FEATURE ARTICLE

# Adventures in Mathematical Knitting

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

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.

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