Peter van Emde Boas

Mechanized Art and the Mad Mathematician

     The mediæval painter may have dreamt about selling his soul to the Devil in exchange for a paintbrush that would producce the perfect painting, enchanting his patrons and producing fame and fortune for himself. In practice this magic never worked out, and therefore the artist had to be trained for a long period, and had to work hard to make a living.
     Human technology provides us with the computer as a tool to be used in all activities man is involved with. Art is such an activity, so art today can be produced by interaction with a computer. The artists in fact may attempt to use computer technology to realize the dream of finding the ideal tools for creating art without jeopardizing the salvation of their souls. In fact, a lazy artist might leave the entire construction of his paintings to the computer, and see how far he can go and get away with it. But chances are high that the observer will discover his cheating by getting bored after only a few pictures.
     So is there any reason to believe that the magic may work this time? In order to answer this question we need to understand more about the power of the computer. And since the computer represents the ultimate operationalization of Mathematics we must approach the artist and his work from the perspective of this mother of all sciences.
      The relation between Mathematics and Æsthetics is non-trivial. History teaches that Mathematics once represented the ideal of all beauty. Mathematics and Art were being taught together among the seven free Arts. And even today you can address an arbitrary mathematician and she will convince you that there is beauty in the formulas or diagrams she has written on the blackboard in her room. Even more æsthetic satisfaction would be gained from the structure she has in her mind, but that feeling can't be communicated. What the observer can discover is Structure. Structure appeals to humans and without Structure there is nothing to feel attracted to.
     There is a fundamental difference between the magic paintbrush of our mediæval artist and the computer program of today's computer artist. The classic painter operates on the visible result, whereas the programmer ultimately produces code in the form of numbers. The ultimate error a computer artist can make, is to consider his program to be only a drawing tool like a paintbrush or a pencil, notwithstanding the fact that this is precisely the impression that today's drawing programs present us with. These programs hide their internal encodings and offer user-friendliness by supporting result-oriented data structures. For a real artist, however, this is the wrong level of abstraction, leading towards an incorrect conceptualization of structure. The artist will be lured into composing art by drawing lines, points and regular regions in a rather randomized fashion on the screen. Which is exactly what was attempted by people creating 'random art', another attempt at inventing a form of modern art which failed in most cases: it turned out to be just boring.


     You may have heard about the idea of putting a large group of monkeys in a room behind typewriters and wait for a Shakespeare sonnet to be produced. Similarly, mathematicians will tell you that by programming a computer to generate all natural deduction proofs starting from the Peano axioms, eventually provable theorems of arithmetic will appear. In both cases the chances of being rewarded during our lifetime (or before Earth is burned to ashes by the expanding Sun) are negligible, and therefore we create literature and mathematics using a more structured goal-oriented approach. Intelligence amounts to producing interesting results: inspiring poems, meaningful theorems and beautiful pictures. Our intuition tells us where to look and what direction not to proceed in. There is no guarantee that, by following our intuition, we have not in fact closed the access road to the results we are after. But we at least are almost sure that by doing so we won't be bored by what we produce.
     It is therefore no surprise to learn that there exists an alternative road towards computer generated art, which was shown to us by, among others, Leo Geurts and Lambert Meertens in the late sixties [1]. Their approach to computer generated art is based on structure. They start from a random pattern of black and white squares on a grid and update it in some fixed but randomly chosen order according to a specific finite automaton rule, until the pattern stabilizes. The resulting screen shows structure in combination with variation: depending on the automaton used, the result will appeal to the human observer or turn out to be boring.
     The idea behind this alternative is simple. Under no circumstances try to randomize at the level of the result. If you use randomness as an input at all, then use it during an initial stage and transform the random seed into a final result by means of a fully deterministic set of rules. These rules will lead to structure and it is structure that the observer wants to see, both in nature and in our artifacts.
     Art always has been subjected to rules. Rules about appreciation for the observers, and rules about creation for the artists. At the academy the students learn rules rather than completed pieces of art. The rules constitute a grammar expressing how to compose art from ingredients, similar to the way in which the grammar of a language expresses how to build sentences and discourses from words and other grammatical atoms. Linguistic competence is rooted in mastership of rules, at least so we are told by one of the paradigmatic theories about language.
     Now evidently there is no proof that this model correctly represents the procedure by which humans produce language and/or understand it. It just happens to be the only model which we are able to represent mathematically and consequently it is the only onewe know how to implement on a computer. And similarly there is no reason to accept the hypothesis that this process correctly represents the way art is created or being observed and appreciated, but again it is the only model which we presently can operationalize and implement on a computer.
     Remko Scha is fully honest about his computer art. By showing not only the resulting picturebut showing how the picture is being produced in time, he gives us insight in the rules of his grammar for art. You will learn everything you might want to know about Artificial by observing it in action. Don't ask for the code - you will not understand it and in any case it would turn out to be boring code.
      The art is not in the programming of the computer but in the selection of the grammar; choosing which operations to include and in which order to apply them. Would we be given the opportunity to inspect the sequence of operations in a symbolic format, the output of the program would be trivial. It is the result of applying these operations on the screen which may or may not impress the observer. But the same holds for the language we speak and write; we are inspired by the resulting sentences and not by the corresponding adorned production trees in some grammar for English.
     While you are observing Artificial in action you may infer the operations invoked. Certainly there exist operations which you could imagine which are not used. Why were they omitted? Presumably they were taken into consideration and subsequently rejected for producing boring results. Or possibly they were simply overlooked or have been left unimplemented because of lack of time. Ask the artist if you want to know what was the reason for doing something and omitting something else. But don't hassle him particular spot occurring at this particular position in some particular picture, since he has no control over his program at this level of detail.


