I am a mathematician who knits as well as a knitter who does mathematics. It has always seemed natural to me to combine mathematics and knitting, and it is inevitable that sometimes the results will be artistic rather than functional. (It is perhaps too much to hope that I could combine artistry and function in addition to knitting and mathematics.)
Because the length of the boundary on a negatively curved surface-in-progress increases as the surface is formed, it is preferable to use crochet (with a hook in one boundary stitch) rather than knitting (with a needle running through all boundary stitches) to create a negatively curved surface. However, there is no known intrinsic-twist construction for a crocheted Mobius band that preserves symmetry near or on the central circle. Moreover, standard crochet stitches are not front-back or top-bottom symmetric, and so a nonorientable surface crocheted in a standard way would have a texture that showed these lack of symmetries. Therefore, this negatively curved Mobius band was constructed using an intrinsic-twist maximally symmetric knitted center, then a round of transferring the knitted stitches to crochet in such a way that the knitting stitch texture was preserved, and then several rounds of symmetrized single crochet with a symmetric increase added on every fifth stitch. (The crochet stitches were symmetrized by mimicking knitted seed stitch.) The regular increases create locally uniform negative curvature.
Because this piece is flexible, it can take on many different forms; a single photo cannot indicate the range of shapes in which it may appear.
The Heawood bound shows that K_7 is the largest complete graph that can embed on the torus. This is an embedding of K_7 on the torus with all vertices centered on the largest longitude. It is the second knitted instantiation of this embedding; this version is larger and has a larger face-to-edge proportion than the first, which was exhibited at Gathering for Gardner 7 in 2006.