Sometimes, when we pose questions of mathematics, its answers are strikingly contrary. Why can't we trisect an angle with the same tools we use to bisect an angle? It's not possible. Why haven't we found the quintic formula? It doesn't exist. Can we at least prove that arithmetic is logically consistent? Nope!

We can view these results as intransigent obstacles to human knowledge, or we can accept them as fascinating illustrations of the boundaries of different mathematical techniques. In this talk, we will explore analogous results for techniques in the fiber arts. For each form of knitting, crochet, embroidery, and so forth, there is a set of limitations on what types of designs they can produce. Sometimes, these limits are broad enough that the the art form can encompass every mathematical possibility. Other times, the craft imposes intriguing restrictions on what patterns we can produce, and we will make the case that these restrictions have their own intrinsic beauty.