Rendering mathematical surfaces and objects in tactile form requires both time and creativity

sarah-marie belcastro

I have known how to knit since elementary school, but I can’t quite remember when I first started knitting mathematical objects. At the latest, it was during my first year of graduate school. I knitted a lot that year, because I never got enough sleep and needed to keep myself awake during class. During the fall term I made a sweater for my dad, finishing the seams right after my last final, and in the spring I completed a sweater for my mom. Also that spring, during topology class, I knitted a Klein bottle, a mathematical surface that is infinitely thin but formed in such a way that its inside is contiguous with its outside (see Figure 1). I finished the object during a lecture. It was imperfect, but I was excited, and at the end of class I threw it to the professor so he could have a look.

Over the years I’ve knitted many Klein bottles, as well as other mathematical objects, and have continually improved my designs. When I began knitting mathematical objects, I was not aware of any earlier such work. But people have been expressing mathematics through knitting for a long time. The oldest known knitted mathematical surfaces were created by Scottish chemistry professor Alexander Crum Brown. (For more about Crum Brown's work, click the image at right). In 1971, Miles Reid of the University of Warwick published a paper on knitting surfaces. In the mid-1990s, a technique for knitting Möbius bands from Reid’s paper was reproduced and spread via the then-new Internet. (Nonmathematician knitters also created patterns for Möbius bands; one, designed to be worn as a scarf, was created by Elizabeth Zimmerman in 1989.) Reid’s pattern made its way to me somehow, and it became the inspiration for a new design for the Klein bottle. Math knitting has caught on a bit more since then, and many new patterns are available. Some of these are included in two volumes I coedited with Carolyn Yackel: Making Mathematics with Needlework (2007) and Crafting by Concepts (2011).

You might wonder why one would want to knit mathematical objects. One reason is that the finished objects make good teaching aids; a knitted object is flexible and can be physically manipulated, unlike beautiful and mathematically perfect computer graphics. And the process itself offers insights: In creating an object anew, not following someone else’s pattern, there is deep understanding to be gained. To craft a physical instantiation of an abstraction, one must understand the abstraction’s structure well enough to decide which properties to highlight. Such decisions are a crucial part of the design process, but for the specifics to make sense, we must first consider knitting geometrically.