Mathematics has a long tradition of interest in visualization methods. The explosive development of geometry in the late nineteenth century was accompanied by intense activity in creating plaster and wire models as well as pedagogical illustrations. The reader need only consult such classic works as Anschauliche Geometrie by Hilbert and Cohn-Vossen to see the influence of this visual approach to mathematics; it is worth noting that the English translation , Geometry and the Imagination, of the German title only barely does justice to the complex nuances of the German anschaulich, which involves other connotations, including ``vivid,'' ``graphic,'' ``intuitive,'' ``demonstratable,'' ``perceivable in the mind's eye,'' or perhaps visualizable: hence our choice of the phrase Visualizable Geometry in the title of this paper.
Despite the fact that pictures scratched on napkins and blackboards never ceased to play an important part in mathematical creativity and intuition, the mathematical literature has been predominantly algebraic for most of the twentieth century. There are good reasons for this: for one thing, it is easy to abuse pictures and convince oneself of false arguments that would not stand up to a formal proof; for another, many interesting current questions involve the properties of spaces so complex that we only know how to treat them algebraically.
It is no accident that the emergence of computer graphics, and especially interactive computer graphics, as a communication medium has coincided with renewed interest in visual mathematics. Many mathematicians harbor the hope that computer graphics technology will have a significant positive influence on the progress of mathematics.