Mathematical knitting

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.

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.

Adventures in Mathematical Knitting [Sarah-Marie Belcastro/American Scientist] (Thanks, Sigma Xi Member!)


  1. I know a few weavers who do not knit much, precisely because it isn’t ‘mathy’ enough for them, and several knitters who did not gravitate toward weaving because there was too much math involved.

    OTOH, I’ve met three engineers (all male) who knit very complicated patterns and will be delighted to purchase any book that further tickles their brain lobes with mathematics.

    Prediction:  All the hipsters will be wearing Klein bottle hats in 2014.

    1. If you will be wearing a Klein bottle hat in 2014, you are in fact wearing a Klein bottle hat right now in 2013.

      Meanwhile, in 2014, your head is the case study:  this is your brain in 2014, this is your brain in a Klein bottle hat ….

  2. I’ve been knitting since I was 9, and this is not even close to how I would knit a more-or-less cylindrical object.  Double point needles exist for exactly this kind of work, and I really do not like seams. 

  3. I feel like I NEED to toss in my hat for Cliff Stoll, the genius over at, who has been making these, and glass zero volume four dimensional projections for many years. Come for the beautiful handblown glass, including the biggest klein bottle ever, but stay for the hilarious math in jokes and ramblingfs

  4. There’s almost always a “knitting” puzzle as part of the MIT Mystery Hunt (this year was no exception) because it’s clearly one of those hobbies that intersects “geek” space in an unusual way.

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