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.