Sarah-Marie Belcastro's long, lavishly illustrated article on the mathematics of knitting and mathematical knitting is a totally fascinating read: chewy, mathy, and inspiring. Makes me want to go and get some yarn.
Adventures in Mathematical Knitting [Sarah-Marie Belcastro/American Scientist] (Thanks, Sigma Xi Member!)
Over the years I’ve knitted many Klein bottles, as well as other mathematical objects, and have continually improved my designs. When I began knitting mathematical objects, I was not aware of any earlier such work. But people have been expressing mathematics through knitting for a long time. The oldest known knitted mathematical surfaces were created by Scottish chemistry professor Alexander Crum Brown. In 1971, Miles Reid of the University of Warwick published a paper on knitting surfaces. In the mid-1990s, a technique for knitting Möbius bands from Reid’s paper was reproduced and spread via the then-new Internet. (Nonmathematician knitters also created patterns for Möbius bands; one, designed to be worn as a scarf, was created by Elizabeth Zimmerman in 1989.) Reid’s pattern made its way to me somehow, and it became the inspiration for a new design for the Klein bottle. Math knitting has caught on a bit more since then, and many new patterns are available. Some of these are included in two volumes I coedited with Carolyn Yackel: Making Mathematics with Needlework (2007) and Crafting by Concepts (2011)...
...Most knitted-in designs are mathematically challenging because of the discretization problem: A smooth line or patch of color drawn on a piece of paper or electronically must be knitted as a sequence of discrete stitches. This harkens back to the mesh shown in Figures 3 and 4. Computer scientists who work on visualization of 3D objects have developed algorithms for imposing a mesh on an idealized object. A finer mesh gives a smoother look, and in fact the use of very fine meshes is what produces realistic computer-generated imagery. In knitting, creating a finer mesh requires both a thinner yarn and substantially more time to complete the project. A great application of meshes to knitting appears in a 2012 SIGGRAPH paper, in which Cem Yuksel, Jonathan M. Kaldor, Doug L. James and Steve Marschner explain how they use a mostly rectangular mesh to produce highly realistic virtual knitted garments.