Mart Art

PCMI 2022 Post 6: Building Party

Yesterday evening at 7pm, a bunch of math teachers and other PCMI folk gathered in the teacher rooms to build math art. Each table had a different thing one could build (slideshow):

It was sweet to see one participant bring their kid, and another bring their partner, and we all had fun creating delicious little things we could take home. I didn’t end up working on any of these set projects because I wanted to continue to learn to knit. One of my PCMI goals was to learn to knit. I remembered Peg was an expert and so I reached out to her before PCMI to see if she would be able to teach me — which she happily did! (And in fact, last weekend, she took me and a few others to Salt Lake City for a yarn/fabric crawl which was so wonderful). I wanted to learn because I’ve been listening to a lot of audiobooks since I’ve had a hard time concentrating on reading since the pandemic. And I figured having something to do while listening would be neat. So during the entire building party, with 80s music blasting, I practiced my knit stitch and perl stich. I made what I thought was a number of mistakes and so I cast and threw out my initial attempts two different times. The third time I was successful-ish. I did three rows of knit stiches, and then I alternated a bunch of knit and perl stitches. I started out with 10 stitches, but somehow ended up with 11 stitches. And a few times, I looked at my needles and didn’t understand what was going on… I had something twisted or I thought I had made a stitch but I didn’t. At those points, I didn’t know what I was looking at so I asked Peg and Rebecca for help — and so I now have a sense of what to look for. However I realize my next step is to learn how to deal with mistakes… What do I do with the 11th stitch that I didn’t want? How do I analyze my knots/stitches to be able to undo them if there is something wrong? In other words, what are ways to deal with errors? In any case, this is what I have so far.

When I wasn’t knitting, I was going to the different tables, looking at all the colorful things people were making. Here’s a collection of some of the objects that were made, displayed the next day… but not everything!

Being surrounded by all this math-art reminded me of the math-art show I helped organize at my school, which we titled Technically Beautiful.

While knitting, I was sitting at the table where people were making the “straw thingy,” which was actually 5 intersecting tetrahedra.

The first time I heard of these tetrahedra was when looking in an math-origami book and saw a connection to a multivariable calculus project. It turns out to get the tetrahedra to interlock perfectly, so they didn’t jiggle around, is a tricky problem. Years ago a student of mine did a project on this:

What I loved is that all I associated with these tetrahedra was this math — finding the coordinates for the vertices of the points, and finding the optimal strut length. However while I was sitting at the table knitting, I was talking with a math professor who shared with me that he sees a “proof without words” with these tetrahedra. He saw something different mathematically than I did. He told me that one could see those interlocking tetrahedra as representing a particular mathematical group. It isn’t quite the permutation group of 5 objects, but rather if you have 5 objects and permute two pairs of two objects (so if you had 12345, you could do a move like 12345–>21345–>21435, or 12345–>21345–>23145). I think he called that group the alternating group of 5 objects. And then he showed me how if you look at the interlocking tetrahedra, and rotated it around a vertex, face, or edge, you get that same group (like the colos of the straws, after a rotation, swap… but in the way of the alternating group). It was fun to have someone way above me in math explain something to me, who would allow me to ask questions, and use hands on manipulatives (we pulled out straws, and did the rotations!) to make things make sense for me. And apparently, this alternating group of 5 objects deals with the insolvability of the quintic equation, something I learned about ages ago in college, but now is faded, distant memory. [Sorry if any of my descriptions are wrong… It was an informal conversation and I haven’t had time to research it yet to flesh it out.]

What’s neat is that now, these interlocking tetrahedra mathematically for me no longer represent only a mathematical question about optimization (the “optimal strut width”) and an interesting problem about how to find the coordinates of the vertices. These interlocking tetrahedra now also represent for me a group, and connects up with the insolvability of quintic equations! Again, I am reminded of the Francis Su quotation from two PCMI posts ago, which talked about how mathematical ideas don’t exist in isolation. Instead, they build up in time and get richer and fuller when they do. I see something different now when I look at these interlocking tetrahedra than I did before the building party.

Lastly, if you want to have your own math building party, or create something that you see a picture of above, here are all the instructions to the creations!

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