How mathematics is done: a conversation

Alice: Hi! Let’s think about symmetrical polygons.

Bob: You mean polygons with identical sides? Like this?


Alice: Hmmm. Not quite what I meant. I think “symmetrical” probably should mean that the corners are all the same too, so that middle one should be a square. In fact, I think the regular convex polygons are what I had in mind.

Bob: What about in three dimensions? I guess the sides should be identical regular polygons?


Alice: Yes, but that’s got the same problem as before – the corners aren’t all the same. The cube and dodecahedron are OK, but the shape on the left is two pentagonal pyramids stuck together.

Bob: Oh, I see what you mean. Two of the corners have five triangles meeting, and the others have only four. The tetrahedron, octahedron, and icosahedron should be OK, though:


Alice: Is that all?

Plato: Without a doubt. There are only five. What’s more, they correspond to the five elements:


Alice: I think there are a few more elements than that, but that isn’t important right now. How can we be certain there are only five?

Bob: Well, look at the shapes that have to meet at a corner. If four squares met, we would have a flat surface, and ditto for six triangles or three hexagons. That means the only possible corners are three triangles meeting, four triangles, five triangles, three squares, or three pentagons. Which are the five cases we’ve got.

Alice: I think you’re right. But we can actually get identical corners with more more than one kind of regular polygon. Look at the prisms – rotate any one whichever way you like, and you’ll see the corners are all identical:


Bob: That middle one is actually just a cube.

Alice: So it is.

Kepler: There’s also something called an antiprism, which has triangles along the side, instead of squares:


Alice: That first one is just an octahedron. And I’ve just realised that the soccer-ball shape works too – the sides are regular pentagons or hexagons, and all the corners are identical:


Bob: is that all?

Archimedes: Actually, I made a list of thirteen that included that last one, but I seem to have mislaid the paper...

Kepler: And with those, that really is all.

Rudy: Unless you’d like to try for four dimensions?