Mathematicians Gather to Play the Numbers Game

Expressionism In Math Mathematicians often talk about the beauty ofa theorem or proof, but that beauty is rarely apparent to nonmathematicians. Now, however, computer graphics is helping to change that. Increasing power and decreasing costs have turned even the least artistically talented mathematician into a Rembrandt with a laser printer. But to mathematicians, the resulting images are more than just a vivid way to advertise their work to the public. In some cases, in fact, the computer's capacity for graphics is changing the direction of mathematical research. Alfred Gray of the University of Maryland, who drew the twin "torus" knots at right using the computer algebra system Mathematica, thinks graphics is driving such a change in differential geometry. Loosely defined as the study of curvature, differential geometry has found applications in fields as disparate as Einstein's theory of general relativity and molecular biology, where researchers have become interested inwhat it has to say about such things as the supercoiling of DNA. In recent years, though, mathematicians' explorations of the subject have begun to turn increasingly abstract, with algebraic symbols playing a bigger role than Geometry mad geometric shapes. But now colored to repre that tide is beginning to flow curvature (top); in the other direction, thanks to the accurate, three-dimensional images that computers can spew forth with ease. "People are looking at more concrete things [now]," Gray says. And as graphics technology gets into more hands, that trend 0e' 5se an is likely to continue. "The revolution is only in its beginning stages," he adds, but already it's aided mathematicians in such things as the discovery, in the mid-1980s, of an entire new class of "minimal" surfaces. Differential geometry is not the only place where pictures help. Michel Lapidus of the University of California, Riverside, is looking to computer graphics to gain extra insight into analytical theories he's been developing over the past several years. Lapidus is interested in the reverberations of mathematical "drums" with fractal boundaries (Science, 13 December 1991, p. 1593). The interaction ofwave phenomena and fractals is an impor< I _ tant aspect of scattering theO ory for applications such as radar ranging, Lapidus points out, but it also contains many problems of purely mathematical interest. Recently, Lapidus has begun analyzing the waveforms produced when the fractal drums reverberate. For help, he turned to Robert Renka and John W. Neuberger at the University of North Texas, who wrote programs to produce pictures of standing waves on a fractal shape known as the Koch snowflake. The picture on the next page shows how the "energy" of a waveform is concentrated at a relatively few peaks, Neuberger notes. That image isn't breaking new ground-it vivid. Knots confirms what the research:nt variation in ers already knew-but it Id torsion. does give them confidence in their numerical methods. And their next steps in computer artistry may well break new mathematical ground. Lapidus is particularly interested in exploring what the standing wave looks like close to the fractal drum's infinitely crinkled SCIENCE * VOL. 259 * 12 FEBRUARY 1993