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.
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