Leonardo

Art, Therefore EntropyAuthor(s): Michel Mendès France and Alain HénautSource: Leonardo, Vol. 27, No. 3, Art and Science Similarities, Differences and Interactions:Special Issue (1994), pp. 219-221Published by: The MIT PressStable URL: http://www.jstor.org/stable/1576055 .

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SCIENTISTS LOOK AT ART

Art, Therefore Entropy

Michel Mendes France and Alain Henaut

Anne Gruner Schlumberger, in memoriam

ENTROPY AND COMPLEXITY If a system can exist in Ndifferent states, then we say that this

system has entropy equal to log N. For example, the entropy of a die is log6, whereas a coin has entropy log2. A system that consists of Ncoins has entropy Nlog2. However, a system that can exist in one state and one state only has entropy logl, or zero entropy. The larger the entropy, the more complex the

system will be. In other words, entropy is a measure of the

complexity of a system. (Since, as Heraclitus says, "everything flows and nothing stays," the words we use may have fluctuat-

ing meanings. We hope this will not offend specialists.) Let us represent each possible state of a system as a point.

A set of N points will have an entropy log N. Extending this idea one step further, let us consider several sets whose aver-

age number of points is N*. This family of sets will have en-

tropy log N*. We are now in a position to define the entropy of any curve in the plane (Fig. 1).

Let N* be the average number of points of intersection of the curve with all possible straight lines drawn through it. It can be shown that N* = 2L/C where L is the length of the curve and where C is its perimeter (i.e. the length of a thread drawn tightly around the curve) [1]. The entropy of the curve is given by log N* = log(2L/C) [2].

Should the curve be a straight segment, then C = 2L, so that its entropy is equal to log(2L/C) = 0 (Fig. 2). In other words, a simple curve has low entropy, whereas a complicated curve has high entropy.

What we would like the reader to retain from this short pre- amble is that entropy is, in fact, complexity. Through this identification, which is common practise in mathematics, ob-

jects and concepts are mutually enriched.

Fig. 1. As the straight line moves around, the number of points where it intersects the curve varies. Its average is equal to the ratio of twice the length L of the curve by the length C of the perim- eter, represented by the dotted line.

.^ / - *,

COMPLEXITY AND INFORMATION A statement such as x is x carries no information. Strictly speaking, it is correct, but we learn nothing from it. Let us imagine for one moment that we find ourselves in a large and crowded hall where a thousand or so people are talking in a very animated fashion on a vast variety of subjects among themselves. The noise pattern generated from this meeting is indeed highly complex. It prob- ably contains over a thousand

ABSTRACT

The authors discuss the idea of entropy and the implications that it has both on artistic creation and on perception of art. Through ex- amples it is suggested that com- plexity, information and entropy are essentially three facets of the same reality. No claim is made re- garding esthetic evaluation.

sentences or statements. Whether they are true or false is of no importance when all we can distinguish is a loud hubbub. This loud hubbub, however, contains all the information in the statements uttered: rich information indeed! Going one

step further, imagine every sound that has been made, that is

being made and that will ever be made (and even that which can never be said) put together; this is known as "white noise." The message in this white noise is so dense that it is

totally inaccessible; it is impossible to extract one single item of information from it.

This situation is not unlike the axiom of choice in math- ematics, which states that from any set one can distinguish an element in it and choose, for example, a countable subset

among the real numbers. Yet some mathematicians refuse to

acknowledge this axiom, and they are perfectly free to do so. For such mathematicians, any sufficiently large set will be so dense and thick that no point in it can be distinguished: a situ- ation entirely analogous to the "white noise" mentioned above.

Let us now go back to the idea of information and see how it can be applied to painting. A plain white canvas would ap- pear to all intents and purposes to contain no information whatsoever. A painting showing a single brush stroke would

appear to convey perhaps a little more. J. Pollock's paintings present us with a great display of complexity: the information contained therein would seem to be enormous.

We could say that, at the limit where colours are mixed ran-

domly on a canvas to the extent that the painting becomes a uniform shade of grey (albeit rather dull in aspect), the com-

Michel Mendes France (mathematician), Dept. Mathematiques, Universite Bordeaux I, F-33405 Talence Cedex, France.

