, 202, 205
iteration, 15, 17, 18, 21, 22, 25, 26, 28, 29, 30,
31*3 4 . 37. 38. 4 5 . 48, 61, 67, 68, 69, 76,
79, 8i, 86-88, 91, 95, 103-104, 1jo- 130,
132-137, 146, 155, 170,172, 176, 210,
2J2, 222, 2370;, 238030, 24207
iterated function systems, 76, 222
ivory sculpture, 62, 63, 65-68
japan,47-48
jew e lry, 5 3 - 5 4
Jews, 99, 101, 200, 202, 207-208, 24m!o
Jola, 162-165
juma, Calestous, 228-229
Kabbalah, 99
Kamil, abu, 205
Karnak, 88
Kauffman, Stuart, 170
kente cloth, 74-76, 226-227
Kenyecta, Jomo, j 88
Kepler, Johannes, 206
Kikuyu.209,23706
King, Martin Luther jr., 199
kinship, 24, 113, 124, 127, 130-131, 145, 164,
186,209, 23503
Kirdi, 29
knot theory, 48
Koch, Helge von, 9-15, 17-18
Kolmogorov, A. N., 152-1 53. t 55
knra, 2! 7-218
Kotoko, 21, 24, 32
Kronecker, Leopold, 208
Kuba. See Bakuba
Knti, Feta, 200
Kwele, 722-123, 127-328
Labbezanga, 3 1-32, 231-232, 234
labor, 24, 39, 113, 187, 189, 196, 227
Leaky, Louis, 23706
Legba, 143-144, '66, 175, 216,23704,
238013
Leibnitz, Gottfried, 100-101
lightning, 9 ’ ~ 9 3
limit cycle, 106, 143, 228
lineage, 24, 124, 127
linearity, 40-42, 71, 74, 76-77, 86, 121,
129-130, 196, 197, 2i i , 222, 237
linguistics, 193
*’•
logic, 4, 28, 70, 98, 111-112, 135, 204, 1 31, 213
Logone-Birni, 21-24
lotus, 335, 137
Lourde, Audre, 24003
Lovelace, Ada, 211-232
Luba. See Bnhiba
Lucas, Edouard, 206
Lull, Raymond, 99—101
tangs, *5-17, 34
Malagasy, 98
Malawi, 196
Mali, 8. 31-32, 71-72, 131, 182
mancala, 101
Mandelbrot, Benoit, 12, 15, 17, 47, 51, 93,
176, 197,208-209, 2 ,4
Mandiack, 44, 52
Mangbetu, 61-68, 70
marriage, 119, 124
masks, 80-81, 84, 121-123
mathematics education, 222, 223-225,
236002, 3
Mauritania, 113, 115,218-219
May, Robert, 159, 168
Mayer-Kress, Gottfried, 23906
Mbuti, 54, 23909
measurement, 4, 9, 12-18,31, 72—74, 79, 89,
122,151,153-155,159-160, 172,
174-175, 239m
medicine, 17, 127, 196, 24205
memory, 34, 97, 156-159, 161, 166, 174, 228,
229, 238113
metalwork, 7, 112, 216. S e e a ls o bronze sculp
tore; iron work
Mezzrow, Mezz, 2 4 7 0 1 0
migration, 121, 227
mimesis, 50-53. 56
Mitsogbo, 127-1 29, 149, 2 10
mod two, 95, 98-99
Mofon, 29-31
morphogenesis. See biology
Morse, Mnrstnn, 97—98, 237114
Mozambique, 222
Mudimbe. V. Y., 149, 180, 189-190, 194,
24005
multiculturalistn, 206, 225
music, 64-65, 143, 149, 154, 174. 193. 194,
