In 1997, as Daina Taimina geared up to teach an undergraduate-geometry class, she faced a challenge. As a visiting mathematician at Cornell University, she planned to cover the basic geometries of three types of surfaces: planar, or Euclidean; spherical; and hyperbolic. She knew that everyone can use intuition to conceive of the first two geometries, which are the realms of, say, sheets of paper and basketballs. The hyperbolic plane, however, lies outside of daily experience of the physical world.

Geometry teachers usually try to explain the hyperbolic plane via flat models that wildly distort its geometry-making lines look like semicircles, for instance. How, Taimina wondered, could she give her students a feel for hyperbolic geometry’s counterintuitive properties? While attending a workshop, the answer came to her: Crochet a piece of hyperbolic fabric.

In a flat plane or a sphere, the circumference of a circle grows at most linearly as the radius increases. By contrast, in the hyperbolic plane, the circumference of a circle grows exponentially. As a result, the hyperbolic plane is somewhat like a carpet that, too big for its room, buckles and flares out more and more as it grows.

In 1901, mathematician David Hilbert proved that because of this buckling, it’s impossible to build a smooth model of the hyperbolic plane. His result, however, left the door open for models that are not perfectly smooth.

In the 1970s, William Thurston, now also at Cornell, described a way to build an approximate physical model of the hyperbolic plane by taping together paper arcs into rings whose circumferences grow exponentially. However, these models take many hours to build and are so fragile that they generally need to be protected from much rough-and-tumble hands-on study.

Taimina realized that she could crochet a durable model of the hyperbolic plane using a simple rule: Increase the number of stitches in each row by a fixed factor, by adding a new stitch after, for instance, every two (or three or four or n) stitches. In 2001, Taimina and her Cornell colleague David Henderson proved that the crocheted objects indeed capture the geometry of the hyperbolic plane. Over the past decade, Taimina has crocheted dozens of these models.

Taimina’s models have made it easy to study hyperbolic lines-the shortest paths between two points on the hyperbolic plane. Given two points, all that’s necessary is to grab each point and gently pull tight the fabric between them. The line can then be marked, for future reference, by sewing yarn along it.

Taimina has used these sewn lines in the classroom to illustrate the hyperbolic plane’s most famous property. The plane violates Euclid’s parallel postulate, which states that given a line and a point off the line, there is just one line through the point that never meets the given line. By sewing lines with yarn, Taimina’s students have observed that in the hyperbolic plane there are, in fact, infinitely many lines through a given point that never meet a given line. Loosely speaking, this happens because the hyperbolic plane’s extreme flaring makes certain lines veer away from each other instead of intersecting as they would in a flat plane.

Because the hyperbolic plane is so hard to visualize, Taimina’s crocheted models are helping even seasoned mathematicians develop a better intuition for its properties. Taimina recalls that one mathematician, upon examining one of her hyperbolic planes, exclaimed, “So that’s what they look like!”

Taimina has crocheted models for many mathematics departments and for the Smithsonian Institution as examples of math teaching tools, but she now thinks twice before agreeing to make someone a model. Because of the exponential growth, crocheting a hyperbolic plane takes a long time. For instance, one of Taimina’s models started with a 1.5-inch row, but the 20th row was already more than 30 feet long. What’s more, the crochet work is hard on the hands, Taimina says, since the stitches must be tight to prevent the fabric from stretching out of its characteristic hyperbolic shape. Luckily for Taimina, many mathematicians “are now enthusiastically making their own models,” she says.

Taimina’s hyperbolic planes have also attracted interest from art lovers. Her models have appeared in art shows all over the United States, and some are currently on display in Latvia and Italy.

“I have met so many people now who don’t have a math background, but who want to understand what these hyperbolic planes mean,” Taimina says. “It makes me happy that people can learn beautiful geometry and not be intimidated.”

Klarreich, Erica. “CRAFTY GEOMETRY.” Science News 170.26/27 (23 Dec. 2006): 411-413. MasterFILE Premier. EBSCO. [Library name], [City], [State abbreviation]. 8 Apr. 2009 <http://0-search.ebscohost.com.millennium.mohave.edu/login.aspx?direct=true&db=f5h&AN=23560436&site=ehost-live>.