JUNE/JULY 1999 NOTICES OF THE AMS 647 The Visualization of Mathematics: Towards a Mathematical Exploratorium Richard S. Palais Let us help one another to see things better.— CLAUDE MONET 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 im- provements in computer technology have made it easy to externalize these vague and subjective pic- tures that we “see” in our heads, replacing them with precise and objective visualizations that can be shared with others. This marriage of mathe- matics and computer science will be my topic in what follows, and I will refer to it as mathemati- cal 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 cur- rent 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 ex- plicit constructions of eversions of the sphere and of embedded, complete minimal surfaces of higher genus. The history of both of these is well docu- mented, and I will retell some of it later in this article. However, my main reason for writing this arti- cle is not to dwell on past successes of mathe- matical visualization; rather, it is to consider the question, Where do we go from here? I have been working on a mathematical visualization program 1 for more than five years now. In the course of de- veloping 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 ex- plain some of them in this article. In particular, working on my program has forced me to think se- riously about possibilities for interesting new ap- plications 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 math- ematical visualization and art that I call the “Math- ematical Exploratorium”. Let me begin by reviewing some of the familiar applications of mathematical visualization tech- niques. 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 clas- sics as Geometry and the Imagination [HC]. The ad- vantage 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 dis- plays quickly and easily, but in addition it then be- comes straightforward to create rotation and mor- Richard Palais is professor emeritus of mathematics at Brandeis University. His e-mail address is palais@math.brandeis.edu. This article is dedicated to the memory of Alfred Gray. Because some illustrated figures become clearer or more impressive when viewed in color or when animated, the author has made available a Web version of the article with links to such enhanced graphics. It is to be found at: http://rsp.math.brandeis.edu/ VisualizationOfMath.html. 1 The program is called 3D-Filmstrip, but I will refer to it simply as “my program” in this article. Later in the arti- cle I will explain how to obtain a copy for personal use.