Osinga launched her crochet project in the hopes of finally getting her hands on a Lorenz manifold, a mathematical object that she had been studying theoretically for years. Meteorologist Edward Lorenz, now an emeritus professor at the Massachusetts Institute of Technology, had set down three equations in 1963 as a highly simplified description of weather dynamics. These Lorenz equations have tremendous mathematical and historical significance. While simulating the equations’ dynamics on a computer, Lorenz found that tiny round-off errors result in hugely different outcomes, a discovery that launched the field of chaos theory.

Osinga explains that Lorenz’ equations describe a flow in three-dimensional space, and the Lorenz manifold corresponds to a certain specific part of a river. “If you throw a leaf in the water and watch it flow downstream toward a rock, the leaf might go to the right or left of the rock,” she says. “But there are particular points where, if you drop the leaf exactly there, it will flow down and get stuck on the rock.” The Lorenz manifold is the two-dimensional surface consisting of all the points where you can drop a leaf and it will flow to the rock, which is represented by the central point, or origin, in a three-dimensional coordinate space.

Since the system is chaotic, the Lorenz manifold twists around with many changes in curvature. To build a computer image of the surface, Osinga and Krauskopf devised an algorithm that starts at the origin and works its way outward in concentric rings. For each ring, the algorithm looks for points from which an object would flow to the origin. The algorithm can’t find all such points, since there are infinitely many, so instead it identifies a collection of prototypical points that are about evenly spaced along the surface and then connects neighboring points by links so that the resulting mesh will resemble the Lorenz surface. In areas where the surface has floppy, hyperbolic geometry, the algorithm will identify many mesh points; where the surface has more tightly curved geometry, the algorithm will identify fewer points.

Osinga realized that the mesh instructions could be read as a crochet pattern: Crochet outward in rings and simply add or remove stitches to suit the mesh pattern. As the fabric grew under her nimble fingers-Osinga has been crocheting since age 7-it automatically took on the curvature of the Lorenz manifold.

“Just local information about where to increase stitches created the entire global shape,” Osinga says. When Osinga had finished crocheting, she and Krauskopf mounted the fabric on garden wire, and it indeed took the shape of the Lorenz manifold, Osinga says.

Unlike Taimina’s hyperbolic planes, whose crochet instructions can be summed up in a single sentence, the instructions for the Lorenz surface fill two pages of a paper that Osinga and Krauskopf published in 2004. “An expert needleworker will be able to [crochet a hyperbolic plane] while having a nice conversation or watching TV,” the pair say in the paper. “Crocheting the Lorenz manifold, on the other hand, requires continuous attention to the instructions in order not to miss when to add or indeed remove an extra crochet stitch.”

Despite the difficulty of making a Lorenz manifold, Osinga hears regularly from crocheters trying to follow her pattern, which is available at a link from her Web site. “I get emails from crafters who are not at all scientifically inclined but want to understand what they are making,” she says. “They ask very intelligent math questions.”

Like Taimina’s hyperbolic planes, Osinga’s Lorenz manifold has taken to the road frequently since its construction, making appearances at mathematical conferences, at art shows, and even on television news. “In my teaching, the students take me way more seriously now,” she says. “This complicated math I do, which seems so useless, gets you on TV.”

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>.