On a Thursday night in Ithaca, New York, Daina Taimina, an ebullient blond mathematician at Cornell University, sits at her kitchen table with her husband, David Henderson, a Cornell professor of geometry. In front of her sits a big Chinese bowl filled with crinkled forms made of gray, blue, red, and purple yarn. Reaching into the bowl, Taimina pulls out a woozy multicolored surface, the likes of which would have delighted Dr. Seuss.
"This is an octagon with a 45 degree angle at every juncture," she explains, displaying a familiar eight-sided shape outlined in white on the curvilinear surface of the wool. "And when you put it together," she continues, folding the material together so that the opposite corners of the octagon touch, "you get this." Just as the ends of a flat piece of paper can be joined to form a cylinder, so the ends of Taimina's woolly octagon can be joined to form a double cylinder. In front of my eyes, the octagon has been transformed into something that is also familiar but even stranger—a pair of hyperbolic woolen pants.
Holding the purple folds of wool in her hand, Taimina pauses to shoo her cat away from a dish of half-eaten vanilla ice cream, then reaches into the bowl again. She pulls out a colorful piece of knitting that looks as if it's been through the wash a few dozen times too many—and transforms it into something like a woolen cube with tubes protruding from the corners. She says it is four joined hyperbolic hexagons; a three-headed creature might be tempted to wear it as a sweater. "If you are a creature living on this plane, let's say an ant, it's still the same hyperbolic space. I can attach four more, like this," she says, assembling a miniature hyperbolic universe before my eyes.
What makes this performance magical is that it should be impossible. Hyperbolic geometry is a mathematical concept so convoluted that just about everyone has given up trying to imagine it. To understand how convoluted, try to remember what you can of the simplest form of geometry, the euclidean, or plane, geometry taught in middle schools. Its forms—triangles, squares—are simple because the rules are two-dimensional: Space does not curve, and the shortest distance between two points is a straight line. The planar world is the world that can be drawn on a flat piece of paper.
Slightly more mind bending is spherical geometry, which describes a world in which space has a constant positive curvature, like the surface of the planet Earth. The shortest distance between two points is still a straight line, but that line curves—imperceptibly, to a person on it—such that it eventually intersects itself. Although spherical geometry is less intuitive, it deals with shapes that are part of the familiar physical world. Using it, ships and airplanes can cross the oceans along "great circle routes" that look circuitous (when displayed on a flat map) but which in fact follow the straightest, quickest way across.
Hyperbolic geometry, conceived by mathematician Carl Gauss in 1816, is stranger still. Like planar geometry, it posits that the shortest distance between two points is a straight line. And hyperbolic space, like spherical space, has a constant curvature—except the curvature is negative rather than positive. Hyperbolic geometry describes a world that is curving away from itself at every point, making it the precise opposite of a sphere, whatever that might look like. (One is tempted to picture an inside-out sphere, but that still describes a positive curvature, since space is curving toward itself at each point.)
Gauss never published the idea, perhaps because he found it inelegant. In 1825 the Hungarian mathematician János Bolyai and the Russian mathematician Nicolay Lobachevsky independently rediscovered hyperbolic geometry. They declared that all the normal rules of euclidean geometry would apply to this geometry except for Euclid's parallel postulate, which states that if you have a straight line and a point not on that line, there exists at most one straight line that passes through the point and is parallel to the line. In hyperbolic space, more than one parallel line runs through that external point; in fact, an infinite number of them do.
The rediscovery of hyperbolic space was not greeted enthusiastically by the analytically oriented German and Austrian mathematicians who dominated mathematics in the West; they dreamed of a logical, orderly universe that could be represented through equations. Not until very recently—after the fall of the iron curtain—did the strange and illogical beauty of hyperbolic forms emerge yet again to claim the attention of mathematicians.
I ask Henderson how it is that shapes that cannot be imagined nonetheless can be found in his wife's knitting bowl. "A hundred years ago, the mathematician David Hilbert proved a theorem that it is impossible to represent the hyperbolic plane in three-dimensional space analytically," he says. " 'Analytically' means 'with equations.' Everybody left off the word analytically later on. They were worried that mistakes or errors would creep into mathematics through geometric intuition, and so they discouraged the study of geometry and everything associated with this weird kind of thinking."
The prejudice against a mathematics that could not be expressed strictly by equations did not exist when Taimina grew up in Latvia under Soviet-style math schooling. "We were taught to start with the picture," she recalls. "You figure out what is happening, and then you set out to prove it."