Alain Henaut (mathematician), Dept. Mathematiques, Universite Bordeaux I, F-33405 Talence Cedex, France.

Manuscript solicited byJacques Mandelbrojt.

Received 19 October 1993.

LEONARDO, Vol. 27, No. 3, pp. 219-221, 1994 219 ( 1994 ISAST

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plexity of the painting itself is infinite. This is analogous to the "white noise" that we mentioned above. Yet, from the observer's point of view, there is no es- sential difference between the original blank canvas and the finished painting. Paradoxically, it follows that the original canvas must therefore be highly com-

plex. The scale of complexity thus ap- pears to be cyclic.

One could probably say that the above procedure underlies the concep- tual creation that was brought about when W. de Kooning did a drawing, R.

Rauschenberg then totally erased it and the resulting "blank" page was signed by both artists. The idea or concept pre- vails; the moment of creation seems fixed at the point where the paper re- became "blank."

At this point we must mention the

quantum vacuum as it is understood in the world of physics: the extraordinarily complex result of the superposition of matter on anti-matter. According to P. Dirac, matter is a hole in a vacuum. To obtain matter one must remove anti- matter from a vacuum. Do time (for- ward time) and anti-time (backward time) coexist in this quantum vacuum? The vacuum fluctuates, but it never seems to evolve, because time and anti- time "superimpose," or, quite simply, neutralise each other.

Before concluding this section we would like to take the liberty of inviting the reader to delve into F.Jullien's beau- tiful book, Eloge de lafadeur (In Praise of the Dull), in which he recalls a story taken from the biography of Tao

Yuanming (fourth or fifth century) in

Songshu.

Tao Yuanming knew nothing about music but he kept at home a very simple lute without any chords. Every time he was invaded by a feeling of wholeness after having had some wine, he would touch the lute to express what his heart was longing for [3].

Silent music, empty music; and yet, how moving.

ART AND ENTROPY

Complexity, information and entropy are hence essentially three facets of the same reality. It would perhaps be appro- priate to add communication to the trio.

All artists try to communicate, to ex- plain, to reveal their ideas or, in some cases, to conceal their goal (but by so

doing they give themselves away). The more information artists are able to in- clude in their work, the more likely they

are to reach and affect their public. Each member of the public has, of course, his or her own personal emo- tional and cultural framework. It is nec-

essary that the information that the art- ist conveys be sufficiently large for there to be an area where it intersects with the viewer's recognition: i.e. large enough for the viewer to be able to construe some of the artist's information from what he or she sees. The viewer's re-

sponse will be one of appreciation, ad- miration or some other kind of strong reaction to the work to the extent that the viewer has a feeling of sharing in the creation with the artist. There is a sym- biosis, or identification of the viewer with the artist in emotional terms. I.

Prigogine claims that he sometimes has the sensation, when listening to music, that he can anticipate the notes before

they are played: as if he himself, at this

privileged moment in the piece, is actu-

ally the composer of the work. It is the duty of all artists to provide

information and hence complexity. Ac-

cordingly, I. Xenakis wrote: "le son beau ou laid n'a pas de sens, ni la musique qui en decoule; la quantite d'intelli-

gence portee par les sonorites doit etre le vrai critere de validite de telle ou telle

musique" (Beautiful or ugly sound is not

meaningful, nor is the music of this sound; the amount of information con-

veyed in the sound should be the real criteria on which to base any judgment concerning the worth of a work) [4]. This idea is also developed by U. Eco in L'oeuvre ouverte [5].

But the question then arises: can and should an artist strive towards the ex- treme limits of complexity? One would be inclined to think not, and is even

tempted to quote Eco's amusing and

highly pertinent observation that no matter how well constructed and com-

plete a dictionary may be, it can hardly be considered a poetic work.

Infinite complexity is randomness, and randomness is certainly not a hu- man creation. Randomness is not a ve- hicle for communication between two human beings. It is up to the artist to discover what the acceptable limits are, and hence avoid being too simple or too

complex. These limits are obviously dif- ficult to define and are variable. They depend on the current cultural climate and change over time. Moreover, any given complex work of art has its own internal dynamism. The dynamic force of the work dictates what the artist may do, and is not entirely within the artist's control. Both S. Mallarme in his "Un

Fig. 2. The perimeter of a straight segment L reduces to a dotted line tightly drawn around it. The length C of the perimeter is 2L.

