In the brief tradition of Chalkdust cover articles there is a developing discussion of how mathematics and art are related.
Art is simply the making of representations. Art happens when a person has an idea or a vision that exists in their imagination (the mind’s eye) and is impelled to communicate said idea by making a visible manifestation (representation) of it in the material world. The idea or vision on its own is not art. Art occurs amid the struggle to make a representation of the idea that the artist can show to other people. Art may be relatively `fine’ or popular, conceptual or objective, highbrow or applied, yet still fall within this definition. Judgements about the quality of art are made largely by consensus among the cognoscenti in a given art milieu. These judgements are subject to change over time as the perception of works of art are always modified by the current `cultural environment’ and fashion.
The image Central Quadratic explains itself, I hope, as a celebration of analytic geometry.
So artists and mathematicians share the `having of ideas’. But what then? Mathematicians communicate their ideas—yes. But ideas in maths take the form of theorems or conjectures about numbers, space or other abstract entities. The quality of these ideas is first assessed by proof. Can the idea be shown to be true? And second, if the idea is true, is it interesting? That is, does it usefully contribute to the mass of existing mathematics? Communication of mathematical ideas may require the invention of new symbols or diagrammatic forms, etc, but these are in the nature of being a new language, not art.
In my view then, art and mathematics share the magical process of `idea getting’ but essentially differ in where they go with those ideas. If maths is to be considered an art, it would have to be a sort of `super-art’ or art `to a higher power’. Easier, I think, to class mathematics as the science of number, space, shape and structure, etc—the abstract entities that exist in our minds.
Imagine an intelligent alien’s perception of our arts and our mathematics. Our art would be more or less incomprehensible, depending on how alien the being was; but our maths would be as true for the alien as it is for us. Furthermore, good mathematics will not diminish with time or go out of fashion.
There is an affinity between some mathematicians and some artists. Certainly, it is a most pernicious error that scientific and artistic talent exclude each other—an idea unfortunately common among school counsellors. The common ground between art and science/maths that leads us to the `getting of ideas’ is the activity we call play. The thing of it—the thrilling thing, the magical thing—is the moment when one discovers a new idea, or pattern, or conceptual framework, or whatever: the eureka moment! And are these moments not usually approached through playing in the mind with new combinations and orderings of existing mental constructs?
In ray tracing, each ray is used to decide the colour of a pixel on the image plane.
Spheres was created using ray tracing.
I had one of my most memorable eureka moments sometime in 1971 while sitting on a dead tree in Epping forest. At that time, I had been collaborating on an automatic projective line drawing program with `hidden line removal’, going where Autocad later arrived. I was considering algorithmic approaches to colouring surfaces in projective drawings. I realised that if I thought of objects in the scene as being represented mathematically as arrays of vertices and planes in some coordinate space, then I could solve for the equation of the line going from an eyepoint through a particular pixel in the image plane and into the scene (as in the diagram). From the equation of the line, I could find the closest surface along the path and then compute the colour and illumination value for that pixel based on the defined colour on the surface, along with its relationship to any light source or other light-emitting surfaces. And so I had invented ray tracing
—the foundation of all computer generated synthetic imaging for special effects in cinema, television and gaming. Of course, I neither invented it first nor alone—and I certainly had neither the persistence nor vision to pursue ray tracing to practical or rewarding development. But its discovery was a thrill, as were the few simple pictures I made using the technique in a primitive manner on the pen plotter available.
As spheres have become my most persistent motif, I will end with two more related works that play on the division and articulation of spherical surfaces: Sphere Architecture and Star Sphere.
[ Pictures: Central Quadratic: Used with permission from UCL Art Museum, University College London; Spheres, Sphere Architecture and Star Sphere: Used with permission from John Crabtree ]