Week 9 Activity + Connections

 

I watched two of the videos (one by accident but couldn't take my eyes away... then accidentally clicked on the second link and couldn't stop watching that). 

Watching De Vries's (2016) Quilting Mathematics video, I collected a few vocabulary words: 

Tessellation or Tiling of the plane: a collection of two-dimensional shapes (tiles) that fill the plane with no overlaps and no gaps. 

Monohedral: tessellations use 1 tile of same shape and size, example: squares (grid), equilateral triangle, hexagon

Dihedral: tessellations use 2 different tiles

Trihedral: tessellations use 3 different tiles

There are 15 known monohedral tilings of convex pentagons.

Periodic tessellation: a tessellation contains translational symmetry.

Aperiodic tessellation: a tessellation that lacks translational symmetry.

Sir Roger Penrose discovered an aperiodic dihedral tiling:

This section of creating symmetry by De Vries connects very well with Belcastro & Schaffer's (2011) dancing mathematics article. 

Cell: smallest element in the pattern that is repeated using the symmetrical operators (T,R,Mirror,G).

Unit: smallest element in the pattern that is repeated by translation. The cell and unit could be the same depending on the translation type.

There are 17 different symmetry groups. 

This example below shows the difference between how a quilter builds the quilt (left) and how a mathematician might analyze the quilt (centre). Their "units" would be different! The symmetry group is on the right.

To conclude her talk, De Vries states "Using mathematical concepts and algorithms in the design of quilts can lead to endless variety. Recognizing mathematical concepts in quilts can surprise, inspire and delight." After she introduced her special set of tapered triangles and the mathematical patterning required to develop her themed quilts, I can see how each variation was constructed with math in mind.

De Vries mentions that she creates and self-imposes rules on her projects to generate something non-random. I wonder if this is what she deems is a mathematical behaviour. Is mathematics the imposition of rules, categorization and structure? Perhaps that should be a piece of the criteria of the Math/Art Manifesto?

I agree with De Vries in that seeing a pattern with some inherent rule is more aesthetically pleasing than seeing something randomly strewn together. Even when observing colours on a virulent tulip there is structure on a cellular level. 

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I tried the Miura-Ori fold that is used by Uyen Nguyen's (2021) fashion design. I really wanted to create myself an origami garment, but unfortunately I ran out of time. I might update this in the future with some sort of attempt. These three shapes answered various questions I had:

Top: Does the paper need to be 'origami paper' (square in shape)? NO

Left: Can I use an entire piece of paper without modifying its dimensions? YES

Right: Can I use the bookmark strip of paper left from cutting a square to do this fold? YES

I asked my students in Earth Science 11 to fold this special fold as a "fun" activity connected to our Astronomy unit. My students did not find this entertaining or fun at all. They struggled hard trying to invert the mountain and valley folds. Following this video by MATU (2020):

This is what remains of their participation for the day. A battlefield of creased papers. Some took the time to complete the shaping and turned out really well. It was much harder than it looked, and I hope it helped them develop more of an appreciation of this skill and its application in Astronomy. 

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Screenshot from Burkard Polster's (2020) Mathologer video What is the best way to lace your shoes? Dream Proof.

I absolutely loved this video. Very entertaining and well explained. I have explored playing with shoe lacings before, but I found that unless they are the criss-cross, they take a substantial amount of time to tighten/loosen to put on/off. The mathematical analysis behind shoe lacings is very interesting. I definitely have more of an appreciation for tight lacings now.

Links to viewing & activities from the week:

Viewing:

1) Carolyn Yackel: How orbifolds inform shibori dyeing (Gathering for Gardner, Oct. 2020, 28 min)

2) Gerda de Vries (University of Alberta) Quilts as mathematical objects (PIMS, UBC 2016, 1 hr)

3) Uyen Nguyen, Origami Fashion Part 1 and Part 2 (20 min total. Uyen recently had a solo show at the Museum of Mathematics – MoMATH – in New York City!) [Optionally, you might also be interested in taking a look at Uyen’s related Bridges paper Uyen Nguyen (Bridges 2020) Folding fabric: Fashion from origami]


Activity:

Choose one of the following to try on your own, or with your students, family or friends this week!

(1) Four Burnaby secondary math teachers (Goeson, Nicolidakis, Gamble and Houghland) developed this curricular work with Coast Salish weaving and mathematics. If you haven't worked with this before (at Indigenous Math Day at UBC, for example), here's a chance to give a try to weaving mathematics.

(2) Try out Miura Ora Origami (the technique Uyen Nguyen uses in her fashion design). Here are two instructional videos (A and B) -- and feel free to find other instructions if that suits you.


(3) Or for something completely different, try a variety of mathematically-interesting (and efficient) ways of lacing your shoes, as described in this Mathologer video!

