Sunday, July 31, 2011

Mathematical Knitting Network

More than 5,000 mathematicians come annually to the Joint Mathematics Meetings (JMM), held in January. In recent years, amidst the usual lectures, poster sessions, job interviews, and much else, these gatherings have also featured an evening event for those particularly interested in mathematical crafting.

Devoted to knitting, crocheting, beading, needlework, paper folding, and more, this informal session is organized by sarah-marie belcastro (Smith College) and Carolyn A. Yackel (Mercer University).

belcastro and Yackel edited and contributed to the book Making Mathematics with Needlework: Ten Papers and Ten Projects (A K Peters, 2007), which contains not only instructions for creating mathematical objects but also insights into the underlying mathematics.

The JMM knitting network event brings together a wide variety of people, both experts and beginners. Participants and projects can vary widely from year to year.

In the realm of counted cross stitchMary Day Shepherd (Northwest Missouri State University) creates painstakingly woven symmetry patterns. For the type of cloth and technique that she uses, the fabric is a grid of squares, and one cross stitch covers one square of the fabric. The only possible subdivision of this square is with a stitch that "covers" half a square on the diagonal, Shepherd says.

These features constrain the number of symmetry patterns that you can weave. Of the 17 possible wallpaper patterns, for example, only 12 can be done in counted cross stitch.

The 12 wallpaper patterns that can be done in counted cross stitch needlework.
Courtesy of Mary D. Shepherd.

Shepherd has also worked on frieze and rosette symmetry patterns. Rosette patterns, for example, give a nice visualization of the symmetries of a square (technically, the group D4 and all its subgroups), she says. See Shepherd's article "Groups, Symmetry and Other Explorations with Cross Stitch" (Word document) and her chapter "Symmetry Patterns in Cross Stitch" in the book Making Mathematics with Needlework.

Rosette patterns for visualizing the symmetries of a square (the dihedral group of the square).
Courtesy of Mary D. Shepherd.

David Jacob "Jake" Wildstrom (University of Louisville) has a passion for crocheting in relief.

One of the few fractals that's amenable to crochet is the Sierpinski triangle. Wildstrom has turned this remarkable geometric figure into blankets, wispy shawls, and even a hat. His instructions for crocheting such figures can be found in the chapter "The Sierpinski Variations: Self-Similar Crochet" in Making Mathematics with Needlework. More information is available at Wildstrom's "crochetgeek" website.

Jake Wildstrom's relief crocheting has turned a fractal known as the Sierpinksi triangle into a shawl.
Photo by I. Peterson.

Tom Hull (Western New England University) specializes in mathematical origami design.

One of Tom Hull's modular origami creations.
Photo by I. Peterson.

Laura M. Shea strings tiny crystal beads to form polyhedra or geometric tilings.

This beadwork bracelet, created by Laura Shea, is based on a triangular tiling.
Photo by I. Peterson.

Creating polyhedra with beads is an interesting way to learn the properties of regular and semi-regular solids, Shea says. In a bead polyhedron, each face becomes open space, each edge becomes one bead, and each vertex becomes a thread void. The resulting structure is light and open.

The "Plato Bead," created by Laura Shea, is a dodecahedron. A bead stands in for each of this polyhedron's 30 edges. Each of the 20 vertices becomes a void surrounded by three beads and thread. The 12 faces of the form become open spaces.
Courtesy of Laura Shea.

The Bridges website shows additional examples of Shea's work.

No mathematical crafts session of the knitting network at JMM would be complete without a Möbius strip—that mind-bending, one-sided, one-edged mathematical object. 

Josh Holden (Rose-Hulman Institute of Technology) displays a Möbius band that he crocheted.
Photo by I. Peterson.

Carolyn Yackel works on her knitting during a JMM knitting network session.
Photo by I. Peterson.

Originally posted Jan. 27, 2007


belcastro, s.-m., and C. Yackel. 2006. About knitting . . .Math Horizons 14(November):24-27.

Klarreich, E. 2006. Crafty geometryScience News 170(Dec. 23&30):411-413.

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