11 March 2005
Vol. 307 No. 5715

Non-Euclidean Potholders

Knot theory is no longer the only branch of mathematics that appeals to the handicrafts set. A big crowd showed up last month at the Kitchen, one of Manhattan's hippest performance spaces, to hear a pair of Cornell mathematicians talk about hyperbolic space. Their main props: crocheted models of objects in the hyperbolic plane, a central concept in non-Euclidean geometry.

In ordinary Euclidean space, a flat plane stretches out forever and parallel lines never meet, explained geo-meters Daina Taimina and David Henderson.

However, as mathematicians discovered in the early 1800s, that's not true in other kinds of space. In the sphere, for example, parallel lines meet at the poles. In the hyperbolic plane, which can be thought of as the opposite of the sphere, parallel lines shy away from each other.

Hyperbolic space is very hard to picture. For more than a century, mathematicians struggled without notable success to make 3D models of it. Then Taimina had the idea of using her crochet hook. She and Henderson use crocheted models in their classes at Cornell and hope that when people create and play with the objects--which look like witches' hats, flamenco skirts, or curly kale--they'll develop an intuitive sense of what hyperbolic geometry is all about. "We all play with balls as children. With a sphere, you have that memory in your hands. But you don't have that with hyperbolic geometry," says Taimina.

CREDIT: CROCHETED MODEL OF A HYPERBOLIC "PSEUDOSPHERE" MADE BY DR. DAINA TAIMINA/IMAGE: THE INSTITUTE FOR FIGURING