Crafty geometry: mathematicians are knitting and crocheting to visualize complex surfaces

Science News, Dec 23, 2006 by Erica Klarreich

During the 2002 winter holidays, mathematician Hinke Osinga was relaxing with some lace crochet work when her partner and mathematical collaborator Bernd Krauskopfasked, "Why don't you crochet something useful?" Some crocheters might bridle at the suggestion that lace is useless, but for Osinga, Krauskopf's question sparked an exciting idea. "I looked at him, and we thought the same thing at the same moment," Osinga recalls. "We realized that you could crochet the Lorenz manifold."

For years, Osinga and Krauskopf, both of the University of Bristol in England, had been studying the Lorenz manifold, a complicated surface that emerges from a model of chaotic weather systems. The pair had created an algorithm to generate 2-dimensional computer visualizations of the surface, but Osinga found the flat images unsatisfying. When Kranskopfasked his question, she suddenly realized that the computer algorithm could be interpreted as crochet instructions. "I had to try it" she says. Eighty-five hours and 25,511 crochet stitches later, Osinga had a Lorenz manifold almost a meter tall and about 25 centimeters in diameter, which now hangs in the pair's house as a decoration. Mathematics has long been an essential tool for the fiber arts. Knitters and crocheters use mathematical principles--often without recognizing them as such--to map the pattern of a cable sweater, for instance, or figure out how to space the stitches when adding a sleeve onto a jacket.

Now, the two crafts are returning the favor. In recent years, mathematicians such as Osinga have started knitting and crocheting concrete physical models of hard-to-visualize mathematical objects. One mathematician's crocheted models of a counter-intuitive shape called a hyperbolic plane are enabling her students and fellow mathematicians to gain new insight into startling properties. Other mathematicians have knitted or crocheted fractal objects, surfaces that have no inside or outside, and shapes whose patterns display mathematical theorems.

"Knitting and crocheting are helping us think about math we already know in a different light,' says Carolyn Yackel, a mathematician at Mercer University in Macon, Ga.

A HYPERBOLIC YARN 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 counter-intuitive 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 Thurstou, 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.