Critique and a Science for the Sake of Art: Fractals and the Visual Arts
Author(s): Noel Gray
Source: Leonardo, Vol. 24, No. 3 (1991), pp. 317-320
Published by: The MIT Press
Stable URL: http://www.jstor.org/stable/1575574
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GENERAL

NOTE

andCritiqueof Art:Fractalsa Science

and

the

for
the
VisualSake
Arts

Noel Gray

ABSTRACTJ^ D rawing attention to the Kantian aesthetic
that underpins Benoit Mandelbrot's [1] argument for 'an
art for the sake of science' does not represent a sufficiently
critical engagement of fractal geometry's entry into visual
discourse (an entry made largely transparent by the suggestion that art acts as a handmaiden to science). Merely
pointing to the extent of this aesthetic debt leaves thenecessity for its adoption unexplored. It also leaves fractal
imagery outside the orbit of contemporary visual critique byaffording an unproblematic status to this geometry's intervention. This status is made possible by the fact that atranscendental aesthetic, in its traditional form, displaces
any notion of a critical visual practice by reducing all engagements of the visual to judgments of taste [2].
The necessity of adopting this aesthetic is apparent in theclaims that first emerge in Mandelbrot's remarkable andimpressive book TheFractalGeometryof Nature [3] and later
reappear, though often only by implication, in his Leonardo
article. Indeed, the title of this earlier work indicates thetruth value Mandelbrot ascribes to his discourse. Namely,
the implication is that nature's truth is fractal geometry: thegeometry of chance and change, the geometry of the vernacular, the measuring of the everyday. (Fractal images
have, in part, a descriptive role in the dissemination of thistruth.) The basic concepts that Mandelbrot advances for
fractal geometry, as points of differentiation from traditional geometry, can be broken down into two categories:
those related to the generative concepts that inform hispractice and those connected to the resultant aesthetic andtruth value conclusions drawn from this practice.
GENERATIVE

CONCEPTS

Mandelbrot positions his practice against the backgroundof what he perceives as the inability of traditional geometry
to describe the everyday forms of nature-forms such as
clouds, rivers and lightning. He states that nature's forms
cannot be reduced to classic geometric shapes. Mathematics
also, he maintains, has been differentiated like traditional
geometry, along conceptual and experiential lines, demonstrating an ever-increasing tendency towards esoteric pursuits and thus creating a widening gulf with nature andfailing to address directly what can be seen or felt. Nature,
for Mandelbrot, is not simply a higher complex version of
Euclidean geometry; rather, it operates at a completely
different level of complexity.
It is important here to note the significance of this claimof a different complexity. For what Mandelbrot appears to besaying is that fractal geometry is nature's owngeometry. The
complexity witnessed in fractal geometry mirrors the self-? 1991 ISAST
Pergamon Press pic. Printedin Great Britain.
0024-094X/91 $3.00+0.00

same complexity evident in nature, evident even at the macroRecently,theauthor
argues,proponents
offractal
level of ordinary perception.
geometrymostnotably,
BenoitMandelbrotIn this argument, fractal geohaveeffectedwhattheyclaimis
is
not
thus
anothernovelintervention
metry
intothedissimply
courseofthevisual.
Theauthor
system of static measurement;
examines
therelationshipbetweenrather, it duplicates to all inMandelbrot's
theoryofgeometrytents and purposes the actual
andhistheory
ofaesthetics
andprocess of the generation of nans
thisteetios isnotesthathat
thisintervention
ture's complexity. Unlike theona displacementofdependent
static reductive geometries ofcriticalvisualcontemporary
practhe past that have concentrated
tice,a practicethat,oncerestored,
affords
theopportunity
fora critiqueon fixed states and/or linearof scientificthevistheory
throughfractalsystems,geometry enualregister.
Theauthor
concludes
thatMandelbrot's
gages the verynonlinear dynambysuggestingics that characterize nature itclaimofnovelty
ofintervention
self. This concept, regarding themayhaveatleastonehistoricalprecedent.
preceenimaging of nature's process ofisessential
to
Mandelchange,
brot's claim of differentiation
offractal geometry from earlier
systems of geometry. In short, the generative concepts
informing Mandelbrot's practice are as follows: (1) Fractal
geometry deals with the real world of everyday perception.
(2) Fractal geometry is nature's own geometry. (3) Fractal
geometry imitates nature's process of change.
RESULTANTCLAIMS

