The color of geometry: computer graphics adds a vivid new dimension to geometric investigations.

The Color of Geometry

Pictures and physical models have long played an important role in mathematics. Nineteenth-century mathematicians, for instance, regularly drew pictures and sculpted plaster or wooden models to help them visualize and understand geometric forms. Their graphic approach represented a way of expressing abstract notions in concrete form. Such sketches and sculptures served as landmarks in the struggle to ferret out the fundamental principles of geometry.

Today's mathematicians are beginning to use computers to create the models they need -- converting equations and mathematical structures into colorful, animated images on a video screen. These modern-day pioneers find computer graphics useful for revealing patterns, communicating abstract ideas and suggesting mathematical conjectures worth testing.

One center of such activity is the Geometry Supercomputer Project, based at the University of Minnesota at Minneapolis-St. Paul (SN: 1/2/88, p. 12). With access to a Cray supercomputer and the aid of a staff of graphics experts, a select group of 18 mathematicians and computer scientists and their associates in the United States and abroad is breaking new ground in exploring geometric forms and creating breathtaking images of mathematical vistas. Members' interests range from knots and soap-film surfaces to the geometry of hyperbolic space.

Some use computer-generated pictures to study fractals -- patterns that repeat themselves on ever smaller scales. Others exploit graphic images to investigate the results of repeatedly evaluating algebraic expressions. Still others look for solutions of geometric problems arising in simulations of a beating heart or a growing crystal.

"Many mathematicians like to sketch things," says Albert Marden of the University of Minnesota, who organized the Geometry Supercomputer Project. The project allows them to go beyond the pencil, he says.

"But it's not as easy as using pencil," Marden adds. "It's hard to write computer programs, and people often don't have the necessary equipment. In this project, mathematicians for the first time can participate in the world of professional graphics and learn what it has to offer."

A flight simulator lets you soar over fields and lakes, dodge mountain peaks and explore exotic terrain without ever stepping into an airplane. It all happens at your computer terminal, and you control all the movements.

Now imagine flying into a three-dimensional mathematical structure -- a surreal, brightly lit landscape representing an abstract world. You're free to examine scenes from different points of view, peek behind objects and zoom in on iteresting features -- all at your own pace.

That's the idea behind the "hyperbolic viewer" developed by computer scientist David P. Dobkin of Princeton (N.J.) University and his colleagues. This computer program illustrates the striking tools members of the Geometry Supercomputer Project are developing to help mathematicians feel more comfortable using computers to investigate mathematical questions.

Different geometries have different rules. for example, in hyperbolic geometry, the sum of the angles within a triangle is less than 180 degrees, whereas the sum is exactly 180 degrees in ordinary, Euclidean geometry. It's relatively simple to program the hyperbolic viewer to show scenes as they would appear in any of a number of different geometries.

Hyperbolic space in particular provides an unusual but rewarding perspective. "As you fly toward things, you get more and more detail," Dobkin says. for example, a pleated surface patched together from hundreds of triangels opens up to reveal still more triangels. That characteristic of hyperbolic space makes it possible to show lots of detail in one part of a scene without cluttering other parts.

The hyperbolic environment may have value as a medium for displaying abstract structures known as graphs, which are simply sets of points connected by lines. When displayed as spheres linked by tubes, graphs resemble the monkey bars found in playgrounds. Mathematician can use the hyperbolic viewer to help lay out three-dimensional graphs -- initially specified only by sets of relationships between adjacent pieces -- in order to seek patterns among the resulting arangements.

The soap bubble is nature's answer to the problem of packaging a given volume in the least amount of wrapping. Its spherical skin represents the smallest surface area attainable for a fixed volume.

The project's answer to the minimal surface probelem is a computer program known as the "surface evolver." Developed by kenneth A. Brakke of Susquehanna University in Selinsgrove, Pa., the program is now available to any mathematician interested in generating and studying least-area surfaces.

Starting with a geometric figure enclosing a certain volume, the surface evolver transform that shape into a new one of the same volume but having the smallest possible surface area. For example, by breaking up its surfaces into successively smaller triangles, the program turns a cube into a close approximation of its least-area counterpart -- a sphere. It can also compute the minimal surface spanning any particular wire frame.

