We started off this week going back to some of the Eulerian circuits we worked on last week and tried out Euler’s Formula on them.
After defining what vertices, edges, and regions are, as defined in graph theory, we counted up those elements in each of the shapes and found that the sum of the number of vertices and edges minus the number of regions always equaled two.
We found, though, that this only worked if we followed a couple of strict rules.
This star, for example, doesn’t work:
But if we make it planar (edges can’t cross without vertices), it works out:
Similarly, every edge needed at least one vertex:
And only one figure at a time. Not this:
But this is okay:
Most of these exercises were taken from Math Lab for Kids by Rebecca Rapoport and were designed by Fan Chung.
I mentioned this is all a branch of mathematics called topology. Most people, when they think of topology, think of Mobius strips. Much to my excitement, most of our Mad Math-ers hadn’t heard of them!
We first saw this clip of Avengers: Endgame that featured a Mobius strip:
And then we made a regular loop from a strip of paper. Then a loop with a half-twist (Mobius strip). Then a loop with a full twist. We saw the former and latter loops had two sides. The Mobius strip only had one. (!!!)
We then cut each of the loops in half. The first loop divided into two separate loops. The loop with the full twist separated into two intertwined loops (of the same size). The Mobius strip just divided into one loop of double the size.
We then watched the following video of Vi Hart and emulated what she was doing:
She mentions a lot of interesting things, but I was able to talk a little bit more about rotational symmetries, frieze patterns, and glide symmetries so that we could follow what she was doing.
I strongly encourage anyone to experiment with Mobius strips, either with paper strips or with fruit leathers! They have such counter-intuitive properties that don’t really make sense until you think of them mathematically.
A little bit of extra credit. Before Avengers: Endgame was released in theaters, this gentleman had a theory about how the movie could utilize time travel in interesting ways using Mobius strips. If the dimension of time was a Mobius strip, you could resolve time-traveling paradoxes with something like quantum superpositioning! This ultimately wasn’t the direction the movie went in, but mind-blowing all the same:
Tom Kim 