200, 204, 209, 23804, 241010, 242012
Mveng, Engelbert, 149-150. 190
Nanknni, 32—34, 148-149. 210
narrative, 93, 95, 96, 133, 137, 146, 148, 149,
179, 186, 202, 206, 23704, 238119
Native American culture, 40-46, 48, 116, 184,
186,229, 23704
In d e x
n a t u r e , 1 7 , 1 8 , 4 7 , 48, 5 0 - 5 3 , 5 6 - 5 7 , 6 2 , 1 4 1 ,
149, 180, 1 8 1 , 1 9 0 , 1 93 , 228 , 2 3 6 0 2 , 2.3902
N a z a r e a - S a n d o v a l , V i r g i n i a , 22 9
“ n e g r i t u d e , " 188, 1 9 0 , 1 9 1 , 2 4 o n 5
n e u r a l n e ts , 1 5 2 , 1 5 4 , 1 65
n e u r o b i o lo g y , 1 5 2 , 1 5 6 , 1 8 7 , 1 9 9 , 2 3 8 , 2400 6,
N e w A g e m y s ti c is m , 187
N i g e r i a , 24 , 94 , 1 3 7 , 1 7 3 - 1 7 5 , 1 8 7 , 200, 2 2 7 ,
230
257
p o w e r law , 7 1 - 7 4 , 8 9 - 9 3 , 1 5 9 - 1 6 1
p r i m i t i v i s m , 53 , 89, 180 , 1 8 8 - 3 8 9 , ! 94>
1 9 6 -19 7 , 224-225
p r o b a b i l i t y , 94. See also c h a o s ; r a n d o m n e s s ;
s ta tis ti cs; s t o c h a s t i c v a r i a t i o n
p r o g r a m m i n g . See c o m p u t e r : p ro g ra m s
pseudorandom n um ber gen eratio n , 9 7 - 9 8
p u s h - d o w n a u t o m a t o n , 1 5 7 —1 5 9
Pythagoras, 203-204
N i l e river, 99 , 2 0 8 - 2 0 9
nomads, 115
n o n d t f f e t e n t i a b l e c u r v e , 239117
Q u e e n L a ti fa , 240115
q u in c u n x ,5 5 ,18 2
n o n l i n e a r i t y , 4 0 - 4 3 , 70, 7 1 , 7 6 - 7 7 , 8 0 - 8 2 , 84,
8 6 - 8 6 , 9 7 , 108, 1 1 3 , 1 1 8 , 1 2 2 , 1 4 3 , 1 6 2 ,
r a ci s m , 180 , 1 8 7 , 188
1 8 2 , 1 9 0 , 200, 2 1 6 , 2 22 , 2 3 6 0 2 , 2 3 7 0 4 ,
ra n do m n e s s, 3 1 , 93—99 , 1 5 2 - 1 5 5 , 1 5 8 - 1 6 : , 1 74 ,
238 08
n u m b er s , 4, 5, 6, 8, 18, 3 1 , 4 1 , 4 2 , 7 6 , 8 6 - 1 0 8 ,
1 22 , 1 5 3 , 1 5 7 , 1 5 9 , 1 8 6 , 190, 2 0 3 - 2 0 6 ,
2 1 2 , 2 2 9 , 235112
n u m e r o l o g y , 4, 20, 9 5 , 1 2 1 —1 2 2 , 1 3 4 - 1 3 5 , 204,
235n2
N um m o , 131, 133, 175
1 8 6 , 1 96 , 1 9 7 , 1 9 8 , 2 3 7 0 4 , 2 38 11 13 , 2 3 9 0 7
ra tio s, 204
r e bi rt h . See bi r th
re c u rs io n , 8 - 1 2 , 1 6 - 1 7 , 34 , 4 3, 4 5 , 4 7 - 4 8 , 5 5 ,
7 7 , 8 6 , 8 9 , 9 3 , 9 5 , 9 8 , 9 9 , 108 , 1 0 9 - 1 4 7 ,
1 4 9 , 1 5 1 , 1 5 5 - 1 5 9 , 1 6 1 , 1 7 6 , 1 8 7 , 190, 1 92 ,
1 9 4 - 1 9 5 , 1 9 9 - 2 0 0 , 2 0 2, 20 5, 2 0 9 - 2 1 4 , 2 1 7 ,
N u p e, 137
2 3 7 0 1 , 238113, 2 3 9 m , 2 4 1 0 7 , 2 4 2 0 0 6 , 7.