Given suitable geometric starting points, the surface evolver computes and displays a wide range of minimal surfaces. Often, there is no equation to express what such surfaces look like, so the computer-generated pictures furnish the best available evidence that a particular form can exist.

"In the past, we simply guessed something would be there or tried it by blowing a soap bubble," says Frederick J. Almgren Jr. of Princeton. "Now it's easy and convenient. You just generate these fantastic geometries on the computer."

The project's minimal-surface team also has a computer program that mimics the behavior of a cluster of bubbles. "You start off with a bunch of points in space, then you devide up the space into cells so that each point has all its near neigbors in its own cell," Brakke says. That creates a jumble of shapes, each having a fixed volume. The program rearranges the bounderies between the shapes, adjusting their geometries without changing their volumes, until the total surface area of all the interfaces reaches a minimum.

The same program now plays a role in efforts to simulate crystal growth, especially the formation of the branching patterns typical of snowflakes. Whether this new approach will work in creating realistic crystal patterns isn't clear yet. "We have no proof yet that the scheme works," says Jean E. Taylor of Rutgers University at New Brunswick, N.J. "But computer experiments will tell us whether we're on the right track."

The heart is a complicated mass of tissue, consisting of bundles of oriented muscles fibers. Some fibers in the heart wall wind around to form shells that look like nested doughnuts. Other fibers take more complicated paths that lace together the heart's right and left halves.

Charles S. Peskin of the Courant Institute of Mathematical Sciences at New York University has spent more than a decade untangling these paths to develop a computer model of a beating heart, first in two dimensions (SN: 9/27/86, p.204) and now in three. "Moving to three dimensions raises interesting and difficult geometric questions," Peskin says. "We are very close to a three-dimensional, anatomical model of how the fibers are laid out. We're now looking for better ways of visualizing it."

Peskin is particularly interested in the problem of reducing the fiber geometry to sets of numbers a computer can use efficiently for its calculations. Especially challenging is the additional problem of displaying a three-dimensional structure in motion while showing the fluid flow that occurs inside it.

"We can look at our results now, but at the moment the pictures are pretty cruue, and we want to improve them," Peskins says. "My hope is that the tools the project is developing will turn out to be useful for this."

When sufficiently refined, Peskin's heart model could allow researchers to study how a normal or diseased heart functions and to use the simulation as a test chamber for experimental devices, such as newly designed artificial valves to regulate blood flow.

While addressing participants' specific mathematical problems, the Geometry Supercomputer Project tackles broader issues as well. "Part of our mission is to develop a graphics programming environment for mathematics," Marden says. Developing computer programs, graphics techniques and tools for other mathematicians to use means establishing compatibility standards so that researchers can readily share software and communicate results. And the search for such standards raises a host of questions concerning how best to represent and manipulate two- and three-dimensional shapes in a computer.

It also means writing computer programs that work on a variety of different machines. "We're trying to save people from having to learn the idiosyncracies of each new device," Dobkin says.

Adds Charles Gunn, director of the project's graphics laboratory, "Part of the dream is to bring graphics tools to the people who can benefit from them. We're just beginning to see how graphics can be used and how it's going to change the way we do mathematics."

The main uncertainty at present is whether the project, now nearing the end of its second year, will continue beyond its three-year mandate. Currently funded by the National Science Foundation (NSF) with additional support from other sources, the project represents a significant drain on the funds available to mathematicians for research. Some critics grumble that NSF should distribute its scarce resources among a greater number of individual mathematicians rather than providing expensive, flashy playground for some of the world's top geometers.

Project participants, who regard their effort as a highly successful experiment, hope to extend the project's lifetime by transforming it into an NSF science and technology center devoted to the computation and visualization of geometric structures. Their proposal calls for a budget of nearly $25 million spread over five years.

Group members also envision an important educational role for the tools they're developing. "Computer visualization offers an ideal approach to the teaching of mathematics," they state in their NSF proposal. "Not only the images, but also thinking how to produce the images, are powerful aids to understanding."

Moreover, novel graphics techniques and large-scale computation allow mathematicians to tackle problems they would otherwise find impossible to solve or even consider. "They increase the playing field in which your ideas can operate," Almgren says.

"In some sense, mathematics is the problems you look at as well as the answers you get," Taylor adds. "This approach extends the imagination and opens up many new questions."