In the 1884 novel Flatland, a square living on a plane bemoans his two-dimensional life. "Life is somewhat dull in Flatland," he writes. "How can it be otherwise, when all one's prospect, all one's landscapes, historical pieces, portraits, flowers, still life, are nothing but a single line?"
The Flatlander knows what he is missing because he has just been carried by a teacher into Spaceland, the third dimension. At
UC Davis, mathematics professor William Thurston takes his students on a similar trip, into the world of multidimensional thinking.
But while the square was impoverished by his new awareness, Thurston's students are enriched. After a few classes of "Geometry and the Imagination," objects as mundane as apple peels or bathroom tiles and as exotic as the expanding universe all assume new character and complexity.
"I just thought this class was going to be on geometry and shapes," said junior Heather Witt, a chemistry major. "I had no idea how much it would apply to everyday things. I've never looked at vegetables, for instance, and thought they were mathematically related. And they are."
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Midway through the quarter, Thurston and co-teacher Ian Agol, a visiting research assistant professor, come to class carrying a box brimming with multicolored cardboard "n-gons" -- polyhedrons shaped like pyramids, cubes, soccer balls, quonset huts and the Astrodome.
Thurston eases into an explanation of how to arrive at the sum of the angles of an n-gon. He begins by calculating the angles of a flat, green pentagon he holds against the chalkboard, then moves quickly to the boxful of 3D shapes. Soon groups of students are counting vertices and multiplying angles, twirling the n-gons around on their fingertips and linking their hands together as they account for all the faces.
As the students' results pile up, they begin to smile and whisper to one another. It soon becomes clear that each student has reached the same sum for every shape -- 720 degrees -- and gained a new grasp of a basic theorem. In the back row, students are now flying a cardboard UFO over a cardboard observatory.
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There is no Nobel Prize for mathematics. Instead, there is the Fields Medal. First awarded in 1936, the Fields Medal is given every four years to mathematicians under the age of 40 for both completed work and the promise of future achievement.
The son of a homemaker and an aeronautical engineer, Bill Thurston thought as early as age 8 or 9 that he might become a mathematician, but then he suffered through a series of badly taught mathematics classes, the kind that "act as a big filter, where each year only half of the class continues to the next level," he says.
When he went to New College in Sarasota, Fla., Thurston was leaning toward biology, neurophysiology or psychology. His professors urged him to focus on math, though, and after graduation he headed to UC Berkeley to study under the esteemed mathematician Morris Hirsch. Thurston earned his doctorate in 1972, taught at the Institute for Advanced Study in Princeton and at MIT, and then joined the faculty of Princeton University.
A flood of fellowships and awards soon recognized his work in the field of topology, the study of properties of shapes that stay the same, even while they are stretched, bent and distorted. He settled longstanding questions on the existence of foliations, which are a type of layered structure for surfaces of different dimensions. He also found a correspondence be-tween three-dimensional topology and geometric symmetry. In the process, he revived some dormant areas of geometrical study.
"It is evident that Thurston's contributions to the field ... are of considerable depth. However, what sets them apart is their marvelous originality," wrote Blaine Lawson, now a distinguished professor of mathematics at State University of New York at Stony Brook. Dennis Sullivan, now also a distinguished professor of mathematics at Stony Brook, said, "Thurston's results are surprising and beautiful. ... [They offer] powerful geometrical estimation on the one hand, and spatial visualization and imagination on the other, which are truly remarkable."
In 1982, at age 37, William Paul Thurston won the Fields Medal.
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At noon Wednesday at the Davis Farmer's Market, the Fields medalist is buying produce for his next class exercise. Later, in the classroom, he starts handing out scissors and vegetable peelers.
Today's assignment is to calculate the total curvature of irregular shapes. "What is it that determines the intrinsic geometry of the surface?" he queries. The students cut peelings from the circumference of apples, bananas and oranges, then tape them flat on paper and calculate angles. The students work in groups; those who have grasped the concept help those who haven't.
"Here's a packet of some other kinds of surfaces," Thur-ston says, holding up something leafy with extravagantly curly edges. "Do you know what they are called? These are called mustard greens. I think." He takes a bite. "Yeah, they are mustard greens. Delicious." Then he shreds them in the name of mathematics.
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In 1991, Thurston returned to UC Berkeley as a faculty member. In 1993, he became director of the Mathematical Sciences Research Institute there. One of his innovations as director was to establish occasional meetings between mathematics researchers and schoolteachers, usually from the high-school level, in his continuing efforts to improve math teaching. He also worked to bring more women and minority researchers to the institute, to redress what he calls the prevailing "cultural homogeneity" in mathematics of white and Asian males.
In 1996, when his wife became a student at the UC Davis School of Veterinary Medicine, Thurston and his unconventional outlook joined the Davis mathematics department.
His current research is largely focused on understanding the topology of "3D manifolds," which are simple structures linked in a system that allows three degrees of freedom, such as a satellite orbiting the moon, while both orbit the Earth, while all three orbit the sun. The work could lead to a theory of the topology of the universe.
Thinking in these dimensions takes a strong sense of the connectivity of mathematics to the world, Thurston says. That connectivity is what he tries to convey to his students by making banana skins into surrogates for black holes. "People need an intuitive sense of mathematics, a mental model of the lay of the land," he said. "I think often the intuition is lost or denatured by a lot of teachers' presentations. It's very mysterious, what makes people catch on or not catch on. I am very interested in trying to get to the bottom of that."
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The fruit-and-veggie-salad class is winding down; the room is filled with the scent of fresh oranges. Thurston looks out the window. "It's early spring, so there are a lot of leaves that aren't out yet. But let me encourage you to look at leaves because they have an interesting variety of shapes," he tells the students. He describes some climate-related differences in leaf geometry, then shifts seamlessly to explaining how sewing the curved shapes of armholes is an exercise in geometry.
After class ends, the Geometry and the Imagination students are still sorting out their curly mustard leaves and flattened apple peels. "I was curious about taking this class, since it was taught by a Fields medalist," says senior Mindy Flanary, a mathematics major. "Now that I've done it, I really see things in a different way." Other students chime in with their own new awareness of geometry in the world around them, in the shapes of wire whisks, bed frames and bicycle tracks.
It's the legacy of a trip to Thurstonland.