Moving from his premises of practice, Mandelbrot's resultant truth claims are, as would be expected, logicallyextended from his generative concepts. For instance, hesays, "Myprevious essays stressed relentlessly the fact thatthe fractal approach is both effective and 'natural'. Not only
should it not be resisted, but one ought to wonder how one
could have gone so long without it" [4].
With this statement, the veracity of the earlier generative
claim, that fractal geometry is nature's own geometry, now
enjoys the status of an undisputed higher truth. Indeed, thisquote suggests not only that fractal geometry's truth is of ahigher order than previous geometric systems by virtue of
its effectiveness and 'naturalness' but also that it is transcendental in character and has merely been awaiting adequate
theorization. In this transcendental register the truth isunproblematic by definition, i.e. it is assumed to be given inNoel Gray (sculptor, graduate student), Box 201, Holme Building, University of Sydney
2006, N.S.W. Australia.
Received 23 July 1990.

LEONARDO, Vol. 24, No. 3, pp. 317-320, 1991317

etched, and fractal art straddles it ver'comfortably. The earliest explicit usesof fractals gave me the privilege ofbeing the first person to tackle in a newway 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 [9].

intuition as a self-evident truth andknown to be so given by recoursetoreason alone [5].
Further, embedding this claim in theauthority of the history of mathematics
and mathematicians, Mandelbrot statesthat "I show that behind their very
wildest creations, and unknown tothem and to several generations of followers, lie worlds of interest to all those
who celebrate Nature by trying to imiEvident in these statements is Mantate it" [6]. This reference to imitation, delbrot's notion of an
unproblematic
coupled with a traditional archaeo- beauty that is assumed to be transsublogical notion of disclosing hidden jective and an a priori truth. Its presencetreasures, leaves little room to doubt in relation to fractals is
undisputed, althe idea of a timeless-truth-to-nature.
unobservedby past mathematithough
Clearly, for Mandelbrot, fractal geom- cians. Thus, Mandelbrot's aesthetic
etry is the practice of this truth. It isemerges as basically Kantian in characworth emphasizing that this notion of ter in that he assumes beauty is an inan unproblematic transcendental truth, herent universal feature of nature andtruth-to-nature, is the pivotal point of isrecognised as such by everyone. As
this practice.
his fractals 'imitate nature', it follows by
Bearing in mind the significance of association that they also will be perthis point, Mandelbrot's aesthetic posi- ceived asuniversally beautiful [10].
tion and the relationship of this aesWhat then is the relationship of thisthetic to the posited truth value of his aesthetic with Mandelbrot's truth claims
geometric practice remain to be ex- concerning fractal geometry? In otheramined [7]. One or two extracts from words,
given his comments that his
Mandelbrot's writings serve to bringpractice is in an essential sense a radical
this relationship into focus. First, in thedeparture from traditional systems of
opening sections of TheFractalGeometry geometry, what then is the significanceof Nature,Mandelbrot states,of attaching a largely conservative aesthetic to his posited radical practice?
This essay brings together a number of
The key to this marriage of the radianalyses in diverse sciences, and it promotes a new mathematical and philocal and the conservative is located in his
sophical synthesis.... Furthermore, it
desire to establish fractal geometry as
reveals a totally new world of plastic
nature's own geometry, the truth ofbeauty....
which has merely been awaiting dis... In addition, fractalgeometry reveals
covery. To make such a claim amounts
thatsomeof the mostausterelyformal
to asserting that fractal geometry itchaptersof mathematicshada hidden
self is transcendental in character-aface: a world of pure plastic beauty
characteristic that is deducible from his
unsuspected till now....
. . . The fractal 'new geometric art'shows surprising kinship to Grand
Masters paintings or Beaux Arts architecture. An obvious reason is that classical visual arts, like fractals, involve
very many scales of length and favour
self-similarity. For all these reasons,and also because it came in through aneffort to imitate Nature in order toguess its laws, it may well be that fractal
art is readily accepted because it is nottruly unfamiliar [8].

