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Stewart Dickson 110 N Whipple Street Fort Bragg, CA 95437 (707)813-0385 MathArtSPD@GMail.com
In the beginning was Plato. A spiritual perfection was attached to geometrical symmetry and proportion. The road from Plato to the present is lined with history.
The computer is a radical novelty in human history - never before have we had such a tool with which we can test the structure of our understanding of ourselves and our world.
Despite its novelty, cyberspace is not without history of its own. There are many stories told on the UseNet every day. Here are three which have caught my attention.
In 1975, Benoit Mandelbrot used a computer at IBM to make a graph of a dynamical procedure which was known to be chaotic. It was a "monstrous" topic of inquiry, because there were no known methods for dealing with such systems - least of which was simply graphing the behavior, because so many - billions - of calculations are required.
What Benoit found was a graph with symmetry, hierarchical self-similarity and infinite detail. No graph of this kind had ever been seen before. He had to invent a new branch of mathematics - fractal mathematics - to describe what he found. In 1982, Alan Norton - also of IBM - extended the mathematics of the Julia set (a "complimentary" graph to the Mandelbrot set) to Quaternion complex 4-space, which can be viewed in 3-space. I have cast the virtual three-space of CyberSpace into the world of physical experience.
The Seventeenth-Century French mathematician Pierre de Fermat wrote in the margin of his copy of Arithmetica by Diophantus, near the section on the Pythagorean Theorem (a squared plus b squared equals c squared), "x ^ n + y ^ n = z ^ n - it cannot be solved with non-zero integers x, y, z for any exponent n greater than 2. I have found a truly marvelous proof, which this margin is too small to contain." This was left as an enigmatic riddle after Fermat's death and it became a famous, unsolved problem of number theory for over 350 years.
Andrew Hanson has made some pictures, and I have in turn made sculpture, of a system analogous to Fermat's last theorem - a superquadric surface parameterized in complex four-space. We think that the mathematics of the n=3 case are similar to Fermat's own proof of the n=3 special case. Our pictures have lent some visual concreteness to the recent news of Andrew Wiles' proof of the Taniyama-Weil conjecture, which implies the proof of Fermat.
By the 1890's the study of minimal surfaces was thought to be exhausted - no new surfaces could be described mathematically which were non-self-intersecting in three-space and which had vanishing mean curvature. However, in 1983 a graduate student in Rio de Janeiro named Celsoe Costa wrote down an equation for what he thought might be a new minimal surface, but the equations were so complex, they obscured the underlying geometry.
David Hoffman at the University of Massachusetts at Amherst enlisted James Hoffman to make computer-generated pictures of Costa's surface. The pictures they made suggested first, that the surface was probably embedded - which gave them definite clues as to the approach they should take toward proving this assertion mathematically - and second, that the surface contained straight lines, hence symmetry by reflection through the lines.
The symmetry led Hoffman to extrapolate that the surface was radially periodic and that new surfaces of the same class could be achieved by increasing the periodicity. He did so by altering the mathematical description of the surface to be the solution to a boundary-value problem constrained by the behavior of a minimal surface at the periodic lines of symmetry. The result: Hoffman proved that Costa's surface was the first example of an infinitely large class of new minimal surfaces which are embedded in three-space.
The technique Hoffman used was to make a picture which caused him to modify his mathematical theory and discover something totally unexpected about that theory. He later extended his techniques to find minimal surfaces of more complex geometry and he also created pictures of them. This is a new kind of experimental mathematics and a procedure not far from creative visual art.
Indeed, minimal surfaces are objects of pure topology - pure abstract form - in which the geometry is simply constrained to be the most minimal and elegant expression of that form. Hoffman is satisfied to leave his objects in the two-dimensional picture plane. I find Cyberspace fundamentally unsatisfying in the lack of tactile presence. I need to bring back artifacts from Cyberspace into the space I physically occupy. This is the basis of the work I am presenting here.
I have to ask myself what is the motivation for this work. The answer has to hark back to the Classical Greeks. There is deep spiritual value in the quest to understand and possess the unknown. My art is deeper than the demonstration of new technology and the novel ideas of the frontier of science. There is the change in creative process which is happening in Cyberspace. There is the collaborative dialogue of the Internet. I am attempting to express these new forces in my work.
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