     What is the true nature of the 'piece of art' which is presented at this exhibition? Clearly there is variation; there exists a single program which produces an infinite collection of pictures. Nobody has control over which picture among the set of possible outputs is constructed. Even worse, by omittng some instructions in the program, the results are irreproducible. Suppose that one run of the program would happen to produce the picture which will miraculously heal all diseases by being loooked at, and that someone would press the Delete button before having stored it; it would be lost forever.
     You can look at the program for a long period and discover more and more structure. But you can be sure that there are pictures which will never appear; if you are waiting for the Mona Lisa, you will wait forever, since she is not contained in teh language generated by Artificial. Instead you might wait for the 'peerfect' in stance of a picture which is generated. But also in that case you will need a long time waiting behind the computer screen.
     In Remko Scha's program there is no place for real-time intuitioin. There is no way the observer can influence the drawing just being made. Neither can the artist or the programmer. The program just generates a single randomly chosen picture from an infinite picture-language, and therefore each particular picture is devoid of meaning and depth.
     The final question is whether this art is boring or not. How should we compare the collection of computer-generated pictures with the œuvre of a standard artist? For this purpose I propose the number of instances till boredom measure as explained below.
     Computer graphics today has given us the Mandelbrot set as an example of an æsthetically very attractive picture which is obrtained by application of very simple mathematical rules. But how many pictures of an enlarged fragment of the Mandelbrot set are required to give the human observer the impression that he has seen all of it?
     Although the details of the set provide us with an unexhaustible source of new patterns with an ever increasing complexity, the answer reads: 'surprisingly few'. Having glanced through one popular book about chaos theory [2], or having explored the possibilities of one good Mandelbrot program on a high resolution screen, the æsthetic feelings of the human observer are saturated. Faced with the next picture of a Mandelbrot fragment the viewer will recognize the structure and find no more surprises.
     Another example of a beautiful mathematical structure, which has been discovered only two decades ago, is the Penrose tiling [3]. How many pictures of a fragment of a Penrose tiling are required for saturating the æsthetic feelings of an observer? Presumably even less than in the Mandelbrot case. The mathematics in this case is quite different: theory shows that there exactly as many different Penrose tilings as there are points in space, but at the same time each finite fragment occurs infinitely often as a subfragment of every possible tiling. So in this case it turns out indeed to be the case that having seen one you have seeen them all, be it in a highly non-trivial way, presumably incomprehensible to the average observer of art.
    We could ask the same question concerning real painters. For example, try to compare Van Gogh, Rembrandt, Rubens and Jeroen Bosch on this measure, and subsequently extend it by including some contemporary artists known to you. If I tell you that in the resulting ranking, in my opinion, Bosch is rated number one, followed by Rembrandt , with Rubens and Van Gogh finishing ex æquo on the last place, this tells you something about my personal taste. And my experience with contemporary art exhibitions is that they score quite low (less than ten pictures in general suffice for having experienced the artist).
    Therefore, for a visitor to this exhibition the real question is how many images being produced by Artificial he needs to see before getting bored and obtaining the impression to have seen everything. Presumably a number between ten and twenty will do. A number which seems to score good in comparison with my experience of visiting an average modern art gallery, and even not too bad in comparison with Rubens. But if Art primarily is a matter of emotion, none of the pictures shown at this exhibition is Art. So if you leave this exhibition with the feeling that you have seen some interesting pictures, thsi tells you where the artist was hidden: in the infinity of the never completed art gallery which is being generated before your eyes.


References

[1] Leo J.M. Geurts, Kristalstructuren, een experiment in computer-kunst, Vacantiecursus 1973, Mathematisch Centrum, Amsterdam, rep. VC 27/73, augustus 1973.
[2] Heinz-Otto Peitgen & Dietmar Saupe (eds.), The Science of Fractal Images, Springer Verlag, New York etc., 1988.
[3] Martin Gardner, 'Extraordinary nonperiodic tiling that enriches the theory of tiles', Scientific American, Mathematical games section, 236 (1), 1977, pp. 110-121.     


Peter van Emde Boas is lector at the Institute for Logic, language and Computation (ILLC), Faculty of Mathematics and Computer Science, University of Amsterdam.