While Taimina’s and Osinga’s models have achieved the most fame, a host of other mathematicians in recent years has started crocheting and knitting mathematical shapes. An exhibit of mathematically inspired fiber arts at the 2005 annual Joint Mathematics Meeting in Atlanta boasted an impressive array of such models. In addition to Taimina’s hyperbolic planes and a Lorenz surface crocheted by Yackel, the exhibit featured Möbius strips, which are twisted rings that have only one side, and Klein bottles, which are closed surfaces that have no inside. There were also crocheted versions of the five Platonic solids-the cube, the tetrahedron, the octahedron, the dodecahedron, and the icosahedron-as well as a bricklike fractal object called Menger’s sponge.

It’s not clear just why mathematical craftwork has suddenly taken off, says Sarah-Marie Belcastro, a mathematician at Smith College in Northampton (Mass.), who organized the exhibit with Yackel. “Part of me says it’s because there are so many more women in math now,” she says. “But every time we give talks, there are men in the audience who say they knit or crochet.”

For a gathering last March in Atlanta to honor mathematics writer Martin Gardner, belcastro and Yackel created doughnut-shaped surfaces, called tori. The patterns on their tori illustrate two well-known mathematical ideas about maps and networks on a torus.

Given a map showing several countries, consider the ways to color each country so that no neighboring countries have the same color. In 1976, mathematicians famously proved that in the flat plane, no such map would require more than four colors. On a torus, however, where there are more ways for a country to wrap around and touch another country, mathematicians showed as long ago as 1890 that as many as seven colors can be required. Yackel’s crocheted torus displays one seven-color map that, remarkably, has only seven countries on it-every country touches every other.

Belcastro’s knitted torus, which can be seen as a companion piece to Yackel’s, displays an intriguing fact about networks on the torus. The torus depicts a collection of points connected by paths. This network is derived from the map on Yackel’s torus by marking one point inside each country and then connecting each pair of points by a path, like a railroad line, that crosses the boundary between their respective countries. Such a network of seven points, each connected to every other by a path, can’t be drawn in the flat plane without some paths crossing. On the torus, however, as Belcastro’s knitting demonstrates, the paths can snake around the hole and avoid each other.

Belcastro and Yackel thought that making the tori would be a simple matter since pictures of the seven-color map and the corresponding network on the torus are readily available. However, it turned out to be “a nightmare,” belcastro says. The challenge was figuring out how to make lines and boundaries look smooth despite the discrete nature of the stitching.

Yackel and belcastro are now editing a book to be called Making Mathematics with Needlework. It will feature patterns and mathematical discussions of 10 craft projects, including knitting, crocheting, embroidery, and quilting. The book isn’t due out until spring. Nevertheless, this holiday season, instead of the ubiquitous gift sweater, you might want to consider knitting a Möbius scarf or a Klein bottle hat, or crocheting some hyperbolic Christmas tree ornaments.

Klarreich, Erica. “CRAFTY GEOMETRY.” Science News 170.26/27 (23 Dec. 2006): 411-413. MasterFILE Premier. EBSCO. [Library name], [City], [State abbreviation]. 9 Apr. 2009 <http://0-search.ebscohost.com.millennium.mohave.edu/login.aspx?direct=true&db=f5h&AN=23560436&site=ehost-live>.