Because the Soviet system also encouraged shortages and the production of shoddy, unappealing goods, every woman learned how to knit and crochet. "You fix your own car, you fix your own faucet—anything," she says with an easy laugh. "When I was growing up, knitting or any other handiwork meant you could make a dress or a sweater different from everybody else's."
The first person to solve the problem of how to construct a simple physical model of the hyperbolic plane for classroom use was mathematician William Thurston, now a colleague of Taimina and Henderson's at Cornell. Unlike most of his American colleagues, Thurston never put much stock in the attempt to represent geometric intuition with mathematical equations.
Henderson's method of constructing a hyperbolic plane involved taping together thin, circular strips of paper. He learned the method from Thurston at a workshop at Bates College in 1978. Afterward, on a camping trip, he constructed his first hyperbolic plane using his Swiss army knife and some Scotch tape.
Twenty years later, Taimina remembers, Henderson was still using the same tattered model. When she was assigned to teach his class on hyperbolic geometry at Cornell, where she had an appointment as a visiting professor, she was forced to confront it.
"It was disgusting," Taimina recalls with a playful shake of her head. "So I spent the summer crocheting a classroom set of hyperbolic forms. We were sitting at the swimming pool with David's family, my girls were learning to speak English and swimming, and I was sitting and crocheting. People walked by, and they asked me, 'What are you doing?' And I answered, 'Oh, I'm crocheting the hyperbolic plane.' "
She begins by crocheting a short row of stitches. Onto that row she adds successive, concentric rows of stitches. The rows, or rings, increase exponentially in length: one additional stitch in every two loops of the previous rows, say, or two stitches in every five. As the number of stitches per row increases, the resulting form becomes wavy and scrunched. Precisely because hyperbolic space expands exponentially, Taimina explains, it requires crocheting rather than knitting. "In knitting, all the stitches you are working with, you have on your needles," she says, adding some stitches to a shape she is completing. "So given the rate of increase, very quickly you cannot move your needles." Crocheting doesn't require all the stitches to be held on the needles simultaneously, enabling Taimina to pack more stitches into a smaller space. Crocheted forms are also stiff enough to hold their shape.
"This is a very interesting one," she says, drawing my attention to a purple flower of yarn that looks like a sea anemone caught on her crocheting needle. The inner row of stitches is an inch and a half long. There appear to be 16 concentric rows in all. Taimina asks me to estimate the circumference of the crenulated purple sea anemone. I guess 24 feet.
"That's close enough," Henderson says. "It's 30 feet—369 inches."
Taimina corrects him: There are 22 rows between the first and last row, not 16. The rate of increase from one row to the next is amazing, and the resulting forms are unusually beautiful. What Taimina's crocheting magically reveals is that hyperbolic geometry is actually part of our everyday universe. Video-game designers can use hyperbolic geometry to create lifelike clothing and hair. Some neurologists even believe that the brain stores information according to the rules of hyperbolic geometry. Although physical applications for hyperbolic geometry are less than two decades old, they represent a profound shift in mathematical thinking, away from the dream of a perfect analytic universe toward a more open and intuitive one.
When I ask Taimina for examples of hyperbolic forms in her own life, she points out the window to the backyard. "It's too dark now," she says. "I can show you tomorrow in the garden. Crinkled parsley. Some lettuce. Wood ear mushrooms. They are all hyperbolic forms."
I dimly remember that astronomers have proposed that the universe may be hyperbolic, displaying a constant negative curvature. Henderson nods and says: "There is evidence now that if you go off in certain directions, you'll come back. Which could be spherical or hyperbolic."
That leads me back to the very intuitive idea that I have been harboring all evening: What better shape for the universe than a pair of hyperbolic pants?
"It's unlikely," Taimina says. To console me, she agrees to model the hyperbolic skirt that she made for a recent talk sponsored by the Institute for Figuring in Los Angeles, after which the film director Werner Herzog took her to dinner and then kissed her good night. The skirt is made of 10 skeins of cotton yarn, each of which is 689 feet long. "The ruffles divide into other ruffles," Henderson says, as Taimina spins around the living room. "That's how you can tell it's hyperbolic." When I suggest a clothing line for mathematicians, Taimina smiles.
"This is one of the hottest silhouettes this season," she says. "It's actually similar to a very old pattern called the godet skirt. It's made with six or eight panels, and it's known to flatter any figure. So you see why hyperbolic geometry is truly important."