Fractals and an Art for the Sake of Science
Author(s): Benoit B. Mandelbrot
Source: Leonardo. Supplemental Issue, Vol. 2, Computer Art in Context: SIGGRAPH '89 Art
Show Catalog (1989), pp. 21-24
Published by: The MIT Press
Stable URL: http://www.jstor.org/stable/1557938
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Fractals

and

an

Art

for

Sake

of

Sciencethe

Benoit B. Mandelbrot

ABSTRACTAT
he artist and the artisan are often hard to tellapart. For example, objects that were in principle meant tobe utilitarian-be it folk architecture, religious imagery, or
drawings and photographs of flowers, birds or water eddies-often end up being regarded as genuine works of art.
It may become hard to distinguish them from works in whichscience was used almost as an excuse for artistic creativity.
Thus, art faces us with broad possibilities. We are presented
with innumerable works of art for the sake of commerce: objects have been commissioned under precise specifications
to be useful-to decorate, to educate, to flatter, to entertain,
to impress or to persuade. We are also presented with a few
works created strictly as art for art's sake. And we also knowof many possibilities that lie, so to say, in-between.
Does mathematics relate in any way to these familiar
forms of plastic art? The classic shapes of geometry are
hailed for their conceptual beauty, but they seem mostly toreside in the imagination of skilled practitioners. Although
the popular poet Edna Saint Vincent Millayproclaimed that
"Euclid Gazed on Beauty Bare" and although Euclid's
geometry was of central importance to painters of the Italian Renaissance during the brief period when perspective
was being 'invented', to the eye of those unschooled inmathematics, the beauty of Euclid's geometry is bare anddry to a fault. At the least it lacks scope and visual variety
when compared with those excesses of either nature or thefine arts, which everyone seems tempted to call 'baroque'or 'organic'.
Today, however, there is more to geometry than Euclid.
During the 1970s it was my privilege to conceive and develop
fractal geometry [1], a body of thoughts, formulas and pictures that may be called either a new geometryof natureor a
new geometriclanguage.And the reason why it is worth discussing here is that I have discovered that, most surprisingly
and without any prodding, this new geometric language has
given rise to a new form of art. I propose here to make a few
disjointed comments on its account. Many readers arebound to be familiar with fractal art, and the volume inwhich this paper appears may well contain some new examples from the 1989 SIGGRAPHshow; nevertheless, closefamiliarity with the subject is not expected from the reader.
The bulk of fractal art has not been commissioned for any
commercial purpose, even though all the early work wasdone at IBM. And it has not necessarily been touched byesthetic sensibility. Therefore, we shall argue that fractal
geometry appears to have created a new category of art, next
to art for art's sake and art for the sake of commerce: art forthe sake of science (and of mathematics).
Benoit B. Mandelbrot (mathematician, research fellow), Physics Department, IBM TJ.
Watson Research Center, Yorktown Heights, NY, 10598, U.S.A.; Mathematics
Department, Yale Universitv, New Haven CT 06520, U.S.A.

? 1989 ISAST
Pergamon Press plc. Printedin Japan.
0024-094X/89 $3.00+0.00

Fractal art for the sake of science is indissolubly based on
the use of computers. It could
not possibly have arisen before
the hardware was ready and the

newformofartredefines
theboundary
between'invention'
and'discovery',as understoodinthesciences,and'creativity',as
understoodintheplasticarts.Can
beperceived
puregeometrybythe'maninthestreet'as beautiful?
Tobemorespecific,
cana shapethat
is definedora
bya simpleequationbesimpleruleofconstruction
perceived
bypeopleotherthan
as having
aesthetic
geometers
as beingatleast
value-namely,
decorative-or
surprisingly
perhapsevenas beinga workofart?
Whenthegeometricshapeis a fracis yes.Evenwhen
tal,theanswer
fractals
aretaken'raw',
theyareattractive.
to
Theylendthemselves
thatis surpris'painting
bynumbers'
eveninthehandsof
inglyeffective,
therankamateur.
Andthetrueartist's
findsthemasensibility
novelandattractive
support.

software was being developed;that is, before the decade of theseventies. What a profoundirony that this new geometry,
which everyone seems spontaneously to describe as 'baroque' and 'organic', should
owe its birth to an unexpected
but profound new match between those two symbols of theinhuman, the dry, and the technical: namely, between mathematics and the computer.
Before we describe the peculiarities of fractal geometry inmore detail, it is good, for thesake of contrast, to commenton examples of similar matches that have arisen in areas
such as the study of water eddies and wakes. In these cases,
the input in terms of reasoning and programs is extremelycomplicated, perhaps more complicated even than the output. In fact, one may argue that, overall, complication does
not increase but changes over from being purely conceptual
to being partly visual, a change that is important practicallyand interesting conceptually. Fractal geometry, however,gives us something quite different. In fractal geometry, theinputs are typically so extraordinarily simple as to look positively simple-minded. The outputs, to the contrary, can bespectacularly complex. Again, while a contribution from anartistic sensibility is not necessary, it is well rewarded.
Let us hasten to raise a question. Since the inputs are so
simple, why is it that fractal art failed to appear earlier andin more traditional ways?The answer lies in a 'Catch 22' situation. To draw the simplest fractal picture 'by hand' would
have been feasible in principle, but would have required
many person-years and would have been ridiculously expensive. Consequently, no one would have considered undertaking this task without having a fair advance knowledge ofthe result; yet the result could not even be suspected until
one actually had performed the task.And a sure wayof being
discouraged from ever undertaking it would have been to
begin with any one of the various definitions of fractals. Hereis one informal definition I often use:
Fractalsare geometricshapes that are equallycomplexin theirdetailsas in theiroverallform.Thatis, if a pieceof afractal is suitablymagnifiedto becomeof thesamesizeas thewhole,it shouldlook

LEONARDO, ComputerArtin ContextSupplementalIssue.pp. 21-24, 1989

21

_.

-:.. ;d-Fig.

t-,j*

*

'-

1. The two
faces of fractal
art. (above) M.
R. Laff and A
V. Norton, FractalDragon, 1982.
(below) R. F.
Von, Fractal
Planeris 1982.
This Fractal
Dragonand

FractalP/anetriseb_eaIofaoe

may be the best
known of all fractals, since they
appear on thetwo halves of thejacket of The

FractalGeomety
ofNature(Ref.
[4]). Their being
set as neighbors
is meant to illustrate the basic
fact that fractal
art straddles theboundary between art that is,
and is not, representational.

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like the whole, either exactly, orperhaps only
after a slight limited deformation.

Are we not right in the middle ofdry geometric principles? An artist
could expect nothing from fractals defined in this fashion, hence no one attempted to draw them carefully. The
few old fractals that had been known
under various names (and depicted
for at least a century) are also the least
interesting esthetically because one
glance shows that everything about
them has been obviously put in byhand; they are orderly to excess. These
images, however, began to grow in

22

of fractals. This happened with my firstbook in 1975 [2].
What were the needs that led me tosingle out a few of these monsters, calling them fractals,to add some of theirclose or distant kin, and then to build
a geometric language around them?
The original need happens to havebeen purely utilitarian. That links exist between usefulness and beauty is, ofcourse, well known. What we call the
beauty of a flower attracts the insects

