Season 3 Episode 0
We're back for Season 3! In episode 0, we had a look at the Platonic solids, and tried to prove that there are exactly five, assuming something mysterious called Euler's formula. This has some deep links to topology, which we didn't really discuss!
There are some nets to make your own models of Platonic solids here, and that website also has links to an interactive visualiser that lets you spin each of the shapes.
If you liked the Platonic solids, then you might also like the Archimedean solids! These have all their vertices the same (just like Platonic solids) and their faces are all regular polygons (just like Platonic solids), but their faces don’t have to all be the same polygon (that’s how they’re different from Platonic solids). There are quite a few, and you stand a good chance to discover some of them yourself if you have some squares and triangles handy. (Hint: consider cutting the corners off a Platonic solid. What happens?). If you’d like a list once you’ve explored for yourself, there are pictures at Wolfram Mathworld, which is a sort of online encyclopedia of mathematics (quite technical, but nice pictures!).
What if we want to think about Platonic solids in higher dimensions?
First, we could instead think about Platonic solids in 2D. Those would be polygons where each side is the same, and where the sides meet at the same angle at each vertex (vertices are all the same). Those are just the regular polygons! The regular polygons what we used for the faces of our 3D Platonic solids, so perhaps for our 4D Platonic solids we should use the 3D Platonic solids. The general term for “polygons” “polyhedra” and so on into higher dimensions is “polytopes”. A four-dimensional polytope is made of 3D polyhedra that have 2D faces in common. This is extremely difficult to imagine!
My strategy for trying to imagine 4D shapes is to use the difference between 2D and 3D as an analogy for the difference between 3D and 4D. Let’s try to imagine a 4D shape made out of 3D cubes. This is a bit like the difference between a cube and a square, but one dimension up.
I can make a 3D cube by starting with five squares in the 2D plane arranged in a + shape, then by folding up the sides and sticking another square on the top. So perhaps I should imagine 3D cubes stacked so that they have faces in common; maybe there’s one central cube and six more stuck to its faces. But I’m imagining that in 3D space. In the analogy, this is like having those squares in a 2D plane – that’s not a cube yet, but it’s the start of a net of a cube. I need to fold those squares into the third dimension and stick another square onto the top.
Back to 3D, I suppose I need to bend the outside six cubes into the fourth dimension (I can’t really imagine that, but I can imagine the middle cube staying still while the other six fade out “somewhere else”). Now I need to stick another cube on "top", so perhaps I can imagine the opposite faces of the outside cubes fading back in. I’ll need to bend or stretch things a bit, but I can bring those faces outward and stick them to another (larger?) cube. Now I’ve got a 4D shape made of cubes – this shape is a tesseract.
If you like playing with dimensions like this, then you might like the books Flatland (Edwin Abbott Abbott, 1884) and Flatterland (Ian Stweart, 2001). The original is a sort of satirical social commentary on Victorian culture that happens to be in 2D, and the one by Ian Stewart is a sort of unoffical sequel that has much more mathematics.
In the livestream we talked a bit about the process that I called “taking the dual” by replacing every face with a vertex and connecting them if and only if the original faces shared an edge. I’m not sure that I made it clear enough that you can do this for any polyhedron, not just Platonic solids!
You might like to think about why the dual polyhedron always has the same number of edges as the original polyhedron, and why taking the dual of the dual returns you to the original polyhedron.
The Wikipedia page on dual polyhedra is, in my opinion, a good example of a Wikipedia page that (1) starts off sensibly, (2) quickly gets too technical, (3) has some nice pictures.
We had some good discussions about whether polyhedra in general might satisfy $V-E+F=2$, and about what a polyhedron even is! There’s a video on V-E+F by 3Blue1Brown that you might enjoy.
If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk.