The Visualization of Mathematics: Towards a Mathematical Exploratorium

Introduction Mathematicians have always used their “mind’s eye” to visualize the abstract objects and processes that arise in all branches of mathematical research. But it is only in recent years that remarkable improvements in computer technology have made it easy to externalize these vague and subjective pictures that we “see” in our heads, replacing them with precise and objective visualizations that can be shared with others. This marriage of mathematics and computer science will be my topic in what follows, and I will refer to it as mathematical visualization. The subject is of such recent vintage and in such a state of flux that it would be difficult to write a detailed account of its development or of the current state of the art. But there are two important threads of research that established the reputation of computer-generated visualizations as a serious tool in mathematical research. These are the explicit constructions of eversions of the sphere and of embedded, complete minimal surfaces of higher genus. The history of both of these is well documented, and I will retell some of it later in this article. However, my main reason for writing this article is not to dwell on past successes of mathematical visualization; rather, it is to consider the question, Where do we go from here? I have been working on a mathematical visualization program1 for more than five years now. In the course of developing that program I have had some insights and made some observations that I believe may be of interest to a general audience, and I will try to explain some of them in this article. In particular, working on my program has forced me to think seriously about possibilities for interesting new applications of mathematical visualization, and I would like to mention one in particular that I hope others will find as exciting a prospect as I do: the creation of an online, interactive gallery of mathematical visualization and art that I call the “Mathematical Exploratorium”. Let me begin by reviewing some of the familiar applications of mathematical visualization techniques. One obvious use is as an educational tool to augment those carefully crafted plaster models of mathematical surfaces that inhabit display cases in many mathematics centers [Fi] and the line drawings of textbooks and in such wonderful classics as Geometry and the Imagination [HC]. The advantage of supplementing these and other such classic representations of mathematical objects by computer-generated images is not only that a computer allows one to produce such static displays quickly and easily, but in addition it then becomes straightforward to create rotation and morRichard Palais is professor emeritus of mathematics at Brandeis University. His e-mail address is palais@math.brandeis.edu.