N y a n g u l a , A l e x , 2 2 0 - 2 2 2 , 242112
Se e also ca s ca d e ; it e r a t io n ; s e l f - r e f e r e n c e
r e f l e x i v e a n t h r o p o l o g y . See a n t h r o p o l o g y :
O du m , H ow ard, 214
O g o n i , 228
O g o t e m m e l i , 131
r e f l e x iv e
r e l i g i o n , 7, 20, 28, 3 1 , 4 7 , 48, 5 3 , 7 8 , 9 0 , 92 ,
9 3 , 9 9 , 1 24 , 1 2 7 , 1 2 9 , 1 3 1 - 1 3 2 , 1 3 5 ,
i/F n o is e , 1 5 9 , 1 6 1 , 1 66
1 4 1 - 1 4 3 , 1 64 , 1 6 6 , 1 7 0 , 180 , 1 8 9 , 1 94 ,
O n y e j e k w e , E g o n d u , 23 0
20 2, 204, 20 5, 2 0 7 , 20 8, 2 1 1 , 2 4 2 n 3
optim ization, 7 3 -7 4
o r i e n t a l i s m , 188
r e p r o d u c t i o n , 1 0 7 - 1 0 8 , 1 24 , 1 2 5 , 1 3 4 , 1 3 8 ,
1 4 0 , 2 0 9 - 2 1 0 , 2 1 2 - 2 1 4 . See also b i r t h
O R S T O M , 25, 29
owari, 1 0 1 - 1 0 8
rite o f p ass age , 34
P a le s t in e , 89
r o m a n t i c o r g a n i c i s m , 1 94
p a r a d o x , 1 2 , 1 1 1 —1 1 2 , 1 64 , 2 0 3 - 2 0 5
R o s i c r u c i a n i s m , 9 5 , 208
p a r t i c i p a n t s i m u l a t i o n , 29, 182 — 1 8 4 , 2 3 5 n s
p e n t a g o n , 204, 2 4 1 m
R o u s s e a u , Jea n J a c q u e s , 1 9 2 - 1 9 3
p e r i o d i c i t y , 103 , 106, 1 4 1 - 1 4 3 , 1 5 3 , 1 5 6 ,
Russell, Bertrand, 211
r it ua l, 3 1 , 68, 9 9 , 1 2 1 , 1 2 3 , 1 2 6 , 1 2 7 , 1 6 2 , 1 6 4 ,
1 6 5 , 1 8 0 , 186
R u c k e r , R u d y, 104, 162
158-160 , 1 7 2 - 1 7 3 ,2 2 8
Pe ter, Rdz sa , 2 1 2 - 2 1 3
S a h a r a , 38 , 71
p h a s e s p a c e , 239116
Sah el, 7 1 - 7 4
p h i l o s o p h y , 1 49 , 1 7 9 , 1 8 9 - 1 9 0 , 2 0 3, 235112
S a m p s o n , Ja ro n , 224
p h y s ic s , 7 , 1 5, 50 , 1 1 3 , 1 5 1 - 1 5 5 , 1 5 8 - 1 7 6 , 1 9 4
Su r o -W iw a , K e n , 228
pi, 206
s c a l i n g , 1 2 , 1 7 - 1 9 , 2 1 , 26 , 28—29 , 3 1 - 35> 3^.
P l a to , 20 3 —205, 2 1 0 , 2 4 m m , 2, 2 4 2 0 3
p l o t t i n g . See g r a p h i n g
4 1 , 4 3 , 4 3 - 4 8 , 5 2 , 5 4 , 5 6 , 6 1 - 6 3 , 6 5 , 68,
7 0 ,7 1-8 5 ,8 6 , 8 9 , 1 0 4 ,1 i o , 112 -114 ,
P o i n c a r e sl ic e , 238118
1 1 6 - 1 1 8 , 12 0 - 1 2 4 , 1 2 6 - 1 2 8 , 1 3 0 - 1 3 5 ,
p o i n t a t tr a c t o r, 106
13 7 , 1 4 1 , 1 4 8 - 1 4 9 , 156, 166, 174 , 175,
p ol ar c o o r d i n a t e s , 2 3 1 - 2 3 3
1 90 , 1 96 , 200, 2 0 2, 208, 2 1 6 , 2 2 5 , 22 6 ,
p o l it ic s , 3 1 , 3 4 , 1 0 1 - 1 0 2 , 120, 1 2 4 , 1 4 5 , 1 7 4 ,
2 2 7 , 228 , 2 3 5 0 2 , 2 3 9 m
1 79 , 180, 1 8 9 - 1 9 0 , 1 9 2 - 2 0 2 , 2 2 7 - 2 3 0 ,
24 0 0 5 , 24111118, 9
