Week 2: Shape Explorations: Beyond Paper Platonic Solids!

 After gobbling up the videos on hexaflexagons, I opted to explore Johannes Kepler's writing and cut up some fruit that I had on hand (which I also subsequently gobbled up). I love hexaflexagons, but it was the topic of my offering for the virtual family math fair last year. I couldn't help but watch the Vi Hart videos, even though I had seen them before. The Hexaflexagon Safety Guide is my favourite of the lot, to let you know. Favorite quote?

A change in chirality could be a sign that your flexagon has been flipped through 4-dimensional space and is possibly a highly dangerous multi-dimensional portal :)

Of further note, I will be attempting the bagel cut as soon as I venture to a proper bagel bakery. The precut bagels I buy for my kids are no good for the exercise.

So! I know that I was supposed to make the paper models of the Platonic Solids before reading the piece by Kepler, but I rebelled and read it before. I'm glad that I did, because I ended up ruminating on Kepler's question about whether the hexagonal structure of the snowflake was an "inherent form" dictated by the nature of the material from which it sprang, or whether it was built from without according to notions of beauty or functional purpose. (pg. 36). Of course--being schooled in the twentieth century--I prescribe to the former idea. Kepler attempted to make sense of the phenomenon by comparing it to geometrical and natural analogues: five Platonic and fourteen Archimedean solids, plus hexagonal honey combs and pomegranate seed cluster arrangements.  He used arguments based on 'material necessity' (p. 62), postulating that rhombic and hexagonal arrangements make the most sense based on their ability to tesselate across a plane and that shapes squeezed into finite space will naturally take these forms. 

I personally was curious about how the structure of individual cells coupled with the pattern of germination might lead to various visible patterns. How many of those patterns would be hexagonal? 

I decided to cut up some fruit and take a look to see what I might find. 

Investigation Part 1: The Pomegranate

I started my little investigation by noting the flower end of the fruit. The bits that had supported petals had mostly dried up and fallen off, however, the pattern of breakage was hexagonal. I sharpied in the missing bits and proceeded to cut it up.

I pictured myself slicing up the pomegranate multiple times, although opted for three cuts like so:

You can see that the pomegranate seed clusters match the 6 sectioned flowerhead of the fruit. This makes sense to me. There is something in the original structure of the flower that mirrors (or perhaps even dictates) the structure of the seed clusters.

Investigation Part 2: Apples and Oranges

When I compare the flower end of the apple to the seed configuration, it appears to match as well. A five petal flower structure mirrors the five seed pockets inside the fruit. I also cut up an orange, and although nothing remains of the original flower, I looked up the number of petals on an orange blossom: 5. The number of sections in the orange? 10...a nice, tidy, multiple-of-5 relationship. Very satisfying.

Investigation Part 3: Beading

I'm in Cynthia's 'Beadwork + Mathwork' project with artist Nico Williams. So far, we've made a beaded triangle and a warped hypersquare. The completed triangle sprang from an initial triangular structure. The hypersquare sprang from an initial rhomboid structure. What might result from an initial pentagonal or hexagonal structure? Today decided to do a little beading and find out. I tried to take pictures, but they didn't tell the story well enough, so I made a quick little video:

If I had enough beads (and time), I would have continued with heptagons, octagons, nonagons, and decagons. It would have been a bit tricky because the angle between the spines that radiate outward from the original polygon gets tighter every time you add another side. Because of the shape and size of the beads, the triangle that forms between the spines is an isosceles triangle with the same angle at the apex every time. This triangle needs space and forces the spines away from the centre in an alternating pattern. I love the idea of making this tighter and tighter ruffled figure. 

Why did I do this? I was thinking about why certain patterns and shapes emerge from an initial structure, the way that 6 seed clusters might emerge from a 6-petalled flower.  With fruits like pomegranates, oranges, and apples, the pattern is constrained by the space inside the fruit and you cannot detect it until you cut it open. With the beads, the pattern structure is free to take shape with the only constraints being the size of the beads, the initial shape, and the pattern followed by the bead master. 

Bead patterns remind me of growing corals. It makes me wonder what the shape of the cells, the initial growth structure, and the growth pattern are for each type of coral!

From PowerPoint Stock Images

Crystals, too:

Also from PowerPoint Stock Images

Finally, it makes me wonder about asymmetrical patterns in the natural world (I'm thinking about sea sponges here) which seem to be outnumbered by symmetrical growth patterns. What kind of patterns create sea sponges? Or what about monstrous, chaotic growth (I'm thinking about cancer here)? What causes that? And why does it take on a life of its own?

Anyway, very interesting! I did make those paper solids in the end, by the way. If I hadn't gone down the beading rabbit hole, I would have explored origami. Instead, I did the old cut and tape Platonic solids. Not super magical, but complete.

How might students respond to this?

Johannes Kepler was curious about the possible reasons for the beautiful hexagonal structure of snowflakes and made connections to other patterned structures such as pomegranates and honeycombs to help him make sense of it. I think that it is important to allow students to develop that feeling of curiosity and wonder first before simply exploring fruit. What do they notice about snowflakes? What other natural structures might be similar? Not sure? Let's explore! What about this apple here? Might it have potential? What might we find if we were to slice it in half? Are all apples like this? What about...(you get the idea). 

The sensory stimulation of touching, smelling, and perhaps tasting the fruit is distracting, but memorable, no doubt about it. When I worked in kindergarten, using the five senses to explore rose petals, runner beans, old crumbling wood, dandelions, whole spices (etc.) was irresistible to them. Here are two blog posts I wrote at the time (both worth a read, I think): Click here and  click here. Intermediate students also love to handle items that feel, smell, and look interesting. They just need to focus on the math and not get too distracted by the materials. They love it, but in my experience, they need help to move from sensory stimulation land to math land. They don't always want to do it, believe me. However, students with sensory impairments might really need this opportunity to explore with other senses and might have interesting insights that others do not. 

I might also bring in Plato's idea that there are only 5 solids that can be made with congruent faces, edges, and vertices. The fabulous Mathigon Polypad allows students to join polygons to create nets and fold them to see what happens:

Do we find these shapes in the natural world? What are some possible avenues of exploration? Perhaps rock crystals (something that all students love).

Finally, some older students would be intrigued by the opportunity to test beading patterns built on central polygon structures like I did. I have done knitting and weaving with intermediate students as opportunities to explore what emerges from patterned ways of working with wool. In my experience, it takes a long while for most students to get the hang of the basics skills associated with each craft and it is hard to get them into a place where you are exploring the mathematical ideas. However, elementary school affords you the flexibility of making the knitting, weaving, or beading an art focus and only making connections to the math once a level of proficiency is gained. 

Personally, I love it despite its challenges :)