References:

Belcastro, S. M., & Schaffer, K., (2011) Dancing Mathematics and the Mathematics of Dance. Math Horizons. 18(3). pp. 16-20. DOI: 10.4169/194762111X12954578042939

De Vries, G. (2016, August 18). Making mathematics with needle and thread: Quilts as mathematical objects [Video]. mathtube.org. https://www.mathtube.org/lecture/video/making-mathematics-needle-and-thread-quilts-mathematical-objects

MATU. (2020, November 9). Miura Ori - Traditionelle Miura-Faltung [Video]. YouTube. https://www.youtube.com/watch?v=EEGmnKKKhrk

Nguyen, U. (2021, January 7). Origami Fashion with Uyen Nguyen Part 1 [Video]. YouTube. https://www.youtube.com/watch?v=i4AoN1DtH6I

Nguyen, U. (2021, January 7). Origami Fashion with Uyen Nguyen Part 2 [Video]. YouTube. https://www.youtube.com/watch?v=bD7vUhdyO34

Polster, B. (2020, June 20). What is the best way to lace your shoes? Dream proof [Video]. YouTube. https://www.youtube.com/watch?v=CSw3Wqoim5M

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.

Culminating Project [DRAFT]: Mathematical Principles in Historical Swordplay

The Geometry of the Blade: Mathematical Principles in Historical Swordplay

By: Kristie Truell & Oliver Podwysocki

"Everyone, please choose your weapon of choice."
Photo by Inna Nasonova on Unsplash

Our Creative Process

Oliver has been teaching sciences, maths and the arts for 11 years. As a hobby, Oliver tries to learn skills that have been lost or are on the decline over time to keep them alive and sustain their presence in the wealth of collective knowledge of society. He has an athletic background and has recently taken up Italian swordplay through HEMA (Historical European Martial Arts). Oliver loves swordplay because it is meditative in that it requires the swordsperson to be fully present (mentally & physically) to deal with threats at high speeds with fluid and efficient motion.

Kristie has been teaching secondary science and math including Chemistry for 4 years. One of the things that made Kristie fall in love with Chemistry was the spatial geometry of molecules and how understanding that made all the other concepts fall into place. Kristie does not have a background in swordplay, but she enjoys yoga and diving which are both activities that rely heavily on body line and alignment. As a learner, her brain works best while moving and being able to look at problems from a different angle. 

Through swordplay, several mathematical connections are explored such as optimizing movement, forcing your opponent into larger geometries which take more time to execute, timing of attacks and striking distances to opponent, creating predictable striking patterns and breaking those patterns for advantage, leverage in sword-on-sword interactions, and so many more! In these lessons, we will explore very foundational concepts that require minimal personal protective equipment and skill to explore safely. All of these concepts could be taught with a pencil and paper, but come very intuitively when felt or embodied such as sword leverage or any concept related to distance (ie. choosing targets, measure/footwork).

Our lessons seek to provide students with a way to physically act out concepts that are typically understood in a very abstract manner. There is a danger of “black boxing” higher level math concepts by rushing to solving with algebra and a calculator. Our lessons allow students to view the math concepts “from the inside”. The shift in perspective required to represent their actions as a drawing, diagram, or notation can lead to deeper understanding (Smith, 2018).

It's both a sword AND a shield. Photo by Chris Linnett on Unsplash

Project Overview

Following the order and structures of Oliver’s teachers of longsword and rapier, the order of the lessons scaffold so that movements, offensive and defensive concepts build upon each other. A few fundamental concepts that are woven throughout are:

1. Footwork: using different types of steps, our footwork provides us with opportunities to get within a particular measure to execute a strike.

2. "The Three Advantages": to maximize advantages over your opponents sword, you must have true edge (edge corresponding to your knuckles) on top of your opponent’s sword, leverage of your forte (blade region closest to hand) against their debole (blade region close to tip), and crossing your blade over your opponent’s to maintain a threatening angle and displace their sword.

We start with footwork to gain a sense of being within range of attack; using the body as its own measurement tool, each individual will have a different amount of displacement with their steps. Then we learn how to strike once we learn how to get within range to attack. After learning how to strike, we can create combinations of sword movements that can flow using permutations that lead to rotational and reflective geometries. Finally, we learn how best to strike an opponent while maintaining control using measure and trigonometry inscribed within a circle. These lessons span a range of mathematical concepts and grade levels. 

Onto this we applied Abrahamson’s framework for embodied lesson design (2014). The components of this framework are:

1. Phenomenalize: Develop an activity that relies on the reasoning or solution strategy connected to the target concept. This requires conceptual anchoring, ensuring the activity is intentionally designed to elicit specific mathematical reasoning rather than simply mining a fun task for incidental connections.

2. Concretize: Introduce the formal models, diagrams, and/or symbolic representation of solution strategy. This step is necessary for reification of the mathematical concept and expanding mathematical vocabulary and notation skills. 

3. Dialog: Guide students in making connections between the embodied activity and formal solution models. This continues the reification process and prevents the embodied portion of the lesson from becoming “disembodied” from the formal notation by drawing out those students’ initial novice observations and bridging between the formal notation.

The Product

Link to instructional slide deck (it is a live working document that will change as we continue to develop it).
View-Only Link

"Armed and ready!"

Photo by Lance Reis on Unsplash