Stephen Shore

Benoit Mandelbrot was a Distin­

guished Scientist lecturer at Bard

on September 11, 1993. He is best

known as the founder of fracta l

geometry. H e was an / BM Fellow

at IBM's Thomas}. Watson

Research Cen ter and Abraham

Robinson Professor of Mathematics

at Yale University. H e has

received numerous awards and

prizes including the 1985 F.

Bernard Medal for Meritorious

SCr"vice to Science, granted by the

National Academy of Sciences and

Columbia University; a Distin­

guished Service A 'ward for Ou.t­

standing Achievement from the

California Institutl' of Technology;

and recently he was the recipient of

the 1993 IVol[ Prize in Physics.

Several years ago, on a sere North

African Ja nua ry afternoon, I was

travel in g t o the pyramids at

Sakkara no rth of Cairo. Climbing

out of the fenile Ni le Valley, head­

ing toward th e desert, I passed

through a t ransitional zone1 a mile­

wide band of tall, stately palms ris­

ing fro m the desert fl oor. Throu gh

these palms was a broad bea ten

path. Slowly advancing up the path

were a herd of water buffalo, with

One buffalo in the lead . Astride

him was a yo un g bo y, wearin g

The Fractal G eometry of Exp erience

only a cloth around his middle, in

his hand a switch . \'V'ith a gentl e

rh ythmic motion he sw un g the

swi tch , alternately ta pping t he

sides of the bu ffalo's hump. Th.is

scene sent an electric shock

th rough my system. T o the finest

detail, it was one that had not

chan ged in four thou sa nd years .

This seemed not simply a vision of antiqui ty, but a vision from antiq­

uity. It was like a physical projec­

tion of somet hi_ng buri ed deep in

my unconsc io us. I had much the

same reaction when [ first encoun­

{Cred [he Mandclbrot Set.

I n t he 1970s at IB M's Watson

Research Cen ter in northern West­

chester County, Benoit Mandel ­

brot developed a new geometry he

called "fractal geometry." It is not

a geometry of idealized forms, of

sq uares, cones, and spheres. It is a

geo metry that describes many of

the bas ic com p lex structures of

nature: the branching of a tree or

of our circu latory system; the

crack in g of cit y pavemen t as it

undergoes the stress of weather,

lightl and wear; the bi llow ing of

cumulus clouds over the plains; the

jaggedness of the coas t of Maine;

the eros io n of the Grand Canyon.

I t is the geometry o f the forms of

13

g rowt h and decay, deal in g with

stru ctures that progress to finer

and finer layers of detail and

yet maintain a similar form. \Xlhen

viewing the Grand Canyon on any

scale-from a satell ite or a high­

fl ying p lane, a hot air balloon or

a seco nd -s tory win d ow at the

Canyon Lodge, from our eye level

o r from the vantage point of a

ch ild on hands and knees examin­

ing a fissure in the baked eanh at

the canyon's rim-we see the same

kinds of forms, the same rugged­

ness, the same degree of complex­

ity. Thi s quality , "scaling," is a

central featu re of both nature and

of fractal geometry . It is the pat­

terns of ever -finer b ranching, of

eddi es within eddies, of organ ic

fragmentation, patt erns that fi ll

our natural world, that a re the

stuff of fractal geometry.

\'V'hil e li ving in Pari s in the 1 950s,

Benoit Mandelb ro t became inter­

ested in the patterns of rh e fre ­

quency of word use. H e had immi­

grated to Fran ce from his native

Po land in the 1930s, fl eei ng th e

H o locau st with hi s family. An

uncle who was a famous mathe­

matician lived and taugh t in Paris

and encouraged h im ill pursuing

mathematics. Th:ll uncle's interest

in the theoretical was, for Mandcl­

brat, ba lanced by his father's love

of the practical. It was the combi­

na tion o f the t heo ret ical and the

practical that fo rmed the basis for

Mandelbrot's in vest iga tions. H e

wan t ed to apply mathematical

ana lys is to th e world he saw

around him. And so, he found

himself, almost by chance, drawn

to the study of word frequenci es.

There was, as he would put it, a

smell about it that intrigued him.

H e recogn ized thi s sme ll yea rs

late r when he was stud ying eco­

no mics.

