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
[6], *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.

Thu Sep 21 19:17:33 CDT 1995