Week 9 Readings: Mathematics & Fibre, Fashion and Culinary Arts

Reference:  

Belcastro, S. M. (2013, March). Adventures in Mathematical Knitting. American Scientist, 101(2), 124. DOI: 10.1511/2013.101.124 

Summary: Belcastro recounts her journey in knitting a Klein bottle - an object known as a manifold where local coordinates are consistent, but not consistent globally. She describes how her prototypes were not mathematically faithful as technique, materials and tools each came with their own limitations adding up to corners that did not line up. Mathematical faithfulness when knitting a proper representation involves (1) making transitions between materials seamless, smooth, and invisible with no edges or bound stitches (2) surface texture considerations that mimic the original object. One major issue with trying to develop a smooth line or patch on a knitted 3D surface is in discretizing - transforming something continuous into discrete points to be knitted within a sequence of stitches.  

Stop 1:

"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." (Belcastro, 2013)

Belcastro mentions being inspired by certain mathematical objects or representations, like JoAnne Growney with poetic inspirations from Week 8 readings, and then goes about trying to figure out how to develop a pattern that could be knit so as to be a flexible manipulable object for investigation. This process resembles that of first foraging through past concepts, then dissecting the concept to discretize and represent the concept in 2D space as a pattern for 3D creation, and (re)enacting or generating the pattern with the materials, tools and techniques to best be faithful to the mathematics of the object. These concepts connect really well to Vogelstein, Brady & Hall (2019). 

Connecting to my experiences of clothing design and fashion, in my twenties I would go downtown to trendy fashionable stores with exorbitant price tags and examine their clothing. Each garment of interest, I would analyze for its construction, layers, fabric, materials like zippers, buttons, etc. then go home and recreate the garment for a 1/5 of the price along with my own personal touches. It was like reverse engineering with flexible materials. The process is something I can relate to! 

Stop 2: 

"All knitting is the generation of global structure via choices made in local stitch creation" (Belcastro, 2013).

The image shows the 2D grid overlayed on the 3D knitted object.

Source: Figure 2 from https://www.americanscientist.org/article/adventures-in-mathematical-knitting from Barbara Aulicino.

The image shows the 2D grid overlayed on the 3D knitted object with increases that would actually be the same size as the other boxes of the grid.

Source: Figure 3 from https://www.americanscientist.org/article/adventures-in-mathematical-knitting from Barbara Aulicino.

Reminds me of small local changes to make huge global ones. And again, talk of grid-like knitting structures, and local changes (increase/decrease stitches) connects to our previous class in which we've discussed butterfly power in association with Chaos Theory (Renert, 2011). Although not contextually similar, but conceptually with local changes to number of stitches, we are changing the dynamic shape of the entire structure in the end and rigid rectilinear thinking would not allow us to accurately predict the final outcome.

Stop 3:

"The way an object is constructed, in any art or craft, highlights some of the object’s properties and obscures others. Modeling mathematical objects is no different: It requires that we make choices as to which mathematical aspects of the object are most important. When it’s possible to do so, I knit objects so that a particular set of properties is intrinsic to the construction" (Belcastro, 2013).

I understand this concept as a hobbyist leatherworker. Depending on which location of the body the leather was obtained, it has certain properties that I would either use to highlight/emphasize in my constructions, or try to mask. For example, any skin from around the joints of the arm pits, neck or, groin tend to be more thin and very flexible which can stretch and warp if pulled - these would not be good for straps or anything that requires tensile forces exerted upon it.   

Stop 4: 

"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" (Belcastro, 2013).

This paragraph sums up Belcastro's entire process. I find this to be valuable as a model for creation: I could replace certain words/ideas with that of the context of poetry, or dance, or baking, etc. and have a template with this mathematical foundation in mind. I have a similar process that I have never tried to articulate in writing before, I just DO, and parts of the process are there to help me remember, understand/conceptualize, etc. like diagramming. I believe I could use this design process and transform it into a checklist of sorts for my students as they develop their own techniques/processes; it may provide supplemental structure that could more often lead to creative success.

Through reading this paper on the fine points and challenges connected with the intricacies of knitting, similarly to JoAnne Growney reading 15 weeks of Walcott's poems, I have more of an appreciation for the craft by "giving it a lot of attention" (Glaz, 2019). I can visualize the adaptations needed to overcome 2D-3D challenges which is a step above what I understood before.

Wonders:

I've been thinking about the variety of ways we have embodied mathematics: We've looked at multi-party, movement based mathematics, literary representations, visual representations, auditory representations, and culinary representations (I may be missing a few). In terms of utility, which of these different artful ways of representing mathematics is most useful? I ask this because of the high amount of planning, understanding and second-by-second effort it takes to generate a knit object, which then yields a manipulable mathematical object that further helps to generate understanding in others. I would consider this high. 

(I know this is a slippery slope to go down, as there's potential for dismissal or devalorizing the ways that sit lower on the scale so I have not put a lot of thought into a hierarchy of experiences as they each have quite a range of uses.)

Separately:
It would be really neat to generate a chart of mathematical concepts (not by grade, as we all have different sets of curricula across Canada) and examples of how they could be represented using the arts. Perhaps, an easily accessible chart with links to instructions would make these kinds of mathematical connections more attractive to all math teachers out there. I wonder how hard it would be for all of us to create a living document with what we've done/tried across the cohorts... across the years... 

References:
Renert, M. (2011). Mathematics for Life: Sustainable Mathematics Education for the Learning of Mathematics, 31(1), 20-26. https://go.exlibris.link/m4ZZHhJC 

Vogelstein, L., Brady, C., & Hall, R. (2019). Reenacting mathematical concepts found in large-scale dance performance can provide both material and method for ensemble learning. ZDM: The International Journal on Mathematics Education, 51(2), 331–346.