Week 8 - Reading Summary

belcastro, s-m. (2013). Adventures in mathematical knitting. American Scientist, https://www.americanscientist.org/article/adventures-in-mathematical-knitting 

 All images are from the article.

 This article in American Scientist is written by sarah-marie belcastro with whom I had the opportunity to interact on Zoom two weeks ago when Susan invited us to the Dance, Movement, and Mathematics group meeting. She has co-published a couple of books with Carolyn Yackel (who was the presenter in “How orbifolds inform shibori dyeing” video that was an option in this week’s viewing): Making Mathematics with Needlework (2007) and Crafting by Concepts (2011).

In this piece, belcastro writes about knitting mathematical models, noting that these knitted models make good teaching aids because they are flexible and are easily manipulated. Additionally, she comments on how a knitter must understand the mathematical concept well in order to craft a model but also how the knitting process itself provides mathematical insight.

Knitting as Geometry

Knitting geometry -
rectangular mesh/grid

belcastro describes the geometry of knitting and likens it to a grid or mesh within a coordinate system. She indicates that shaped knitting is a manifold, meaning that the coordinate system is only consistent in a local section of a piece, but not over the whole piece. Knitted objects can “stand in for 2D objects” just as paper is often used, and therefore can act like the skin of 3D objects.  In this way, belcastro is able to make knitted Klein bottles, a mathematical object that in which the “inside is contiguous with the outside.” She has been working on improving knitted Klein bottles for over 20 years.

Left: image of a Klein bottle; Right: Knitted Klein bottle

Getting the Math Right

belcastro insists that the mathematics in knitted models be “mathematically faithful” to the mathematical object. The model may end up highlighting some mathematical features over others and belcastro takes this into consideration when she plans the construction; she makes her patterns in order to feature certain properties. When knitting surfaces, which is the bulk of her mathematical knitting work, she has criteria for the projects in order to maintain the mathematics: there must be no edges or bound stitches, any transition between yarns must be invisible, and textures must be consistent to the mathematical object. For this last criterion, this means that if a surface transitions from outside to the inside, as in a Klein bottle, then the texture on the inside and outside of a knitted model must be the same. Conversely, if more than one texture is needed, the knitted model must reflect this.

Some of belcastro’s models have one mathematical design knitted into another. The example she gives of this is having a Mobius band knitted into a knitted Klein bottle. This leads to a set of challenges such as the difficulty of creating smooth lines out of rectangular stitches, the inherent difficulty of the rectangular mesh created by knitted stiches (as opposed to square mesh, that might be more ideal to work with), knitting a line into a bumpy surface, and knitting multiple line segments into a curved surface. Some of these challenges are illustrated through discussion and photos of models of Torus and Torus knots.

Knitted torus with curved lines embedded in the design

 Design Process

The article concludes with a description of the design process that belcastro uses. First she decides on the model and what the goals are for the mathematics (what mathematics should be featured or highlighted). Next, she plans both the large structure and the fine structure. Finally, she produces a pattern as the last step before knitting.