van Emde Boas
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
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
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 . 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
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 uvre of a standard artist? For this purpose
I propose the number of instances till boredom measure as
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 , 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 . 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
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
 Leo J.M. Geurts, Kristalstructuren, een experiment in computer-kunst,
Vacantiecursus 1973, Mathematisch Centrum, Amsterdam, rep. VC 27/73,
 Heinz-Otto Peitgen & Dietmar Saupe (eds.), The Science
of Fractal Images, Springer Verlag, New York etc., 1988.
 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.