Po pp e r, K a r l, 6, 1 7 9 , 2 3 9 m
p o p u l a t i o n , 5, 2 5 , 4 9 - 5 0 , 9 7 , 1 5 9 , 1 6 8 , 1 96 ,
1 9 7 , 20 5, 22 9 , 236116
S e h i n n a k e r , E. F., 228
S c h y l e r , G e o r g e , 1 94
s c u l p t u r e , 7, 5 2 , 6 3 , 6 6 , 68, 79 , 80, 8 1 , 84,
i t 2 , 113 ,12 7 ,13 3 ,1 3 4 ,1 3 8 -1 3 9 ,2 1 6
se cr et s , 9 3 , 9 7 , 1 2 1 - 1 2 2 , 200, 204
P o r tl a n d B a s e li n e Essays, 1 8 8 - 1 8 9
s e lf - g e n e r a t io n , 9 5 , 9 7 , 100, 1 3 5 , 140, 20 6 , 209
p o s i t i v i s m , 1 79
s e lf - o r g a n i z a t io n , 1 0 1 , 104, 1 0 7 - 1 0 8 , 1 6 1 ,
p o s t m o d e r n i s m , 1 9 3 - 1 9 4 , 1 99 , 2 1 6 , 2 3 6 n s ,
241 n n y , 1 1 , 242116
1 6 4 - 1 6 6 , 168 , 1 70 , 1 7 6 , 1 9 5 - 1 9 7 , 2
22 o , 22 6 , 2 2 8 - 2 3 0 , 2 4 2 0 6
j 8,
Index
258
self-organized criticality, 1 6 1 , 1 7 0 , 2 2 6
Tnhwa, 127, 130,237118
s e l f - r e f e r e n c e , 1 1 0 - 1 1 2 , 1 3 5 , 1 3 7 - 1 4 0 , 1 46
ta ll ie s , 1 2 1 - 1 2 2
s e lf - s im il a r it y , 4 , 1 8 - 1 9 , 2 1 > 2 4 > 2 9 >3 1 - 34 > 3 ^.
4 2 , 4 3 , 9 3 , 1 0 0 , 1 2 4 , 1 2 5 , 1 40 , 1 7 6 , 1 9 5 ,
T a n g , C h a o , 161
209, 2 1 8
S e n e g a l , 8, 5 5 , 8 1 , 9 3, 1 4 0 , 1 6 1 - 1 6 2 , 1 7 4 , 18 2,
t a r u m h e t a , 8 6 - 8 7 , 106, 108 .,,
1 83 , 1 9 0 , 1 1 7 - 1 1 8 , 2 3 7 0 5
T a n z a n i a , 8 9 , 1 95
ta tto o patterns, 47
t e x t i l e s , 7, 1 7 2 —1 7 3
S e n g h o r , L e o p o l d , 7, 190
T h o m p s o n , D ' A r c y , 1 90
s e x u a l i t y , 2 0 9 - 2 14
T im e , A x e l, 23704
S h a m m a s , A n t o n , 200, 202
t i l i n g , 1 72
S h a n g o , 90 —9 3 , 1 7 5
T o g o , 124
Shaw, C a ro ly n M artin, 2 0 9 -2 10 , 23706
to u r is m , 3 4 , 2 1 7 - 2 1 8
S i e r p i n s k i g a s k e t (o r t r i a n g l e ) , 1 1 3 , 1 1 5 ,
triangular num bers, 8 6 - 8 7 ,
t r ib e , 4 0, 1 8 9 , 203
218- 219
tr ic k st er , 9 9 . i t 6, 1 3 7 , 1 7 4 , 1 7 5 , 182 , 2 1 6
S i m s , j o h n , 2 22
s in u s o i d a l w a v e s , 1 4 1 - 3 4 2
t r i g o n o m e t r y , 68
s la v e s , 108 , 1 2 2 , 200, 235 114 , 2 3 7 0 5
S o l o m o n o f f , R a y , 15 3
T r i n h , M i n - h a , 3 2~3 3
T s w a n a , 200
S o n g l i a i , 3 1 - 3 2 , 1 95
T u r i n g , A l a n , 2 1 3 - 2 14
S o t h o , 200
Turing m achine,
s o ul , 3 3 - 3 4 , 1 2 4 , 1 26
tw ins, 8 9 -9 0 , 1 8 1 - 1 8 2
S ou th A frica,
t 5,
, 0 '^
5 7 — 1 5 9 , 238112
t
184 , 200
S o u t h P a c i f i c , 3 9 , 4 7 - 4 8 , 186
S o w , F a t o u , 18 3
