the fractal geometry of experience - stephen shore · 2019-04-21 · ple fractal formula. fractal...
TRANSCRIPT
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.