In LeonardoMandelbrot states,
What we call the beauty of a flower
attracts the insects that will gather andspread its pollen. Thus the beauty of aflower is useful-even indispensableto the survivalof its species....
... .What happened next to fractal art

asit evolvedbringsus to the traditional
dichotomy between representationaland nonrepresentational art. In thewell-recognized forms of art, this dichotomy no longer seems so strongly

318

generative

claims. It therefore

follows

that in order to avoid a coherency problem at a fundamental level, namely, atranscendental system of change [11]married to, say, a critical visual theoryof the image, Mandelbrot is led into
the marriage of his transcendental geometry to a transcendental aesthetic.
On the other hand, if a critical element were to be introduced into theimages generated by his geometry, or ifa critical reader/viewer were to enter
into any engagement of his images,
then two developments would conceivably result. First, the necessary separation between image and theory would
be open to critique, a separation imperative to his claim that, first and foremost, it is fractal theory and not theactual existence of fractal images thathas been the deciding factor in establishing fractal geometry as nature's own

Gray,Critique and a Science for the Sake of Art

geometry [12]. Second, if any criticalengagement of the fractal images
should produce a reading that is at odds

with the theoretical ground that generates these images in the first place (inother words, should fractal geometry
enter into the author-reader dlebate),then not only would the hierarchicalstructure in the separation of theory
and image be at risk, but it would also
severely weaken the claim that fractal

geometry is the truth-to-nature. This
truth relies to some degree on the assumption of a correspondence between
certain fractal images and visible nature. This correspondence is groundedin Mandelbrot's claim that the viewer
will experience an unproblematic recognition of the visual similaritybetween
nature and particular fractal images,
along with an unproblematic acceptance of the beauty of these images.
If fractal geometry, through a criticalreading of its images, should be shown
to be open to critique in its claim to
speak nature's truth over and above
other discourses, then not only does
the very idea of an objective-geometrictruth-to-nature become open to critique, but, by extension through thiscritique, scientific practice in general
will be seen to be as much involved withthe construction of truths as it is involved in the search for them.
Thus, Mandelbrot's intervention invisual discourse (an intervention forcedon him by the very claim that fractal
geometry mirrors nature) provides theground for visual theorists to critiquea major theoretical development inscience. The dependency on images
that is at the heart of the practice of
fractal geometry, made transparent by
the image's relegation to the role of aservant to science,

is, in Nietzscheanterms, a relationship of promise that
can turn and bite its own tail. The promise of the theoretical veracity offinallydiscoveringthe truth-to-nature by themathematization and geometrizationof creation will itself become increasingly difficult to sustain as the images ofthis geometric creation are increasingly
shown to be problematic. For instance,
for a fractal image to be recognised
as mirroring nature (or its processes),
there must be an existing image of na-

ture with which to compare it. However, as this other image is by necessity
also a representation (even if confinedsolely to the imagination), the fractal
image thus emerges as a representation mirroring another representation.
By reduction back to the generatingtheory, this establishes fractal geometry
as simply another technique for producing representations. Thus, fractal
imagery can differ from what it is mirroring only in the means of production.
But if two different forms of production
can produce images that mirror eachother, then what determines the privileging of one means of production
over the other? The answer is a privileging that is necessary to make sense of
the idea of chaos theory finally discovering the truth-to-nature. As the 'real'
image of nature must alwaysfall within
representation, any claim to represent
what ultimately generates all images(i.e. nature) will therefore always be aclaim to have represented representation. (Note that this claim entails
the necessary contradiction of beingoutside of representation and hence nolonger capable of representing anything.) The only way around this impasse is to claim that it is possible tohave access to 'real' nature in an unmediatedfashion, which of course denies
all existing knowledge, history and cultural embedment.

the sake of commerce: art for the sakeof science (and of mathematics)" [ 14].
To place this claim of the productionof novelty in context, it is relevant toreview a small fraction of the history ofan art for the sake of science, an earlier
moment of novelty, incidentally, that
also claimed geometry as the guiding
producer of art:
The name KALEIDOSCOPE,
which I
have given to a new Optical Instrument, for creating and exhibiting
beautifulforms, is derived from theGreek words kalos,beautiful;eidos,ato see....
form;and skopein,... The fundamental principle, there-

fore, of the Kaleidoscopeis, that it
produces symmetricaland beautiful
pictures, by converting simple into
compound or beautiful forms, andarrangingthem, by successivereflections,into one perfectwhole ....

