Algorithmic Art

This is the beginning of a page that attempts to sketch the emergence of algorithms as a new medium in art and how it is related to mathematics as a way of understanding growth and form in nature.

This page is itself an experiment in new media. It is a Surfers Guide to the Web, a trail of things I picked up myself and a starting point for a similarly rewarding tour you can make.

I announced this page as an attempt of a sketch. A sketch of what? It could be a course of study, or it could be a catalog of an exhibition. Or it could grow out to a virtual exhibition, a contemporary (dis)embodiment of André Malraux's notion of Musée Imaginaire. The ever-vigilant French Ministry of Culture and Communication has celebrated the centennial of Malraux's birth by their guess at what Malraux would have liked as an update to his choice of the art book as virtual museum.

What makes something Art? One might be tempted to answer "Beauty". But try that next time you talk to a serious painter whose work you admire. A sure way to make her cringe with embarrassment is to say: "I find your paintings so beautiful." She is after something else; beauty is a side effect.

The leaf of a plant, a work of engineering, a landscape, a proof of a theorem, ... All these can be experienced as beautiful. None of these was intended to be beautiful.

I'm interested in the side effect as it may occur in visualizing a mathematical phenomenon or when an artist subjects himself to constraints that can be expressed mathematically or as an algorithm. Hence "Mathematical Art" or "Algorithmic Art".

For the time being just a listing of books and other items. Chronology helps. The earliest I can think of is certain decorations that have been used for centuries. A compilation can be found in "The Grammar of Ornament", recently re-issued in a beautiful production.

Life forms are often beautiful. "Growth and Form" by D'Arcy Wentworth Thompson was motivated by the desire to show that this beauty need not be ascribed to capricious creativity of a creator, but can be explained by mathematics and by the assumption that economics forced optimal solutions under constraints in the form of physical law. A recurring pattern in the plant and animal kingdom is five-fold symmetry. Other paths lead to the pentagram: the golden-ratio and the Fibonacci sequence. It is so pervasive that there is, alive and well after decades of publication, the The Fibonacci Quarterly, the official publication of The Fibonacci Association. If you don't mind downloading this big ppt file The Physics of Foams, then you'll be rewarded with Nature at its most fascinating.

In the quest to track down that elusive quality that is Beauty, Symmetry often comes up. It certainly has something to do with it, but it is not so that more of either leads to more of the other. Symmetry is not necessarily simple.

Mathematics has done a great job in elucidating symmetry by distinguishing various types as being associated with the algebraic concept of Group. For each type of symmetry, there is the corresponding symmetry group. There are seventeen ways in which a two-dimensional pattern can be symmetric. Most, if not all, are illustrated in "Grammar of Ornament". Mathematically more challenging, and also of more practical importance, is the situation in three dimensions. Then there are 230 symmetry groups. The practical application is crystallography.

Many books on group theory or crystallography treat this. One that is equally concerned with the aesthetic aspects is "Symmetry" by Hermann Weyl. Reason for listening to him first is that he was a great mathematician, who also contributed to relativity theory.

No crystal seems to exhibit five-fold symmetry. However, if one stretches the concept of "crystal" from periodic to almost-periodic, then one includes the three-dimensional analogues of the amazing Penrose tiles. Amazing, because their discovery is only half a century old.

An example of beauty arising from working hard at something else is "Funktionentafeln mit Formeln und Kurven" (Tables of Functions with Formulae and Curves) by E. Jahnke and F. Emde. These authors tried hard to visualize complex (literally and figuratively) functions as clearly as possible. The resulting diagrams and typography are a work of art. Hard to buy; try a good library. I believe I have seen editions going back to the 1920s, which are rare. Most copies on the market are Dover re-issues starting at 1945. (Dover had good taste when it was young).

Sooner or later the conversation will turn to Richard Buckminster Fuller. It is a huge topic. It is a cult, it is a movement. Ask a sober scientist, and he will often react with terms like "crackpot" and "crank". These are applied not only to crackpots and cranks, but also to geniuses who make mere scientists nervous.

