FEATURE ARTICLE

# Adventures in Mathematical Knitting

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

# The Design Process

So, what exactly does the design process for a mathematical object entail? Here is how I proceed. After deciding on an object to model, I articulate my mathematical goals (in practice, I often do this unconsciously). The chosen goals impose knitting constraints. This gives me a frame in which to create the overall knitting construction for the large-scale structure of the object. Then I must consider the object’s fine structure. Are there particular aspects of the mathematics that I can emphasize with color or surface design? Are particular textures needed? While solving the resulting discretization problem, I usually produce a pattern I can follow—my memory is terrible and I would otherwise lose the work.

A recent mathematical creation can serve as a case study. A diagram in Allen Hatcher’s *Algebraic Topology* had caught my eye, and I thought it would look fantastic knitted. The object shown is an equilateral Y extruded to be a three-finned thing with one end rotated by 1/3 and glued to the other end. Although, unlike a Möbius band, the object is not a manifold, it is a generalization of a Möbius band. So I thought I could use a similar construction—if only I could devise a way to knit outward from the central circle (the center of the Y). I wanted the knitted object to be created from a single strand of yarn, because the mathematical object has a single edge. Thus, I had to create a way to use a single strand of yarn to produce three interlocking sets of free stitches. (Ordinary knitting has only two sets of stitches, upper and lower, per strand of yarn.) Once I had solved that problem—and it took me a while—I decided to use a texture that would look the same from all viewpoints, so that the central circle would be less visible. For my first attempt at the object, I decided to keep things simple and add no more requirements. The result is shown in Figure 12. After my first attempt was done, I took one look at it and realized that it resembled a cowl. I resized the next version to produce a garment. A wearable mathematical object is a rare, and welcome, practical result.

Although I have worked on various knitting projects, I’m still not finished fiddling with designs for the Klein bottle—and it’s been about 20 years since I began. I have been asked to adapt my construction into a wearable hat. It’s one among many mathematical knitting challenges I look forward to completing.

- belcastro, s.-m. The Home of Mathematical Knitting. http://www.toroidalsnark.net/mathknit.html.
- belcastro, s.-m. 2009. Every topological surface can be knit: A proof.
*Journal of Mathematics and the Arts*3:2, 67–83. - belcastro, s.-m., and C. Yackel, eds. 2007.
*Making Mathematics with Needlework: Ten Papers and Ten Projects.*Natick, MA: AK Peters. - belcastro, s.-m., and C. Yackel, eds. 2011.
*Crafting by Concepts: Fiber Arts and Mathematics.*Natick, MA: AK Peters. - Dayne, B. 2003. Geek chic.
*Interweave Knits,*Fall, - Doyle, W. P. 2011. Past Professors: Alexander Crum Brown (1838–1922). University of Edinburgh School of Chemistry website. http://www.chem.ed.ac.uk/about/professors/crum-brown.html.
- Hatcher, A. 2002.
*Algebraic Topology*. Cambridge: Cambridge University Press. Also available at http://www.math.cornell.edu/~hatcher/AT/ATpage.html. - Reid, M. O. 1971. The knitting of surfaces.
*Eureka–The Journal of the Archimedeans*34:21–26. - Walker, J. 1923. Obituary notices: Alexander Crum Brown.
*Journal of the Chemical Society, Transactions*123:3422–3431. - Yuksel, C., J. M. Kaldor, D. L. James and S. Marschner. 2012. Stitch meshes for modeling knitted clothing with yarn-level detail.
*ACM Transactions on Graphics*(Proceedings of SIGGRAPH 2012) 31:3. Available with video at http://www.cemyuksel.com/research/stitchmeshes/. - Zimmerman, E. 1989.
*Knitting Around.*Pittsville, WI: Schoolhouse Press.

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