A critique of fractals could be precededby a critique of the co-extensive relationship the computer has with bothparts of fractal geometry: fractal theory
and fractal imagery. This is a coextension that severely weakens the axiomatic claim of separation demandedby the privileging of fractal theory as
the generative condition for the possibility of fractal images.
Hence the question is to what extent
the computer and its operator function
as the condition of possibility of both
fractal theory and fractal images. (Thisis a point, incidentally, that Mandelbrot raises, but not, understandably, asa critique of his sole authorship of fractal geometry [13].) Would this critiqueof association establish the computer/
operator as the actual truth-to-nature?
Or does being the possibility of truth
relate more to the notion of the creator
than to the process of creation?
The critical sensitivity of the notionof an art for the sake of science is further highlighted by Mandelbrot's connection of his 'art' to the significanceafforded the notion of novelty in thesciences and the arts: "Therefore, weshall argue that fractal geometry appears to have created a new category ofart, next to art for art's sake and art for

Ifwe examinethevariousobjectsof art
which haveexercisedthe skilland ingenuity of man, we shall find that they
derive all their beauty from the symmetry of their form, and that one workof art excels another in proportion as
it exhibits a more perfect developement [sic ] of this principle of beauty.
Even forms of animal, vegetable andmineral derive their beauty from thesame source....
. . When we consider the immense
variety of professions connected bothwith the fine and the useful arts, inwhich the creation of symmetrical ornaments forms a necessary part, wecannot fail to attach a high degree of
utility to any instrument by which theoperations of the artist may be facilitated and improved [19].

... The property of the Kaleidoscope,

which has excited more wonder,and
thereforemore controversythan any
other, is the number of combinations or changes which it is capableof producingfrom a smallnumberof
objects....

Finally, as a prior historical moment
pertinent to the relationship of chaostheory and geometry, Brewster writes,"Lethim now take a Kaleidoscope, anddirect it to the same object: he will instantly perceive the most perfect orderarise out of confusion" [20].

... It will create, in a single hour, what

CONCLUSION

This second condition informs Mandelbrot's idea that great art imitates nature. The unproblematic transpositionof this idea between the discourses ofscience and art characterizes Brewster's
work as well [18]:a thousandartistscould not inventin
the courseof a year;andwhileit works
with such unexplained rapidity, it
worksalsowitha corresponding
beautyand precision[15].

References and Notes

These fragments of history are from thenineteenth-century polymath Sir David
Brewster.While Brewster'swork did not,as has Mandelbrot's, reshape much of
the wayin which geometry is conceived,
he nevertheless also relied on a Kantian
aesthetic in describing the nature of
the images produced by his inventions.
Also, like Mandelbrot, Brewster expressed the possibility of art assisting in
the appreciation and understanding of
science and its principles [16].
However, what is even more striking
than this shared aesthetic and its connection to theories of discourse dissemination [17] is the similarity of theirrespective ideas regarding the universal
character of visual discourse. This
conceptual similarity, though involving
works separated by a wide expanse oftime, highlights the question of parallelism in the theoretical conditions informing their ideas.
One of these conditions is the assumption that changes in the development of artistic practice can be reduced
to the production of visual novelty and/or changes in the means of production.
Another condition, with relation to
ground, is the Kantian transcendentality: 'nature giving the rule to art'.

1. B. Mandelbrot, "Fractalsand an Art for the Sakeof Science," Leonardo,Supplemental Issue Computer
Art in Context (1989) pp. 21-24.