Coup de ds jamais n'abolira le hasard"

(A throw of the dice never will abolish chance) and J. Joyce in Finnegans Wake were well aware that this principle was at work, that what they were writing had a life of its own. Conscious of this underly- ing drive inherent in their creative activ-

ity, they exploited it to the full. P. Cezanne, J. Pollock and F. Bacon,

among others, have all been preoccu- pied by the degree to which randomness was an integral part both of their cre- ative activity and of the life of the work itself. Bacon has perhaps given the clearest description of this:

When I was trying in despair the other day to paint that head of a specific per- son, I used a very big brush and a great deal of paint and I put it on very, very freely, and I simply didn't know in the end what I was doing, and suddenly this thing clicked, and became exactly like this image I was trying to record. But not out of any conscious will, nor was it anything to do with illustrational painting. What has never yet been ana- lyzed is why this particular way of paint- ing is more poignant than illustration. I suppose because it has a life com- pletely of its own. It lives on its own .... [emphasis added] [6]

It has a life of its own. We find a simi- lar phenomenon in music in the works of such masters as K. Stockhausen, H. Pousseur, P. Boulez in his Marteau sans maitre, Xenakis, of course, and finally, A. Boucourechliev, who takes us on a magi- cal ride through Archipels.

There is also composerJ. Cage, who held a musical stave drawn on transpar- ent paper up to the night sky and traced in the stars as notes on this stave. The

composition obtained from these "star" notes can be performed as clearly celes- tial music. In the case of Cage, the idea is the work of art.

In the world of jazz, composers and

performers alike acknowledge that there exist certain rules that prevent them from playing just any old thing. Yet they all agree that a vast range of free expres- sion is still possible within their rules of improvisation. J.L. Chautemps has per-

220 Mendes France and Hinaut, Art, Therefore Entropy

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haps spanned the most spectacularly broad range of possibilities concerning the length of a work. His shortest, "PAQLPD Rasoir d'Occam PEENM," lasts but a quarter of a second, whereas his

piece entitled "From a Saxophonological Point of View" lasts 200 million years.

An artist draws material from the sur-

rounding world, from the past and from the anticipated future. He or she can extend and increase the message by sprinkling it with rand6mness. In The Name of the Rose, Eco played this card by loading his text full of Latin quotations that were totally incomprehensible to us lesser beings, "the unhappy nonfew." The reader, feeling overwhelmed and

inadequate when faced with this massive dose of information that excludes all but the most erudite of readers, is bullied into reading through the whole work even though he grasps but fragments of it. This probably results in more read-

ings than Eco could have ever foreseen. The work of art has a life of its own, as

we have already stated, but one usually associates life with the animal and veg- etable world. A work of art may also be viewed as a mountain about to be scaled

by a mountain climber; it holds itself in- accessible but can be reached if ap- proached with the necessary sensitivity and skill.

THE MOUNTAINEER AND THE MOUNTAIN The above idea leads directly to the dis- cussion of work that is being undertaken at the present time by Michel Mendes France along with P. Cordier (a moun- tain guide), P. Bolon and J. Pailhous

(cognitive scientists). In this work, we are trying to measure the level of skill of mountaineers. A given number of mountaineers are invited to climb a

given rock face. Their respective path- ways are recorded. It turns out that (if we simplify rather radically) the better the mountaineer, the straighter the

pathway will be. The entropy log(2L/C) of the pathway of the good mountaineer will thus be low, while that of the weaker mountaineer will be higher.

This analysis can be approached from a different angle. At the bottom of the rock face the mountaineer is free to move in whatever direction he or she

pleases; there are no constraints. We con- sider that the mountaineer is free to use

any of N muscles (all the muscles in his or her body). Once the mountaineer is on the rock face he or she can only use M muscles where M is less than N. The

mountaineer is bound to maintain an ad-

equate number of holds on the face, at the risk of falling off otherwise. At the bottom of the rock face the moun- taineer's entropy was log N, and on the rock face it decreases to log M. We postu- late that the loss in entropy log N- log M is equal to the entropy log(2L/C) of the

pathway. In other terms

log N= log M+ log(2L/C)

This expression of the principle of the conservation of entropy is not unlike the theories of Prigogine.

