My Favorite Theorem

Kevin Knudson: Welcome to MFT. I'm Kevin Knudson, your host, professor of mathematics at the University of Florida. I am without my cohost Evelyn Lamb in this episode because I'm on location at the Banff International Research Station about a mile high in the Canadian Rockies, and this place is spectacular. If you ever get a chance to come here, for math or not, you should definitely make your way up here. I'm joined by my longtime friend Justin Curry. Justin.

Justin Curry: Hey Kevin.

KK: Can you tell us a little about yourself?

JC: I'm Justin Curry. I'm a mathematician working in the area of applied topology. I'm finishing up a postdoc at Duke University and on my way to a professorship at U Albany, and that's part of the SUNY system.

KK: Contratulations.

JC: Thank you.

KK: Landing that first tenure-track job is always

JC: No easy feat.

KK: Especially these days. I know the answer to this already because we talked about it a bit ahead of time, but tell us about your favorite theorem.

JC: So the theorem I decided to choose was the classification of regular polyhedra into the five Platonic solids.

KK: Very cool.

JC: I really like this theorem for a lot of reasons. There are some very natural things that show up in one proof of it. You use Euler's theorem, the Euler characteristic of things that look like the sphere, R=2.

There's duality between some of the shapes, and also it appears when you classify finite subgroups of SO(3). You get the symmetry groups of each of the solids.

KK: Oh right. Are those the only finite subgroups of SO(3)?

JC: Well you also have the cyclic and dihedral groups.

KK: Well sure.

JC: They embed in, but yes. The funny thing is they collapse too because dual solids have the same symmetry groups.

KK: Did the ancient Greeks know this, that these were the only five? I'm sure they suspected, but did they know?

JC: That's a good question. I don't know to what extent they had a proof that the only five regular polyhedra were the Platonic solids. But they definitely knew the list, and they knew they were special.

KK: Yes, because Archimedes had his solids. The Archimedean ones, you are allowed different polygons.

JC: That's right.

KK: But there's still this sort of regularity condition. I can never remember the actual definition, but there's like 13 of them, and then there's 5 Platonics. So you mentioned the proof involving the Euler characteristic, which is the one I had in mind. Can we maybe tell our listeners how that might go, at least roughly? We're not going to do a case analysis.

JC: Yeah. I mean, the proof is actually really simple. You know for a fact that vertices minus edges plus faces has to equal 2. Then when you take polyhedra constructed out of faces, those faces have a different number of edges. Think about a triangle, it has 3 edges, a square has 4 edges, a pentagon is at 5. You just ask how many edges or faces meet at a given vertex? And you end up creating these two equations. One is something like if your faces have p sides, then p times the number of faces equals 2 times the number of edges.

KK: Yeah.

JC: Then you want to look at this condition of faces meeting at a given vertex. You end up getting the equation q times the number of vertices equals 2 times the number of edges. Then you plug that into Euler's theorem, V-E+F=2, and you end up getting very rigid counting. Only a few solutions work.

KK: And of course you can't get anything bigger than pentagons because you end up in hyperbolic space.

JC: Oh yeah, that's right.

KK: You can certainly do this, you can make a torus. I've done this with origami, you sort of do this modular thing. You can make tori with decagons and octagons and things like that. But once you get to hexagons, you introduce negative curvature. Well, flat for hexagons.

JC: That's one of the reasons I love this theorem. It quickly introduces and intersects with so many higher branches of mathematics.

KK: Right. So are there other proofs, do you know?

JC: So I don't know of any other proofs.

KK: That's the one I thought of too, so I was wondering if there was some other slick proof.

JC: So I was initially thinking of the finite subgroups of SO(3). Again, this kind of fails to distinguish the dual ones. But you do pick out these special symmetry groups. You can ask what are these symmetries of, and you can start coming up with polyhedra.

KK: Sure, sure. Maybe we should remind our readers about-readers-I read too much on the internet-our listeners about duality. Can you explain how you get the dual of a polyhedral surface?

JC: Yeah, it's really simple and beautiful. Let's start with something, imagine you have a cube in your mind. Take the center of every face and put a vertex in. If you have the cube, you have six sides. So this dual, this thing we're constructing, has six vertices. If you connect edges according to when there was an edge in the original solid, and then you end up having faces corresponding to vertices in the original solid. You can quickly imagine you have this sort of jewel growing inside of a cube. That ends up being the octahedron.