Mandelbrot,Fractals and an Art for the Sake of Sciencethat will gather and spread its pollen.
Thus the beauty of a flower is usefuleven indispensable-to the survival of
its species. Similarly, it was the attractiveness of the fractal images that firstbrought them to the attention of manycolleagues and then of a wide world.
Let me tell how this started happening. In the 1960s, the basic idea of thetheory of fractals was already present
in my mind, having been devised to
study such phenomena as the erratic
behavior of stock prices, turbulence influids, the persistence of the discharges of the Nile, and the clusteringof galaxies, which manifests itself withthe presence of great intergalactic
empty spaces. But society seemed tothink that my theories, their mathematical techniques and their goals
were strange, as opposed to simply
new. As a result, my attempts to makemy thoughts accepted as soundseemed always to encounter a wall of
hostility that words and formulas
failed to circumvent.
One day it became necessary to convince Walter Langbein, the editor of awater resources journal, to accept apaper I had co-authored. He was askilled and able scientist, but not one
to gamble on wild, unproven ideas. I
decided to resort to a tactical detour,
presenting him with two images in thehope that Langbein would find it impossible to distinguish between reality
and 'forgeries' that were based solely
upon an early fractal theory. If thiswere to happen, he would no longer
be able to view this theory as irrelevant
to his work, he could not and would
not reject our paper outright, and hemight eventually accept fractals. This
is indeed what happened: the detour
through the eye turned out to be
successful, and its offspring grew beyond expectation.
What happened next to fractal art
as it evolved brings us to the traditional
dichotomy between representationaland nonrepresentational art. In thewell-recognized forms of art, this dichotomy no longer seems so strongly
etched, and fractal art straddles it very
comfortably. The earliest explicit usesof fractals gave me the privilege ofbeing the first person to tackle in a
new way some problems that must be
among the oldest that humanity had
asked itself: how to obtain 'figures'that represent the shapes of mountains, clouds and rivers? It turns out
that, when the representation of nature by fractal is perceived as successful, it also tends to be perceived asbeautiful. Unquestionably, the fractal'forgeries' of mountains and clouds
are examples of representational art.
The skeptic will immediately raise
another question. Is it not true that thecolors used to render these mountains
and clouds are chosen by rules thathave nothing to do with any geometry?
If this is so, these 'forgeries' are notpurely fractal. What precise role, then,does color play in the acceptance ofwhat you call 'fractal art'? This may
sound like a very strong objection, but
in fact it is easy to answer.
First of all, the question did not andcould not arise with the first fractal
pictures, simply because they were inblack and white. I might also add thatin many cases this supposedly obsolete
palette is the one I continue to favor.
When the use of color did arise,Richard Voss and I worried that it
might detract from our primary concern with the geometry. Thus, initially
he decided to color his art simply, as
in the London TimesWorldAtlas, but inlandscapes viewed from an angle instead of the zenith, this proved to bevisually unacceptable. However, we
continued to avoid excessive artistic
intervention, and Voss kept his esthetic urges under the tightest of control.
This, in my opinion, helped fractal
geometry make its intended point.
Once that point was achieved, how-ever, a completely different situation
was created in which reserve was nolonger an overriding obligation. In
the recent crop of pictures by F. Kenton Musgrave, 'SIGGRAPHtricks' areallowed, but one absoluteconstraint remains. Every surface that is depictedmust be a fractal surface, and all commands that are used to improve therendering must be global commands.
To 'fix' an unsatisfactory corner of apiece by a local patch is not permitted.
Many computer artists would find thisconstraint to be quixotic, but it is essential if fractal art is to preserve its
integrity.
While dealing with fractals intended as forgeries of nature, we
found that cases soon began to multiply in which this intent failed. The result, however, remained just as beautiful, and occasionally even more so.
Happy errors! Furthermore, a person
fascinated by shapes could not avoid
forgetting on occasion the original
goals of the fractal geometry of nature
and would play on with fractal algorithms just to find where they mightlead. Thus as a fractal model of mountains is deformed by changing thevalues given to one or a few numbersthat characterize the fractal's form,
the picture becomes less and less 'realistic' as a mountain and gradually becomes altogether 'surreal'.

Even more striking surrealism prevails within the second major aspect offractal geometry. Fractal 'dragons', ofwhich the 'oldest' is reproduced here
(see Fig. 1)-and of which millions
seem to have been drawn since-havenever been meant to represent anything in nature. Their intended usefulness concerned mathematics, since
they helped me investigate a process
called the 'dynamics of iteration'.
Early in the century, the mathematicians Pierre Fatou and Gaston Julia
had found that this process presents
a deep and surprisingly intellectual
challenge. Then for 60 years hardly
anyone touched the problem becauseeven the most brilliant mathematicians, when working alone with theproverbial combination of pencil-andpaper and mental images, found thatits study had become too complicatedto be managed. My fresh attack on iteration could rely upon the help of thecomputer, and it was effective: the newmathematical order was spectacular.
For the purposes of this discussion,
this does not matter at all, of course;
but a side result does matter a great
deal: the resulting balanced co-existence of order and chaos was foundalmost invariably to be beautiful.
As in the case of the fractal mountains, the new iteration-generated
fractals were already perceived to be

Fig. 2. Fractal landscapes. These illustrations
exemplify three of many successive stages in thedevelopment of fractal landscapes. One may callthese stages, respectively, 'archaic', 'classic' and'romantic'.
The archaic wire model illustration (above) was
done by S. W. Handelman (1974), who was then myprogrammer at IBM at a time when our work was
dominated by the extreme crudeness of the tools.
The classic illustration (below) is by Richard F.
Voss of IBM (1985). It is an improved form of one ina series he prepared for my book of 1982 [4]. By
then, the computer tools had become less obtrusive,
and allowing fancy to take over was a genuine temptation. But fantasy had to be resisted because these pictures were primarily tools of scientific discourse. The
wonder is that these extreme constraints should haveallowed the emergence of Voss's masterpieces of subdued elegance.
The romantic illustration (see back cover) is by mypresent Yale student, F. Kenton Musgrave, andmyself (1989). Today, wire models that are better
than the archaic one take 1 second to be computedand drawn on a workstation, and the number of available colors has changed from being unmanageably
small to being unmanageably large. The most innovative use of fractals now is to serve as support for an
artist's inspiration and skill.

-It

?

?.

