Knitting mathematical objects

Reading:

Belcastro, S. M. (2013). Adventures in mathematical knitting: rendering mathematical surfaces and objects in tactile form requires both time and creativity. American Scientist, 101(2), 124-134.

Summary:

In this reading, the author tells us about her process of constructing mathematical objects using knitting. She relates how she started and the challenges she has faced to translate abstract features from math objects through knitting techniques. She talks not only about her process of understanding the features of mathematical objects she wants to highlight but also about how she has developed different knitting techniques to express these mathematical features.

Source: Image generated by Bing.com using the prompt "knitting mathematical objects".

Long stop:

“You might wonder why one would want to knit mathematical objects. One reason is that the finished objects make good teaching aids; a knitted object is flexible and can be physically manipulated, unlike beautiful and mathematically perfect computer graphics. And the process itself offers insights: In creating an object anew, not following someone else’s pattern, there is deep understanding to be gained. To craft a physical instantiation of an abstraction, one must understand the abstraction’s structure well enough to decide which properties to highlight. Such decisions are a crucial part of the design process, but for the specifics to make sense, we must first consider knitting geometrically”.

[...]

“So, what exactly does the design process for a mathematical object entail? Here is how I proceed. After deciding on an object to model, I articulate my mathematical goals (in practice, I often do this unconsciously). The chosen goals impose knitting constraints. This gives me a frame in which to create the overall knitting construction for the large-scale structure of the object. Then I must consider the object’s fine structure. Are there particular aspects of the mathematics that I can emphasize with color or surface design? Are particular textures needed? While solving the resulting discretization problem, I usually produce a pattern I can follow—my memory is terrible and I would otherwise lose the work”.

This time I had a long stop connecting two parts of the text. My first stop was in the passage where the author explains her motivations to design knitting mathematical objects: i) build a manipulative and ii) understand the abstraction while building the physical object. While the first reason can be achieved just by buying a manipulative that was made for someone else, the second reason caught my attention. Dealing with the challenges of building a physical object of an abstraction can give you a better understanding of what is essential in the concept.

This led me to the second passage, where the author describes her challenges in translating abstract features through different materials and techniques. I got fascinated by thinking how her decisions may change, highlight, or hide the abstract features of the mathematical object. It is a great puzzle and work of engineering! I ended up thinking about the question below:

Question:

What should we consider when translating abstract concepts into physical objects? 

Source: Image generated by Bing.com using the prompt "knitting mathematical surfaces and objects in tactile form".