KK: You join two vertices when the corresponding dual faces meet along an edge. So the cube has the octahedron as its dual. Then there's the icosahedron and the dodecahedron. The icosahedron has 20 triangular faces, and the dodecahedron has 12 pentagonal faces. When you do the vertex counts on all of that you see that those two things are dual. Then there's the tetrahedron, the fifth one. You say, wait a minute, what's its dual?

JC: Yeah, and well it's self-dual.

KK: It's self-dual. Self-dual is a nice thing to think about. There are other things that are self-dual that aren't Platonic solids of course. It's this nice philosophical concept.

JC: Exactly.

KK: You sort of have two sides to your personality. We all have this weird duality. Are we self-dual?

JC: I almost like to think of them as partners. The cube determines, without even knowing about it, its soulmate the octahedron. The dodecahedron without knowing it determines its soulmate the icosahedron. And well, the tetrahedron is in love with itself.

KK: This sounds like an algorithm for match.com.

JC: Exactly.

KK: I can just see this now. They ask a question, “Choose a solid.” Maybe they leave out the tetrahedron?

JC: Yeah, who knows?

KK: You don't want to date yourself.

JC: Maybe you do?

KK: Right, yeah. On our show we like to ask our guests to pair their theorem with something.

JC: It's a little lame in that it's sort of obvious, but Platonic solids get their name from Plato's Timaeus. It's his description of how the world came to be, his source of cosmogeny. In that text he describes an association of every Platonic solid with an element. The cube is correspondent with the element earth. You want to think about why would that be the case? Well, the cube can tessellate three-space, and it's very stable. And Earth is supposed to be very stable, and unshakeable in a sense. I don't know if Plato actually knew about duality, but the dual solid to the cube is the octahedron, which he associated with air. So you have this earth-sky symbolic dualism as well.

Then unfortunately I think this kind of analogy starts to break down a bit. You have the icosahedron, the one made of triangle sides. This is associated to water. And if you look at it, this one sort of looks like a drop of water. You can imagine it rolling around and being fluid. But it's dual to the dodecahedron, this oddball shape. They only thought of four elements: earth, fire, wind, water. What do you do with this fifth one? Well that was for him ether.

KK: So the tetrahedron is fire?

JC: Yeah, the tetrahedron is fire.

KK: Because it's so pointy?

JC: Exactly.

KK: It's sort of rough and raw, or that They Might Be Giants Song “Triangle Man.” It's the pointiest one. Triangle wins every time.

JC: The other thing I like is that fire needs air to breathe. And if you put tetrahedra and octahedra together, they tessellate 3-space.

KK: So did they know that?

JC: I don't know. That's why this is fun to speculate about. They obviously had an understanding. It's unclear what was the depth or rigor, but they definitely knew something.

KK: Sure.

JC: We've known this for thousands of years.

KK: And these models, are they medieval, was it Ptolemy or somebody, with the nested?

JC: The way the solar system works.

KK: Nested Platonic solids. These things are endlessly fascinating. I like making all of them out of origami, out of various things. You can do them all with business cards, except the dodecahedron.

JC: OK.

KK: It's hard to make pentagons. You can take these business cards and you can make these. Cubes are easy. The other ones are all triangular faces, and you can make these triangular modules where you make two triangles out of business cards with a couple of flaps. And two of them will give you a tetrahedron. Four of them will give you an octahedron. The icosahedron is tricky because you need, what, 10 business cards. I have one on my desk. It's been there for 10 years. It's very stable once it's together, but you have to use tape along the way and then take the tape off. It's great fun. There's this great book by Thomas Hull, I forgot the name of it [Ed note: it's called Project Origami: Activities for Exploring Mathematics], a great origami book by Thomas Hull. I certainly recommend all of that.

Anything else you want to add? Anything else you want to tell us about these things? You have all these things tattooed on your body, so you must be

JC: I definitely feel pretty passionate. It's one of those things, if I have to live with this for 30 years, I'll know the Platonic solid won't change. There won't be suddenly a new one discovered.

KK: Right. It's not like someone's name, you might regret it later. But my tattoos are, this is man, woman, and son. My wife and I just had our 25th anniversary, so this is still good. I don't expect to have to get rid of that.

Anyway, well thanks, Justin. This has been great fun. Thanks for taking a few minutes out of your busy schedule. This is a really cool conference, by the way.

JC: I love it. We're bringing together some of the brightest minds in applied topology, and outside of applied topology, to see how topology can inform data science and how algebra interacts in this area, what new foundations we need and aspects of algebra.

KK: Yeah, it's very cool. Thanks again, and good luck in your new job.

JC: Thanks, Kevin.

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