In Part 1 we started by looking at the five Platonic solids:
We saw that each Platonic solid has a bunch of symmetries where we reflect it across planes called mirrors, which all intersect at the center of the solid. If we take a sphere and slice it with these mirrors, it gets chopped up into triangles, and we get a pattern called a Coxeter complex.
Let’s pick a random Platonic solid and see how it works. For example, the dodecahedron:
gives this Coxeter complex:
In simple terms: we puff up our Platonic solid into a nice round sphere, and then chop its faces into triangles drawn on this sphere. That’s how it always works. And you’ll notice that each triangle has:
• one corner at the center of a face of our Platonic solid,
• one corner at the center of an edge of our Platonic solid,
• one corner at a vertex of our Platonic solid.
So any triangle determines a vertex, edge and face of our Platonic solid! But we don’t get just any old vertex, edge and face this way. The vertex has to lie on the edge, and the edge has to lie on the face.
That’s how it always works—check and see! So, let’s make up a definition:
Definition. Given a Platonic solid, a flag is a vertex, edge and face where the vertex lies on the edge and the edge lies on the face.
In fact, a Platonic solid always gives a Coxeter complex with one triangle for each flag.
Next, in Part 2, we looked at operations that flip these triangles around. I showed you how it works for the cube. I was too lazy to puff it up into a sphere, but I chopped its surface into triangles, one for each flag:
This is the lazy man’s way to draw the Coxeter complex of the cube.
Then I studied operations that flip triangles over to triangles that touch them, One operation, called V, changes which vertex of the cube our triangle contains. For example, if we start with this black triangle:
the operation V changes it to this blue one:
Another operation, called E, changes which edge of the cube our triangle touches:
And a third, called F, changes which face of the cube our triangle lies on:
Using these operations, we can get to any triangle starting from any other. But the cool part is that these operations obey some equations! For any Platonic solid, they always obey these equations:
These say flipping twice the same way gets you back where you started. But there are also three equations that are more interesting. Some of these depend on which Platonic solid we’re looking at. They can be seen using pretty pictures:
• For the cube we get this relation:
from this picture:
Get it? We flip our triangle to change which vertex of the cube it contains, then flip it to change which edge of the cube it touches… and after 8 flips we’re back to where we started, so:
or for short:
We get a 4 in the exponent here because each face of the cube has 4 edges and 4 vertices. So, you can easily work out how this relation goes for any Platonic solid.
• For the cube we also get this relation:
from this picture:
We get a 2 in the exponent here because each edge of the cube contains 2 vertices and touches 2 faces. This is true no matter what Platonic solid we have!
• And finally, for the cube we get this relation:
from this picture:
We get a 3 in the exponent here because each vertex of the cube touches 3 edges and 3 faces. So, you can easily work out how this relation goes for any Platonic solid, too.
The operations V, E, F, together with these relations, generate something called the Coxeter group of the Platonic solid. To know exactly what I mean here you need to know a wee bit of group theory, which is the study of symmetry. But even if you don’t, I hope you get the idea: the Coxeter group consists of all the ways we can flip one triangle in our Coxeter complex over to another by a sequence of V, E, and F flips.
Coxeter diagrams start out being a cute way to record the equations we just saw. Later, they’ll become an amazingly system for creating and classifying highly symmetrical structures: Platonic solids in all dimensions, the solids we get by truncating these in various ways, and more.
Today I’ll draw these diagrams a bit differently than the ways you usually see in books. That’s okay: there are lots of different ways to draw them, depending on what you’re using them for.
In my current way of doing things, the Coxeter diagram of the cube looks like this:
There’s an edge from the letter V to the letter E, labelled by the number 4. This means
And there’s an edge from the letter E to the letter F, labelled by the number 3. This means
But there’s no edge from the V to the F. This means
In other words, we make up an extra rule: if an edge would be labelled by the number 2, we leave it invisible. The reason is that relations of this sort are ‘boring’: they show up a lot. For example, we’ve seen that (VF)2 = 1 for all Platonic solids.
We also don’t bother to record these relations:
because these too are ‘boring’. We’ll assume relations of this sort always hold.
Now, I’ll leave you with a puzzle:
Puzzle 1. What are the Coxeter diagrams of the five Platonic solids? What patterns do you see in these?
But so you don’t think I’m wimping out, I’ll do an example myself. The triangular tiling of the plane is not a Platonic solid:
But it’s what you’d get if you tried to build a Platonic solid where 6 equilateral triangles meet at each vertex! And we can still define a Coxeter complex for it. It looks like this:
There’s one triangle here for each flag in the triangular tiling. And if you stare at this picture, you can read off the equations for the Coxeter group, just as we did for the cube. We have the boring equations
but also:
since each face of the triangular tiling has 3 vertices and 3 edges, and:
since each edge of this tiling contains 2 vertices and touches 2 faces, and:
since each vertex of the this tiling touches 6 edges and 6 faces.
So, here’s the Coxeter diagram of the triangular tiling:
Whoops, now I can’t resist giving you another puzzle! There are two more regular tilings of the plane, so you should try those:
Puzzle 2. What are the Coxeter diagrams of the square tiling:
and the hexagonal tiling:
?
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