fea-palais.qxp 5/12/99 12:54 PM Page 647 The Visualization of Mathematics: Towards a Mathematical Exploratorium Richard S. Palais Let us help one another to see things better.— CLAUDE MONET Introduction However, my main reason for writing this arti- cle is not to dwell on past successes of mathe- Mathematicians have always used their “mind’s matical visualization; rather, it is to consider the eye” to visualize the abstract objects and processes question, Where do we go from here? I have been that arise in all branches of mathematical research. working on a mathematical visualization program1 But it is only in recent years that remarkable im- provements in computer technology have made it for more than five years now. In the course of de- easy to externalize these vague and subjective pic- veloping that program I have had some insights and tures that we “see” in our heads, replacing them made some observations that I believe may be of with precise and objective visualizations that can interest to a general audience, and I will try to ex- be shared with others. This marriage of mathe- plain some of them in this article. In particular, matics and computer science will be my topic in working on my program has forced me to think se- what follows, and I will refer to it as mathemati- riously about possibilities for interesting new ap- cal visualization. plications of mathematical visualization, and I The subject is of such recent vintage and in would like to mention one in particular that I hope such a state of flux that it would be difficult to write others will find as exciting a prospect as I do: the a detailed account of its development or of the cur- creation of an online, interactive gallery of math- rent state of the art. But there are two important ematical visualization and art that I call the “Math- threads of research that established the reputation ematical Exploratorium”. of computer-generated visualizations as a serious Let me begin by reviewing some of the familiar tool in mathematical research. These are the ex- applications of mathematical visualization tech- plicit constructions of eversions of the sphere and niques. One obvious use is as an educational tool of embedded, complete minimal surfaces of higher to augment those carefully crafted plaster models genus. The history of both of these is well docu- of mathematical surfaces that inhabit display cases mented, and I will retell some of it later in this in many mathematics centers [Fi] and the line article. drawings of textbooks and in such wonderful clas- sics as Geometry and the Imagination [HC]. The ad- Richard Palais is professor emeritus of mathematics at vantage of supplementing these and other such Brandeis University. His e-mail address is classic representations of mathematical objects [email protected]. by computer-generated images is not only that a This article is dedicated to the memory of Alfred Gray. computer allows one to produce such static dis- Because some illustrated figures become clearer or more plays quickly and easily, but in addition it then be- impressive when viewed in color or when animated, the comes straightforward to create rotation and mor- author has made available a Web version of the article with links to such enhanced graphics. It is to be found at: 1The program is called 3D-Filmstrip, but I will refer to it http://rsp.math.brandeis.edu/ simply as “my program” in this article. Later in the arti- VisualizationOfMath.html. cle I will explain how to obtain a copy for personal use. JUNE/JULY 1999 NOTICES OF THE AMS 647 fea-palais.qxp 5/12/99 12:54 PM Page 648 of only a few seconds can generate trillions of floating point numbers. While there are statistical techniques for making sense of such huge data sets, displaying the velocity field vi- sually is essential to get an insight into what is going on. Also, scientists who need and use mathematics but are not completely at ease with abstract mathematical notations and formulas can often better understand the mathemati- cal concepts they have to deal with if these concepts can be given a vi- sual embodiment. Finally, there is no denying that mathematical visu- alization has a strong aesthetic ap- peal, even to the lay public—witness the remarkable success of coffee Figure 1. Symmetries of the Costa surface. The Costa surface (left) is cut by the table picture books of fractal im- three coordinate planes into eight congruent tiles, fundamental domains for ages! the symmetry group. The horizontal plane cuts the Costa surface along two straight lines; the upper and the lower half are moved apart so that they do not Mathematical Visualization 6⊆ overlap. The vertical planes are planes of reflectional symmetry and the Computer Graphics symmetry lines are emphasized as gaps in the top and bottom part. The eight fundamental domains, one per octant, can each be represented as a