The Color of Geometry: Computer Graphics Adds a Vivid New Dimension to Geometric Investigations

Article excerpt

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. …