?.

c

'i

_??J
:?'3

r
'

Mandelbrot, Fractals and an Art for the Sake of Science23'Fig. 3. Two fragments of theMandelbrot set.
_|f~' ^The Mandelbrot~T- 'set is explained_- *?- ^in Refs [4], [7]and [8]. The firstfragment (above;'
R F. Voss, 1988)..R * '
was selected so
.-^
^~
'
_ '^
~ as to include,near its center, asmall replica of
the whole, with
its obvious symmetries and repe'': m'^
titions, and evento incltudeadditional symmetries that are notpresent in theseetset. Thist
wcetwholefragment, there' ......mpute"fore, leans too
far towards orderliness. Thes es,cond fragmenti(below",B. B.
:
Mandelbrot),iwhich is from aEo ogeneralized, notthe 'ordinary',
Mandelbrot set,
was selected toprovide contast.M since it is devoidof obvious: symmetries.

beautiful in their original black andwhite. More precisely, the output of
my work was a collection of numbersthat in the early stages had to be reduced to two possibilities, to be represented by black and white. After color
became involved, these numbers werefirst represented
by colors chosen
more or less at random by color-blind
hackers. (An awful case of painting by
Yet even these fractals
numbers!)
were, in a way, beautiful. But when thecoloring was placed in the hands of atrue artist, we began to see true
wonders.
Our skeptical critic will come backat this point to remind us that fractals

24/ s~ ~

should share the credit for this art withboth the computer and the programmer-artist who frames the object andselects the colors. These last two factors are the ones usually consideredcentral to computer art; hence thecritic's point concerns the significanceof the fractal's additional input. In
some cases (as in one of the illustrations of this paper) fractals' most obvious contribution is an obtrusive symmetry that may in fact be found to bevery objectionable. In other cases,
however, when the symmetry is hidden we see an interplay between
strong order and just enough change
and surprise. My readings on the

Mandelbrot,Fractals and an Art for the Sake of Science

meaning of art suggest that such an interplay is one of the basic prerequisitesof plastic beauty.
To summarize, the altogether new
feature brought in by fractal art is thatthe proper interplay between orderand surprise need not be the result
either of the imitation of nature or ofhuman creativity, and it can result
from something
entirely different.
The source of fractal art resides in therecognition that very simple mathematical formulas that seem completely
barren may in fact be pregnant, so to
speak, with an enormous amount ofgraphic structure. The artist's taste
can only affect the selection of formulas to be rendered, the cropping andthe rendering. Thus, fractal art seems
to fall outside the usual categories
of 'invention', 'discovery' and 'creativity'.
All this seems to have happened
long ago, and today fractal geometry
is so well establishedthat young
people are astonished to find that the'father of fractal geometry' (as I am
delighted to be called) is still alive. But
I hope to live long enough to really understand what has happened.
References

and Notes

1. The first three books on fractals are Refs [2],[3] and [4]. Among later books, I recommend
Refs [7] and [8], which are of higher graphic
quality than mine, though of narrower focus. Of
the approximately 40 books that have by now
been written on fractals, nearly all are for mathematicians and/or physicists, and Refs [4], [7] and[8] are the only books written in English for abroad audience. Ref. [5] belongs here because of
thejournal in which it appeared and because it is
my earliest statement on the issue of fractals andesthetics, and Ref. [6] can be viewed as a commentary on the present paper.
A quandaryand an Apology.It would have been
nice also to recommend works with which I am
less closely associated, but this would be very hard.
Each quote creates one happy person and many
unhappy ones. The time when my close associates
were the only people involved with fractals passed
years ago, and to write a balanced survey is not atask I enjoy.
2. B. B. Mandelbrot, Lesobjetsfractals(Paris:Flammarion, 1975,1984, 1989).
3. B. B. Mandelbrot, Fractals:Form,Chanceand Dimension(San Francisco: W. H. Freeman and Company, 1977).
4. B. B. Mandelbrot, TheFractalGeometryof Nature(New York:W. H. Freeman and Company, 1982).
5. B. B. Mandelbrot, "Scalebound or Scaling
Shapes: A Useful Distinction in the Visual Arts
and the Natural Sciences", Leonardo14, No. 1,4547 (1981).
6. F. K. Musgrave and B. B. Mandelbrot, "Natura
ex Machina," IEF. ComputerGraphicsand Applications9, No. 1, cover and 4-7 (1989).
7. H. -0. Peitgen and P. H. Richter, TheBeautyof
Fractals(New York:Springer, 1986).
8. H. -0. Peitgen and D. Saupe, eds., The ScienceofFractalImages(New York:Springer, 1988).