His first interest in economics was

income distribution. He then stud­

ied com modi t y p ri ces, finding

himself attracted to ph enomena

that we re not described by bell

curves (in which the greatest fre­

quency of occurrences clusters

arou nd the average with diminish­

ing frequency as one moves away

from th e average). H e began to

reali ze that what drew him to these

fi elds of stud y, what was behind

the smell tha t attracted him, was a

common pattern and this was the

pattern of scaling. To demonstrate

how scal ing applies to commodity

prices, take, for example, the mar­

ket price of cotton . A graph o f

Erosion, 1956

14

hourly price fluctuations would

look the same as a graph of daily

price fluctuation s, which, in turo,

would look the same as a graph of

weekly pri ce flu ctuations, which,

in turn, would look the same as a

graph of monthl y price fluctua ­

tions. This strucrural self-similarity

on different scales is scaling. Man­

delbrot knew that this was signifi­

cant. What follows from thi s

understanding? he asked himself.

He explored other disciplines

including fluid dynamics and

hydrology and kept running into

self-similarity-scali ng.

Mandclbrot had always th ought

visually. He tells a story of once

having a geometry professor who

used a textbook witham illustra­

tions. The professor sa id that

images lied; that a circle was best

described as x2+y2=-1. For Mandel­

brat, a circle was described by that

equation and by: O. They are dif­

feren t aspects of the same real ity.

Underlying his analytic percep­

tions were mental images, which

he felt helped unlock his intuition.

He would have a visual under­

standing of an aspect of the world

and then, unlike the artist for

whom visual understandings arc

communicated visually , he would

work to devise a formula that

would delineate the phenomenon,

turning mental im ages into num­

bers. With the advent of computer

grap hi cs in t he lat e 1960s, the

numbers were turned back into

images. The comp uter grap hi cs

would, in turn, refresh his intu­

ition, to use Mandelbrot's words.

Working with graphics paved the

way to fractal geometry. This same

interplay of intuition, analysis, and

computer graphics led years later,

in the late _1970s, to the discovery

of the Mandelbrot Set.

Mandelbror was working with the

formulae of two early [wentieth­

century French mathematicians,

Pierre Fatou and Gaston Julia. Their

work involved iterated formulae, in

whjch the result of an operation is

fed back into the formula, whose

operation in turn is repeated, and so

on. Their formulae are plotted on

the complex plane-a rnathernatiCJ.I

plane whose two axes are th e real

numbers and the imaginary num­

bers.1 The work of Fatou and Julia

couldn't be taken furth er until the

development of the compu ter,

because the com pu ter enables

almost endl ess iteration. It was

while playing with what arc called

Jul ia Sets and searching for a basic

15

formula of which the Julia Sets are

only special instances tbat M-andcl­

brot discovered the set that bears his

name.

The Mandelbror Set is a visual rep­

resentation of a boundary within a

specifi c mathematical mapping. It is a mathematical object of beaury,

power, and, literally, infinite com­

plexity. The set begins with an

island like shape. At one end the

twO sides curve in toward the cen­

tral axis. On the opposite end [here

is a spire projecting out along the

central axis. Floating on the spire is

a much smaller version of the orig­

inal island. Shooting like so lar

flares from the perimeter of the big

island are whorls and sea horse

tails, paisleys and lightning bolts.

No matter how much any segment

of rhe perimeter is magnified-a

hundred times, a thousand times, a

million times-similar forms con­

tinue to appear, s imilar but not

identical. Every now and then a

shape like the original island reap­

pears and on its edge th e process

begins all over again.

What is especially compelling for

me abo U[ the Mandelbrot Set is a

quality of recognition, th e sense

that I've seen it before. Mandelbrot

reports thar many people have had

this same experience. It is far less

surprising that fractals in general

and the phenomenon of scaling

should spark recognition, they so

fill the world in which our species

has evolved and in which we, as

individuals, have grown up. Very

few shapes 111 nature look

Eucl idean. It would take a compli .

cated Euclidean formu la to

describe a complex shape such as a

lun ar mountain. Yet this same

shape could be described by a sim­

ple fractal formula. Fractal geome­

try brings the discipl ine of geome­

try back to the meaning of it s

Greek root, geometria, to measure

the world. Fractal geometry did

not spri ng from -abstract thought,

but from observation of the real

world. This very fact made it

unappealing at first to mathe-mati·

cians, but engaging to J lay public.

