Adventures in Mathematical Knitting
Rendering mathematical surfaces and objects in tactile form requires both time and creativity
The Case of the Klein Bottle
As mentioned earlier, a Klein bottle is an abstract, infinitely thin mathematical surface, formed in such a way that its inside is contiguous with its outside. That is, if it were thickened enough have an inner skin separate from its outer skin, you could run a finger along the surface from any point on the outside to the corresponding point on the inside. Figure 5 shows a line drawing of a Klein bottle in three dimensions, along with a true 2D representation as a marked rectangle; we bend and stretch the rectangle through additional dimensions to match up the arrows and glue them together. As the 3D representation shows, the surface appears to pass through itself. At this place there must be a self-intersection or a hole.
In the terms of the 2D representation, my first Klein bottle was made with the knitting needles parallel to the single arrows in the rectangle, and the rounds of knitting formed in the direction of the double arrows (see Figure 5). It had a lot of flaws, but at the time, I only noticed one of them: It was immobile, stuck in one configuration in 3-space, because I had made it with a self-intersection. To address this problem, I promptly made a crappy Klein bottle with a hole (see Figure 6).
The shortcomings of my early Klein bottles fell into two categories: aesthetic and mathematical. Aesthetics first: These things were ugly! One reason is that I used terrible materials. I have always preferred to use circular knitting needles, which are composed of two pointed tips connected by a thin, flexible cable. Back in graduate school, I often used hacked-together needles, shortening the cables of needles from the 1960s by melting the plastic (which promptly fell apart) or constructing my own circular needles from wire and packing tape (which caught on the yarn incessantly). These efforts to make needles that were both short enough and flexible enough for the projects resulted in a frustrating knitting experience and less than lovely results. Later I was able to get bamboo needles with hollow cables, which can be successfully modified. And I have learned how to use standard knitting paraphernalia to achieve the results I desire.
It wasn’t just the needles, though—the yarn was a problem, too. That first Klein bottle was made from respectable but low-grade acrylic yarn. The second was made from a yarn of unknown provenance, which I found at a garage sale. It was awful to work with and had a sickening plastic-plus-fiberglass feel. These days, I tend to use high-quality rather than random yarns, and I choose the color, type of dyeing and texture of the yarn with the intended mathematics in mind. Finally, in designing the objects, I improvised without much thought for details, so the first Klein bottle is misshapen and the second has a weird vertical line on it (hidden in the figure).
Now for the mathematical issues. The approach I used in both cases was to knit something like a cylinder, pass one end of the cylinder through its side, and then graft the two ends of the cylinder together. One can build in a side slit (as I did with the second Klein bottle), or create a self-intersection by passing live stitches through the side of the cylinder (as I did with the first Klein bottle). Either way, using this method, one basically creates the Klein bottle from an identified rectangle, with the rows parallel to one pair of sides of the rectangle (single arrows in Figure 5) and the columns of stitches parallel to the other pair of sides of the rectangle (double arrows in Figure 5). It is very easy to graft the ends of the cylinder incorrectly. Unless the knitter introduces a half-twist before grafting, the four corners of the rectangle will not meet. Without the twist, the result is still a Klein bottle, but the coordinates suggested by the rows and columns of stitches do not match the coordinates of the underlying rectangle, and the geometry of the finished object is not as desired. A knitted cylinder, however, resists twisting—thus the difficulty in this step.
I eventually saw a solution to this problem: Create the twist intrinsically rather than adding it at the end. I accomplished this by starting with Miles Reid’s Möbius band construction. This method effectively places the needles parallel to the double arrows shown in Figure 5, and knitting proceeds up both of the single arrows at once, each row wrapping twice across the rectangle. Because the twist is already present in the knitted fabric, the desired joining of the underlying rectangle happens automatically, with coordinates that match the rows and columns of stitches. Skeletal instructions for knitting a Klein bottle in this fashion are available at my mathematical knitting website, and a photograph of one is shown in Figure 7.
That solution was the product of years of thought. Once I realized how flawed my first Klein bottles were, I wanted to do better, to create a design that was more faithful to the mathematics. Thus the question arose: What does it mean for a knitted object to be mathematically faithful?