More than Knitted Eyebrows
For twenty years, mathematician sarah-marie belcastro knits mathematical surfaces to help others visualize three-dimensional mathematical objects. Her surfaces are tactile, can be manipulated...and often worn.
A deep understanding of the structure is necessary before she can figure out how to best knit a given surface. For example, Klein bottles are impossible three-dimensional shapes, yet can be "approximated" by knitted structures.
In the February (2013) issue of American Scientist, the feature article focuses on belcastro's knitting process and knitted products. Her examples of knitted Möbius bands and Klein bottles abound.
Plus, belcastro describes the process of using knitting to create mathematical manifolds, mentioning considerations such as yarn used, stich choice/pattern, and start-to-end creation of the structure. Her frustrations and triumphs in improving her knitted Klein bottles led to an interesting question: "What does it mean for a knitted object to be mathematically faithful?"
Part of her response focuses on the choices the knitter makes in deciding "which mathematical aspects of the object are most important." To illustrate, she continues: "Most of the objects I make have both topological and geometric aspects. That is, these objects have an overall shape that is preserved when the objects are bent or stretched, and they have a specific form and structure in space. Sometimes the topology takes precedence and sometimes the geometry; this difference dictates whether and where curvature is placed on the object. Often I knit surfaces. These are 2D and mainly smooth, which means that the constructions should have no seams, visible or otherwise."
Other issues complicate the design and creation process: surface texture, mesh, and the inclusion of lines or illustrative designs (e.g. complete graph K7 on a torus). And, she has the examples to document both the struggles and the successes.
The article is an enjoyable read....so try to read it while is still available FREE on-line. Or, buy a copy...both will save you trying to hunt it down in a library at a later date.
Note: My thanks to R.B. (Bellingham) for teling me about this article. Also, if you want to explore these ideas further, consider a visit to the ... The Institute for Figuring.