Mandelbrot believes that people

love mathematical structure even if

they don't realize they do. This is

another way of sayi ng that there is

a basic analytic componem to om

nature. We need (Q find order in

the wo rld, .in some rudimentary

way, to be able to function in it.

On the simplest le vel this may

entail the conceptual ization of a

fact: my dog learning the wo rd

Image 1:

The Mandelbrot Set, shown 111 Its entl rety, IS

contained in th is irn;'lge, which is rough ly two

inches on each side. Bound between -2 .25 and

0.75 o n the x axis, it ex tends on ly 1.5 units

above and below on the y axis.

Image 2:

As the imagc is magnified by a factOr of tcn,

we sec d crai l cmcrging from the needle ­

shaped forms ncar the center of the set. The

seem in gly circu lar shapes on each side arc

repeated in an infinitely decreasing pattern.

16

Image 3: Another zoom by ten allows closer inspection

of a circle. Rather than becoming smoother,

more detail is revealed as the set is examined

wirh grcate r magnification. If rhe origin:tl

image were shown at this scale, it would bt'

nearly seventeen feet on each side.

Image 4: Here, a radial pattern is displayed. Spiraling in infinite deta il tow.1fds the e(,!Hcr, this image

exhibits a co mplex it y onl~' hinted at in the original po rtra it of the M.lndclbrol Sct. whose

sides wou ld now be more than one-half the

length of a football fi eld.

Image 5:

After another 700 m by a factor of tell, lIu:

sym lll l'try of the previous image is rcfleered in

two snull er po ni o ns of the sct. Although

th ese .1ft' min i:l.lurc ve rsio ns of th e paltern,

there is no b rea kdow n of detail as we con ­

t in ue to look more closely at the set.

17

Image 6: TIH.' or iginal image: h:l.') now become mon..'

t han tlllTC miles on :l side. If displayed at the

.')ca ic of the fir.')[ image, this image wou ld be

microscopic. Ilow('\,cr. the level o f detail 11.15

nut diminished, ;1nd, in i;l e t, a rcp lic;l of the

origin:t l M,lIld elbro l SCI Ins 'lppcarcd tint

cont.lin s ,\11 t he detail of it s :tllcesto r. Th l'

:tppca r:lll (e of this mini ature illustrates the

se lf-simiLtrit y of the Mandclbrot Set, which

will never ((,,1se to provide infinitc v:lri:1.tion :It

:lny Ill:t!-:nific,nion.

" road " (as in "Get out of t he

road!") an d recognizing each

instance of a road he encounters.

On a more profound le ve l thi s

o rderin g reflects ou r world view,

be it simplistic and rigid or com­

plex and flu id . And it is the ana­

lytic pan of us that is stim ulated

by a visual rep resentation of a geo­

metrical st ructure that so we ll cor­

relates with our experience of the

world.

Ph eno men a so pervasive in the

world as fractals and scaling must

scat themselves at the core of the

struc[Ure of o ur unders tandi ng.

Artists through the ages have con­

sciously o r intuitively understood

the visual aspect a nd pow e r o f

these phenomena. From medieval

illum inations and Pers ian mini a­

(tI res ro the app lication of paint on

a Cezan ne la ndscape, arti sts have

depicted fractal fo rms. Art ists were

attracted to these forms no t sim ply

because they describe natu re, but

beca ll se th ey" mean somethin g."

For the Chinese landscape painter,

fractals descr ibed the effe rvescence

of a life force in mountains or the

poignancy o f a lone pi ne. For the

builders of the Gothic cathedrals,

frac tals were th e language of the

complex ity in uni ty of God 's cre-

- - ----- --- - --- -- -

Dedicatory page of t.he H oughton Shah-N emah (16th century)

18

a rio n. Fo r Dur er , in " Kni ght.

D eath and the Devi l," the frac tal

cliffs and exposed roots loom i n~

ove r the sce ne al luded to tim e,

death, and decay. But they are nOl

simply symbols, they are ana logs

of deeper meaning and experience.

They exist for what they arc and <11

the same time resonate on psycho­

log ica l, emo tional, and spiritll;l]

levels.