2. The expression Kantian aestheticis used here torefer especially to two central elements of this aesthetic schema: (1) disinterested beauty, i.e. thejudgment of beauty as ifthejudgment is universallytrue and (2) aestheticjudgment in all its modalities
acting to reconcile the theoretical (cognitive) withthe practical (moral) and the world of nature withthe world of freedom (imagination acting as the
exemplar of this freedom). The expression traditional transcendental aesthetic is used here to refer tothe schema that argues that all art objects containor exhibit in varying degrees a set of a priori aesthetic elements, universal in character. Hence therole or judgment of the informed observer (thecritic) is to determine the presence of these elements in the object and to discuss this presence inrelation to the notion of 'good art'. Clearly, in thissystem the artist's intentions, the critic's tastes(selections of objects to discuss) and the a priori
elements are all intricately linked. Thus, the apriori elements together with the significance accorded to the artist's intentions tend to confine all
discussion by the critic to definitions of the object
as 'good' or 'bad' art.
3. B. Mandelbrot,

The Fractal Geometry of Nature(New York:Freeman, 1983).
4. Mandelbrot [3] p. 3.
5. See N. K. Smith, Immanuel Kant's Critique of Pure

Reason (London: Macmillan, 1986) esp. A5052/B74-76, A23-A41/B37-B58, A19-A22/B34B36, B59-B72/A42-A48, B89-B91/A65-A66. See
also S. Korner, Kant (Middlesex: Penguin, 1987)
p. 35; R. C. S. Walker, Kant (London: Routledge &Kegan Paul, 1978) pp. 14-27; W. H. Werkmeister,Kant: The Architectonic and Development of His Philos-

ophy(La Salle, IL: Open Court, 1980) chaps. 3-5.
6. Mandelbrot [3] p. 4.

Gray,Critique and a Science for the Sake of Art

319

7. See Mandelbrot [3] pp. 2-6, 9, 19, 21-24, 59,
154, C2-16, 406-407. See also H.-O. Peitgen and
P. H. Richter, eds., The Beauty of Fractals:Imagesof ComplexDynamic Systems(New York: SpringerVerlag, 1986) pp. 152, 159. Also in this collection,
see the chapters by G. Eilenberger, "Freedom,
Science, and Aesthetics", and H. W. Franke, "Refractions of Science into Art".
8. Mandelbrot [3] pp. 2, 4, 23.
9. Mandelbrot [1] pp. 22-23.
10. For a comparison with Kant's aesthetic claims,
see Smith [5] A1-49, B1-73; and J. C. Meredith,
The CritiqueofJudgment(Oxford: Clarendon Press,
1986) bk. 1, secs. 1-22. See also Werkmeister [5].
11. It is precisely the lawful character of Mandelbrot's claims for his geometry that makes possible
the cohabitation of the term 'transcendental' withthe term 'change'. Change here refers merely to a
notion akin to an unfolding-as-law. Even a change
into something completely different does not,theoretically speaking within the domain of chaostheory, necessarily constitute a violation of these
supposed laws. Indeed, such an event is tantamount to an expression of the 'naturalness' and'obviousness' of these laws. As law here has beenshifted to refer to process rather than principle, theidea of a transcendental system of change nolonger in itself constitutes an incoherence in terminological usage. The law is things changing.This
move from principle to process, including the attempts at the imaging of process, represents for me
the precise point at which the innovative brillianceof Mandelbrot's work affords an entry for criticalvisual practice to engage science.
12. See Mandelbrot [3] pp. C16, 22. Even a cursory
reading of Mandelbrot's text quickly establishes
the privileging of theory over image. What I havein mind here is the problem of recognition. The
idea of a one-to-one correspondence between fractal images and nature presupposes an already existing capacity for the recognition of this visualcorrespondence. If recognition pre-dates fractal
theory, then fractal theory is dependent on an
already existing image of nature. Hence theprivileging of theory over image is dependent on a
presupposed image. With such a dependency, it isdifficult to see how a clear separation between
theory and image can be sustained. With both parts