graph, and One important lesson I have learned Hoffman-Meek’s theorem that the Costa surface is embedded follows easily from my own experience is that from this fact. mathematical visualization pro- gramming should not be approached phing animations that can bring the known math- as just a special case of 3-dimensional (3D) graph- ematical landscape to life in unprecedented ways. ics programming. While the two share concepts and Even more exciting for the research mathe- algorithms, their goals and methods are quite dis- matician are the possibilities that now exist to use tinct. Indeed, there are peculiarities inherent in dis- mathematical visualization software to obtain fresh playing mathematical objects and processes that insights concerning complex and poorly under- if properly taken into account can greatly simplify stood mathematical objects. For example, giving programming tasks and lead to algorithms more an abstract mathematical object a geometric rep- efficient than the standard techniques of 3D graph- resentation and then displaying it visually can ics programming. Conversely, if one ignores these sometimes reveal a new symmetry that was not ap- special features and, for example, displays a math- parent from the theoretical description. Just such ematical surface with software techniques de- a hidden symmetry, disclosed by visualization, signed for showing the boundary of a real-world played a key role in the Hoffman-Meeks proof of solid object, many essential features of the surface the embeddedness of the Costa minimal surface that a mathematician is interested in observing will [H]. (See Figure 1.) Similarly, a morphing animation end up hidden. The mathematician’s fine catego- in which a particular visual feature of a family of rization of surfaces into parametric, implicit, al- objects remains fixed when certain parameters gebraic, pseudo-spherical, minimal, constant mean are changed can suggest the existence of a nonob- curvature, Riemann surfaces, etc., becomes blurred vious invariant. The helicoid-catenoid morph that by the computer graphics notion of surface, and we discuss later is an example of this kind. (See Fig- one quickly learns that not only are off-the-shelf ure 2.) computer graphics methods inadequate for cre- Applied mathematicians find that the highly ating and displaying all of these various types of interactive nature of the images produced by re- surfaces but also that a special method designed cent mathematical visualization software allows to optimize the display of one type of mathemat- them to do mathematical experiments with an ical surface may not be appropriate for others. ease never before possible. Since very few of the One corollary of this is that it is not a good strat- systems they deal with admit explicit, closed form egy to base mathematical visualization on some solutions, this ability to investigate solutions vi- small fixed number of predefined, high-level graph- sually has become an essential tool in many fields. ics routines and expect that one will be able to shoe- For example, in studying fluid flow close to the horn in all varieties of mathematical objects. Of onset of turbulence, the description of a velocity course, one needs a number of low-level graphics field in a small 3-dimensional region over a period primitives to get going, but instead of the Pro- 648 NOTICES OF THE AMS VOLUME 46, NUMBER 6 fea-palais.qxp 5/12/99 12:54 PM Page 649 Figure 2. Helicoid-Catenoid Morph. Shown here (using three rendering methods) are six frames of the associate family morph joining the helicoid and catenoid minimal surfaces. Patch rendering (top) and wireframe rendering (middle) expose the isometric quality of the deformation, while ceramic rendering conceals it. Note how the automatic hidden lines feature of the painter’s algorithm makes the patch version visually superior to the wireframe one. crustean approach, attempting to fit each mathe- they are highly intuitive and so require less ex- matical object to one of a few high-level display planation. But it is important to realize that almost methods, it is better to use the low-level routines all of the same points could be made in relation to design optimal display algorithms for each spe- to the display of conformal mappings, solutions cial kind of mathematical situation. This entails of ordinary and partial differential equations, or more effort for the programmer, but the superior visualizations associated to almost any other cat- results warrant the extra effort. A second corollary egory of mathematical object. is that one or more mathematicians must play a Multiobject vs. Single-Object Graphics Worlds central and ongoing role in the planning and de- I claimed above that mathematical visualization has velopment of any serious mathematical visualiza- features that set it apart from general computer tion software project.