We will now attempt to justify the above equation. A good mountaineer is

very much at ease on a rock face and moves as he or she wishes. This mountaineer's M is close to N, so the variation log N- log M is small, as is

log(2L/C). A weak mountaineer is virtu-

ally a prisoner of the rock face, where his or her cultural universe is greatly re- duced; it is highly unlikely that such a climber will be able to read the informa- tion available there. This mountaineer will probably even miss the point (!). In this case both the variation log N- log M and the entropy log(2L/C) are high.

The analogy that we suggest is that the rock face is similar to a work of art. A

very rugged rock face is rich in informa- tion, and any person endeavouring to climb such a face may well succeed ac-

cording to his or her personal cultural universe. When the rock face is very smooth there is very little information available, and practically no mountain- eer-however talented and cultivated- will undertake its conquest.

CONCILIATORY NOTES It is clear that the study we have under- taken tends to oversimplify and that we have no pretentions whatsoever of ven-

turing into the world of esthetics. Some

people have overstepped the mark and tried to measure the worth of a work of art in scientific terms using scientific pa- rameters. It is most alarming to find the

following kind of statement, penned by Soviet physicist A. Kitaigoroski: "Any de- viation from artistic truth in literary works of mediocre authors can be con- sidered as being a transgression against order" [7].

We have talked a lot about entropy, complexity and information, and we have tried to show by examples that these three concepts cover one and the same reality. One therefore asks why three words are used when one would suffice. Fortunately, we feel we have a

rather powerful answer to this ques- tion. Each of these three words is asso- ciated with a cohort of attendant im-

ages that are specific to that word. By associating these images with each other, we are able to enhance the con-

sistency of the words, fleshing out and

coloring their skeletal forms. For ex-

ample, when one thinks of the word en- tropy, the word randomness immediately springs to mind. Similarly, the word

complexity triggers the word difficulty, and the word information leads us head- on to the media. There is a subtle dis- tinction between the way a mathemati- cian and an ordinary person in the street will use the word information. The

meaning of the word for one person does not have to be perfectly matched in every detail to the meaning that word has for another. It is most fortu- nate that this is so, and that people recognise this fact when they speak to each other. Remove the "I-get-the-gist- of-what-you-say" aspect of the game of lexical exchange, and communication would grind to a halt: communication is only possible if the meanings of words fluctuate to fit the context. A de-

gree of uncertainty, vagueness or messi- ness is both desirable and necessary: Without entropy there would be no

possibility of exchange, and without en- tropy there would be no art.

Acknowledgment Nous remercions chaleureusementJoan Mendes France pour l'adaptation anglaise de notre premiere version "Sans Entropie point d'Oeuvre."

References

1. H. Steinhaus, "Length, Shape and Area," Colloquium Mathematicum 3 (1954) pp. 1-13.

2. M. Mendes France, "Chaotic Curves," in Rhythms in Biology and Other Fields of Application: Deterministic and Stochastic Approaches, Proceedings of the Journees de la Societe Mathematique de France, held at Luminy, France, 14-18 September 1981; Lecture Notes in Biomathematics, No. 49 (Berlin, New York: Springer-Verlag, 1983) pp. 352-367. See also M. Mendes France, "The Planck Constant of a Curve," in Fractal Geometry and Analysis, J. Belair and S. Dubuc, eds. (Dordrecht, The Netherlands: Kluwer Academic Publishers, 1991) pp. 325-366.

3. F. Jullien, Eloge de la fadeur (Paris: Editions Philippe Picquier, 1991) p. 60.

4. I. Xenakis, "Musiques formelles," La Revue Musi- cale, No. 253/254 (Paris: Richard-Masse, 1963) p. 10.

5 U. Eco, L'oeuvre ouverte, Chantal Roux de Bezieux, trans. (Paris: Editions du seuil, 1965).

6. David Sylvester, The Brutality of Fact: Interviews with Francis Bacon (New York: Thames and Hudson, 1987) p. 17.

7. A. Kitaigoroski, L'ordre et le desordre dans le monde des atomes (Moscow: MIR, 1980) p. 237.

Mendes France and Henaut, Art, Therefore Entropy 221

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