320of the fractal binary having in common an alreadyexisting image of nature, what then is the basis ofseparation, other than different modalities?
13. Although to the best of my knowledge Mandelbrot does not ascribe any formative role to thecomputer in the initial theoretical development of
fractal geometry, he does pay considerable credit
to the assistance of several computer operators, e.g.
[3] pp. 151-160. The following passage brings in
the image as a possible generative source, though
I think the implication is directed to future changes
to fractal geometry rather than to the initial theoryof Mandelbrot's geometry: "Graphicsis wonderful
for matching models with reality. When a chancemechanism agrees with the data from some analytical viewpoint but simulations of the model do notlook at all 'real', the analytical agreement should
be suspect. A formula can relate to only a small
aspect of the relationship between model and reality, while the eye has enormous powers of integration and discrimination" [1] p. 22. On the other
hand, in his Leonardoarticle where theory generation is not an issue, Mandelbrot does signal thesignificance of the computer, albeit in relation to
the possibility of fractal art (see [1] p. 21).
14. Mandelbrot [1] p. 21.
15. D. Brewster, Treatiseon the Kaleidoscope(London: Hurst, Robinson, 1819) pp. 1, 17, 111, 116.
For a more extensive analysis of the kaleidoscopewith regard to its relationship to philosophy,
geometry and the visual, see N. Gray, "Laughter inthe Ruins",unpublished honours thesis, Power Research Library,Univ. of Sydney, Australia, 1989.
16. Like Mandelbrot, Brewster was a tireless advocate of the value and significance of retaining abroad and innovative approach to his practice. For
instance, between the years 1800 and 1862, hedelivered no less than 299 papers, ranging in topics
from the principles of the kaleidoscope and stereoscope to the optical illusion of the conversionof cameos into intaglios and, vice versa, to theChinese Mirror and its ability to reflect from its
polished face the figures embossed on its back. The
sad fact that Brewsterwent blind while conducting
his optical experiments directed at the sun makes
the extent of his work and his public lectures all themore impressive.
17. The expression 'theories of discourse dissemi-

Gray,Critique and a Science for the Sake of Art

nation' refers here to the long-established use ofimagery in explaining or disseminating the complexity of scientific practices. Clearly,the idea of an
unproblematic aesthetic greatly aids this dissemination as it assumes that the range of visual responses is largely the same for everyone. Indeed,
the use of diagrams in mathematics and geometry
is predicated on this principle. For a challenge tothis idea of image purity versus specific perceptual
responses, see M. Serres, Hermes,Literature,Science,
Philosophy(Baltimore, MD: Johns Hopkins Univ.
Press, 1982).
18. For more on the notion of an unproblematic
correspondence between the discourses of scienceand art or 'an art for the sake of science', see F.
Nietzsche, TheBirthof Tragedy,W. Kaufmann, tran.
(New York:Vintage, 1967) pp. 52-92. The Socratic
promise and science are to a large extent interchangeable in this text. For a concise thesis thatuses Greek art to emphasize certain features of
Greek geometry(and vice versa), see W. M. Ivins,

Art and Geometry,a Study in SpaceIntuitions (NewYork:Dover, 1946). See also M. Andreose, ed., The
ArcimboldoEffect (London: Thames & Hudson,
1987) pp. 155,185,187,189. This volume discusses
Giovanni BattistaFonteo's (1546c-1580c) and Gregorio Comanini's (1550c-1608) poems, which comment about the ways in which Arcimboldo's paintings (such as the Four Seasons) aid the viewer inunderstanding the underlying lawful principles
governing nature. For an analysis of the relationship between Arcimboldo's paintings and Renaissance science (ordering systems), see N. Gray,"Facing the Mask, Science as Ground in Arcimboldo's Paintings", in RenaissanceEssays (Sydney:Power Research Publications, Univ. of Sydney,1990). For discussions on the significance of images assisting in the practice of science (specifically,
the medical sciences), see Hugh W. Diamond's
extensive use of photography in diagnosing casesof madness in the nineteenth century, in L. G.
Sander, ed., TheFaceofMadness,OriginsofPsychiatricPhotography(Secaucus, NJ: Citadel Press, 1976);
and for similar ideas in the twentieth century, see
M. Webb and L. Szondi, TheSzondiTest(Montreal:
Lippincott, 1959).
19. Brewster [15] pp. 113-115.
20. Brewster [15] p. 149.