On my first day of graduate school at Princeton University in 1979, an older student warned that I would encounter a lot of strange characters in the mathematics department. The strangest of all, he said, was a phantomlike figure known as Nash. In the weeks that followed, I occasionally caught sight of Nash shuffling down the hall in a shabby coat and bright red sneakers or sitting in the cafeteria by himself, staring off into space. But I felt his presence most from the nonsensical messages he wrote on blackboards at night. These often took the form of mathematical equations, but they had as much to do with mathematics as a cat walking across the piano has to do with music.
One day, as a group of students gathered to chat with a professor after a class, someone asked about this mystery man. Lowering his voice, the professor told us that Nash had once been John Nash, the brightest light in the greatest group of students that ever studied math at Princeton. In the late 1940s and 1950s, John Nash had made discoveries that his peers still use every day—the Nash equilibrium, the Nash embedding theorem—even as they averted their eyes from the man himself. But then, somewhere along the line, he lost his grip on reality. He began to believe he was receiving messages from outer space and that there were great and hidden conspiracies against him.
Now, four decades after John Nash was lost to mathematics, mathematics itself may hold the key to treating schizophrenia, the mental illness that held his mind hostage. A new way of analyzing shapes, called morphometrics, may allow doctors to tell what changes occur in the brains of schizophrenics before they lose contact with reality. Morphometrics is also providing clues to the development of fetal alcohol syndrome and Alzheimer’s disease and is improving the ability of brain surgeons to map out the routes they will take to perform delicate operations. In the study of the brain, the shape of things to come is, quite literally, shape.
Fred Bookstein, a statistician at the University of Michigan, has spent more than two decades turning morphometrics into a quantitative science. The basic idea, he says, dates back to the sixteenth century and the work of the German artist Albrecht Dürer. Inspired, perhaps, by the recent discovery of perspective geometry, Dürer tried laying grid lines over the faces in his portraits. By moving the lines, while keeping the features of the face in the same position relative to the grid, he could transform the face any way he wanted, turning a bluff forehead into a sloping one, a weak chin into a lantern jaw.
Bookstein’s modern variations on Dürer’s theme—four little fun house faces—peer down from a bulletin board outside his office. One is a photograph of Bookstein looking like a dour version of Billy Crystal; the other three are “not-Freds”—computer-generated caricatures of the first photograph. To make the caricatures, Bookstein first scanned his photograph into his computer. Then he attached a grid to 13 “landmarks” on the face, such as the top of the forehead and the tips of the ears. When he was done, he simply moved a few of the landmark points around, thus forcing the grid to warp and bend as if a thin metal plate were attached to it. Engineers, it turns out, have used such “thin-plate splines” for years. But it was Bookstein who realized that these composite images are a perfect way to represent changes in what morphometricians call “shape space”—and to detect shape differences both large and small.