Given the protean nature of this phenomenon, it is hard to know where to start. Much easier to handle is what little I have seen myself. Important events in one's life tend to be etched in memory. In this case it was visiting my student friend Jón Kristinsson (Architecture) where I spotted on his shelf "R. Buckminster Fuller" by John McHale (New York, 1962), which I recommend as an introduction. Another good introduction is "A Buckminster Fuller Reader". There is also a more recent anthology.

Next thing to look at is books about RBF. Hugh Kenner is a story in itself. He is a literature critic. In addition he wrote a book on the mathematics of geodesic domes, and was an early computer hobbyist. And he wrote a life of RBF.

RBF was a hugely successful speaker. There were no prepared notes. He would close his eyes, concentrate, and then started telling what he knew. By all accounts, he held his audiences spell-bound for hours. His writings were like that as well. He never compared what he was writing with what he had written before. Every time he started from scratch. When criticized for the big overlaps between his books, his answer was that he was like a pole vaulter: even though the bar had been set higher, every time he needed to make the same run-up.

John Applewhite grew up as a friend of the Fuller family. When he retired from the CIA, he took up the task of helping RBF to organize his writings and lecture videos and get RBF to write a systematic account of everything. Applewhite wrote a lovely little book about the saga, which led to the two volumes of "Synergetics".

Buckminster Fuller is often considered an architect. There are two kinds of architects: traditional and modern. As Fuller clearly did not belong to the former category, it seemed natural to ask about influences on his work. In response to one such inquiry he wrote (Chapter 3 in the the earlier anthology): "Many people have asked me if the Bauhaus ideas and techniques have had any formative influence on my work. I must answer vigorously that they do not. Such a blunt negative leaves a large vacuum and I would like to eliminate that vacuum by filling in with a positive statement ..." He could just have said: the "modern" architects found a nice slogan ("Form Follows Function"), but used it only to get rid of some of the ornaments. But instead Chapter 3 is an extensive statement of his design philosophy.

Tent structure by Frei Otto. See his Complete Works.
Another example of an engineer-architect is Pier Luigi Nervi.

Fuller is an example of engineering merging into mathematical sculpture. When a sculptor is subject to severe constraints, then the work takes on a mathematical flavour. With Buckminster Fuller tensegrity was intended as a dome-building technology.
Kenneth Snelson is considered a sculptor because he was pursuing tensegrity as a phenomenon in its own right.
In sculpture, the high-tech flavour can be more subdued than it is in Snelson's work. Some of the works of Elias Wakan consist of large numbers of identical wedges manufactured out of hemlock with standard, albeit modern and high-grade, carpentry equipment. The mathematics goes into predicting how a given algorithm with play out in 3-space. No other sculpture, mathematical or otherwise, smells and feels as good.
In one sense Wakan has created a new medium -- one cannot regard his work as an outgrowth of cabinet making. But he is using traditional materials and tools for his algorithmic structures. In that respect he is following Maurits Escher, who used the same strategy: show the finished work, then leave it to the mathematicians to explain it.

To the left, see Escher's "Square Limit", a 1964 woodcut print from two blocks, 34 by 34 cm, see plate 25 in "The Graphic Work of M.C. Escher".

Mathematical explanations have been given by Bruno Ernst and by Douglas Hofstadter in a literary fashion. Peter Henderson is remarkable in that he undertakes a literally algorithmic interpretation of Escher's "Square Limit": by analyzing its structure he arrives at an algorithm in the form of a program that causes a computer to print an image such as the one to the right. What is the work of art: the woodcut print or the program?

Henderson published his analysis and his program; see here for a recent paper.

The Escher/Henderson team gives what is perhaps the clearest illustration of the notion of Algorithmic Art. If it is indeed clearest, it is only that in the visual realm. Sound, by its nature, is more abstract than image. So one is likely to find even purer Algorithmic Art in music. This is one of the three strands in the "Eternal Golden Braid" that Douglas Hofstadter has in mind. What makes his book great is that he not only ties together the visual and aural dimensions of Algorithmic Art, but also connects it to deep computer science.


Thanks to Mantis Cheng, Belaid Moa and Frits Swinkels for the various pointers they supplied.