spectru m , 5 - 6 , 49, 5 1 - 5 2 . 56, 1 7 2 - 1 7 3 , 176,
231-234
S p i l l e r s , H o r c e n s e , 194
spiral, 2 3 - 2 4 , 29 , 3 1 , 4 5 , 4 7 - 4 8 , 7 6 - 7 9 , 8 1 ,
8 6, 1 0 4 - 1 0 5 , 1 0 7 - 1 0 8 , i i 2 , 1 2 9 - 1 3 0 , 148,
1 62 , 364, 2 j o , 2 1 6 , 22 4 , 226, 23808,
242113
sp i ri t, 4, 28, 3 1 , 8 9 - 9 0 , 1 1 3 , 1 1 9 , 1 2 1 , 1 2 4 ,
1 2 6 , 1 2 7 , 1 2 9 , 1 3 1 , 1 4 1 , 148 , 1 74 , 1 7 5 ,
1 8 6 , 18 8 , 1 9 3 , 194, 200, 2 0 4, 2 3 7 0 7
U l a m , S t a n i s l a w , 102
V a n W y k , G a r y , 200
v i d e o , 99 , 2 2 6 - 2 2 7 , 2 2 9 , 2 4 2 0 3
v i r t u a l c o n s t 1 net i o n , 2 1 , 29 , 1 83— 1 8 4 , 2 1 3 ,
230 , 235115
v o d u n , 9 0 - 9 3 , 9 4 - 9 5 , 1 4 1 - 1 4 3 , 1 44 , 1 6 6 , 170,
■74. ' 7 5 1
t 83,
190, 1 94 , 2 1 6 , 2 3 8 0 1 3 ,
24004
v o n N e u m a n n , J o h n , 1 0 1 - 1 0 2 , 108
v o o c l o o . See v o d u n
voting, 1 6 4 - 1 6 5 , 229 -230
S p i v a k , G a y a t r i , 184
W a sh b u r n , D orothy, 48, 187
s q u a r e ro o t , 205
W e s t , C o r n e l , 194
s t a t e , 3 9 - 4 0 , 5 1 , 189 , 2 3 6 m
w h i t e n o i s e , 1 5 4 — 1 5 5 , 1 5 8 - 1 6 1 , 1 7 3 - 1 7 4 , 228,
s ta ti s t ic s , 18, 2 4 1 0 9
sta tus , 2 6 - 2 9 , 5 5 , 68, 2 3 5 6 5
2 39n 7
W i e n e r , N o r h p t t , 21 4
s t o c h a s t i c v a r i a t i o n , 9 3 , 24 111 9
W o l f r a m , S t e p h e n , 1 0 6 , 1 5 s . 1 58
S t o l l e r , Pa u l, 3 1 , 1 95
S t o n e w o r k , 29 , 1 0 3 , 1 1 3 , 1 3 5 - 1 3 7 . 1 8 5 , 1 96 ,
2 10
s t o o ls , 5 5 - 5 6
s tr u c t u r a l i s m , i 8 t , 188
S u d a n ,8 1,1 3 5
s y m b o l s , 6, 7, 8, 20, 24, 3 4 , 4 2, 4 3, 5 5 , 7 1 ,
7 7 —7 8 , 9 3 - i o t , 1 0 8 - 1 0 9 , 120, 1 2 6 —1 2 8 ,
W o l o f , 162
w om b, 34, 133, 212
w o m e n , 24 , 3 2 - 3 4 . 9 0 , 1 2 4 , 1 9 5 , 2 0 0 , 204,
2 1 2 - 2 1 3 , 2 2 2 - 2 2 7> 24 ° n 5 , 2 4 2 0 4
Y o r u h a . 8 1 . 8 2 , 1 1 2 , 1 1 3 , 1 1 8 , 1 7 4 . 1 8 3 , 190.
1 96 , 240114, 241117
1 3 1 , 139, 1 4 5 , 1 4 7 , 1 5 1 - 1 5 2 , 1 5 6 - 1 5 8 ,
Z a i r e . See D e m o c r a t i c R e p u b l i c o f C o n g o
1 6 4 ,17 9 ,18 1-18 2 ,18 6 ,18 8 ,19 2 -19 4 ,
Z a m b i a , 8, 26 , 2 2 0 - 2 2 2
1 96 , 2 0 8 , 2 1 1 , 2 4 0 0 1 , 2 4 2 n3
Z e n o o f E l ea , 2 0 3 - 2 0 5
sym m etry, 7 , 3 1 , 4 2 - 4 3 , 4 5 - 4 7 , 79, ' ' 3 , ' '8 ,
1 8 6 - 1 8 7 , ' 9 0 , 1 9 7 , 2 2 2 , 236113
S y r i a , 89
Z h a h o r i n s k y r e a c t i o n , 1 0 4 , 162
Zim babw e,
Z u l u , 222
io o ,
196,200