Crocheting Bour’s Minimal Surfaces

Mathematics and crochet might not appear the most likely pairing for most people. However, crocheting is an inherently mathematical process. You can create various shapes using stitches with different heights and increasing or decreasing the number of stitches in certain places. Crochet also makes it possible to create many shapes that are very difficult to make with any other technique. One can crochet shapes such as the Klein bottle and Seifert surfaces of knots in freestyle, without following exact instructions, since these surfaces do not have a strict shape. A Klein bottle can be short and wide or tall and narrow. But there are many surfaces, for example spheres and disks, that have a specific shape. To crochet such models, you need crochet instructions, and to create such instructions, you need a good understanding of the underlying mathematical model.

The idea of knitting or crocheting mathematical or scientific models is not new, though it has not been used very widely. The Scottish chemist Alexander Crum Brown knitted several interlinked surfaces to visualize the ideas presented in the late nineteenth century in his paper “On a Case of Interlacing Surfaces” [2]. Miles Reid wrote a paper on knitting mathematical surfaces in the 1970s [13] that inspired several new patterns, including a Möbius scarf and a Klein bottle. The crocheted hyperbolic surfaces were introduced by Daina Taimina in 1997 [5], and her idea led to a bloom of so-called hyperbolic crochet. A few years after the paper on hyperbolic crochet appeared, Hinke Osinga and Bernd Krauskopf described how to crochet an approximation of the Lorenz manifold [10]. See also [14] for further examples of mathematical crochet.

Both the hyperbolic plane and the Lorenz manifold require precise crochet instructions. The hyperbolic plane has constant negative Gaussian curvature, and so it looks the same at every point. This allows for a rather simple pattern that can be worked in rounds in which after a few setup rounds, every nth stitch is doubled. The Lorenz manifold is a less-regular surface, and it requires a much more complex pattern of stitches. The model is also worked in rounds, but unlike the hyperbolic surface, it requires detailed instructions on when to add or remove stitches. It takes full advantage of the versatility of crocheting, requiring three different types of stitches, which allows different parts of a round to have different heights. In this paper we consider Bour’s minimal surfaces
, which are “crochet symmetric,” allowing for simple crochet instructions (excluding possible intersections) and requiring only one type of stitch, with the added or removed stitches evenly spaced across a round.