Th ere arc also so me artist ic crc ­

at ions that bear a similarity to the

Ma nd elbro t Se t in whole or in

part. This is harder to explai n rhan

an art ist 's attract ion [Q fractals ,

since the Mandclbrot Se t is a math­

ematical o bj ect and unlike frac t:ds

doesn't exis t in the world of Oll r

ex per ience .1 Perhaps the most

obvious example is the pais ley. Its

compl ex curving a nd sw irling

forms, their densi ty and repetition

and scaling propert ies arc all strik­

ingly similar to the ac tivity deep

withi n the perimeter of the set.

Another exa mple is the architt'v

tu ral dome topped by a sp ire .

Instances are found in many cul­

tures . In most cases rhe sp ire is

inter rupted by a small sphere o r

other shape much as the spire of

the Mandelb rot Set is interrupted

by a small repetition of the island.

T he sc hcmatic arc hiteclUral rOO tS

of this Il13Y be found in Stupa # I at

Sanc hi , 'nelia (3rd to I st century

B.C. ) . Co ns ider S1. Ba s il 's in

MoscoW w ith its on ion domes

( [(,til ce ntury). Here the bu lgin g

sidcs and int errupted spires arc

clcart y l"cmini scent of the set. To

th is is add ed the complex pattern­

ing on the domes' surfaces. Com­

pl<.'x patt ernin g o n spired domes

also appears t hro ugho ut Is lamic

;Hchitect ure. It is as thoug h th e

promine nces and w horl s of th e

M:lIldclbro t Set arc co llapsed into

arabesques on the domes' surfaces.

Fr l'cd from th e phy sica l con ­

st raims of architecture, Islamic cal­

liJ;rap hers and arti sts illu mina ted

manu sc rip t s an d Ko ra ns wi t h

spi red med all ion s a nd rose ttes

comp let e wi th co mple x ed ges .

"fakc the dedicato ry page of the

Ii o ughton Shah -Ne mah ( 16th

(emury) reproduced on the oppo­

s ite page. H ere is t he delin eated

(emral axis, the spires interruplCd

by small rosettes, the prominences

fro m the edges of both large and

small rosettes, the ri chl y intrica te

b yer in g of pattern on different

scales. Each small section presents

a world dense with structure intO

which o ne ca n enter. Th e upper

Masjid-i-Shaykh, Isfahan

car touc he reads: " In Il is Nallle,

th e Most Prai sed a nd Mos t

Exal ted! " That these im ages and

dom es ap pear repeatcd ly in a

sac red co ntext shoul d u nderlin e

rhe centrality of their meanin g to

their cu ltures.

19

For the artist, fracta ls do not sim­

ply represent rhe look of nature,

the}' are an expression, a manifes­

ta ti on, of a force d eep wi thin

n:nure, and, by extension, within

o urse lv es, from our circulato ry

sys tems and bronchial passages to

o ur nervous systems and brai ns.

And to our minds. The image of

the Mand elbrot Set was a source

fro m w hich so me a rti s t s and

craftsmen of different cul tures and

per iod s d rew in the crea t ion of

the ir sacred bui ldings and pictures.

We arc attrac ted to it and find it

strangely fam ili ar because it is atl

image that we hold in our uncon­

sc iolls minds.

Stephen Shore is the chairman of

the Phologl'aphy Department at

Bard. I-Ie has had one-man sho .... :s

at the Metropolitan Museum of

Art, the Mu seum of Modem Art in

New Yo rk , and the A rt Institute of

Chicago.

Notes

I . All inl.!gin,lry numhl'l' is the squ.He root of .\ nq!"uive re,l[ number, c.g. ,<- .J. = 2/.

1. It dOl'S bCJr J rclati~![IShip to somcthing (,l lled the hifu re~tion di,lgram. T he bifurca­lion (li,lgl',lI11 i llu~lr.lIes the Ol1sel of eh,llls ill

,I S~'lcm and ll.l~ applic,nions in biology. l'IN'lronies. op ti cs. c hcmi st ry, ,lIl d othl'r fid ,ls. The bifurclIiun (Iiagram is a slice on !Ill.' n·,t l pl.l11C through !Ill' !\hndclbrot Set. \'</fll'r,· duos is shown u n t he bifurc.H ioll d i.l ;;r.\lI1, t he .\l.in.Jdbro t SCI dis pl .l~· l> it­pll.lnt'I~IIl.lh"ric,\[ forms.