African Fractals: Modern Computing and Indigenous Design 0813526140, 9780813526140

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— 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

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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

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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.

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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.

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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

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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

.

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.

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♦ ♦ ♦ 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 . ^

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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

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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

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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

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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 ,

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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

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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

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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

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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.)

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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.

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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.

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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

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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 .

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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 ,

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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

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( 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

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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

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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.

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B ■ ■ — =► ■■■ BBB BBBB BBS ■ B B B B B B B fl

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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 .

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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

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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

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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 .

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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 .

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

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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

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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

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Si

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FIGURE

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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

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“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

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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

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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

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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.)

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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.

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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.

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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-

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) 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.

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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.

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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

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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

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This print from West Africa suggests the full spectrum from order to disorder. (From S i e b e r 1972.)

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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

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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 • 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. 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.

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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