Coxeter groups are a huge amount of fun. Normally their delights are reserved for people who have already studied group theory. But I don’t think that’s fair. You don’t need to know music theory to enjoy a piece by Bach. And you don’t need to know group theory to enjoy Coxeter groups.
In fact, it’s probably better to learn theories after falling in love with some examples. Imagine a world where you had to learn music theory before listening to music. A world where everyone studied music theory in elementary school, high school and college, but only people who majored in music were allowed to listen to the stuff. In that world, people might say they hate music… just as in this world they say they hate math.
So, here goes:
Last time I showed you that any Platonic solid has a bunch of symmetries where we reflect it across planes. These planes, called mirrors, 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.
If we start with the cube, here’s the Coxeter complex we get:
For artistic reasons, half the triangles are colored blue and half are colored black. But it’s not just pretty: there’s also math here. If we take any black triangle and reflect it across any mirror, we get a blue triangle… and vice versa.
Instead of taking a sphere and slicing it with mirrors, we can start with the cube itself. Here’s what we get:
It’s not quite as pretty (especially because I drew it), but it makes certain games easier to play. These games involve picking one triangle and calling it our favorite. It doesn’t matter which. But we have to pick one… so how about this:
Each different symmetry of the cube sends this triangle to a different triangle. This instantly lets us count the symmetries: there are 48, since each of the cube’s 6 faces has been chopped into 8 triangles.
But even better, we get a vivid picture of the symmetries of the cube! Let’s see how this works.
Any triangle in the Coxeter complex has three corners:
• one corner is a vertex of the cube,
• one corner is the center of an edge of the cube,
• one corner is the center of a face of the cube.
Here’s how it works for our favorite triangle:
Now the real fun starts. We can move from any triangle to a neighboring one in three ways:
1) We can change which vertex of the cube our triangle contains. Starting from our favorite triangle, we get this blue triangle:
Note: the blue triangle touches the same edge of the cube as the black one. It also lies on the same face. Only the vertex has changed!
What have we actually done here? We’ve reflected our triangle across a mirror in a way that changes which vertex of the cube it contains. Let’s call this way of flipping a triangle V.
2) We can change which edge of the cube our triangle touches. Starting from our favorite triangle, we get this yellow triangle:
Note: the yellow triangle contains the same vertex of the cube. It also lies on the same face. Only the edge has changed!
What have we actually done here? We’ve reflected our triangle across a mirror in a way that changes which edge of the cube it touches. Let’s call this way of flipping a triangle E.
3) We can change which face of the cube our triangle lies on. Starting from our favorite triangle, we get this green triangle:
Note: the green triangle contains the same vertex of the cube. It also touches the same edge. Only the face has changed.
What have we actually done here? If you can’t guess, you must be asleep: we’ve reflected our triangle across a mirror in a way that changes which face of the cube it lies on! Let’s call this way of flipping a triangle F.
By repeating these three operations—changing the vertex, edge or face—we can get to any triangle starting from our favorite one. We can even use this trick to label all the triangles. Let’s call our favorite triangle 1:
Let’s call its neighbors F, E and V, since we use those three reflections to get to these new triangles:
Starting with these, we can get more triangles by changing the vertex, edge or face:
See what I’m doing? We get the triangle VE by starting with the triangle V and then changing which edge of the cube it contains. We get EF by starting with E and then changing which face it lies on. And so on.
However, there’s a ‘problem’. See the triangle VF? We got there from the triangle V by changing which face it lies on. But we could also get there another way! We could start at F and then change which vertex this triangle contains. So we could equally well call this triangle FV.
Luckily, in math nothing is really a problem once you understand it. This is why math is more fun than real life: merely understanding a problem makes it go away. We’ll just say that
So, we can use either label for this triangle: it doesn’t matter.
More deeply, if you start with any triangle, change the vertex it contains and then change the face it lies on, you get the same triangle as if you first change the face and then the vertex. That’s what the equation VF = FV really means. It’s a fact of geometry: a general fact about Platonic solids.
Let’s go a bit further:
I’m using the same rules; check to make sure I did everything right! There’s another little ‘problem’, though: see the triangle labelled FEF? We got there from FE by changing which face of the cube our triangle lies on. But we could also get there starting from EF by changing the edge. So really we have
But this is not a general fact about Platonic solids: it shows up because in the cube, three faces and three edges meet at each vertex. That’s why the equation has three F’s and three E’s.
We can go on even further, but you can already see where the next problem will show up. See that unlabelled triangle in the front face of the cube? At the next stage we’ll want to label it VEVE, but we’ll also want to label it EVEV. So:
Again, this is not a general fact about Platonic solids! It shows up because the cube has square faces, so four vertices and four edges touch each face. That’s why the equation has four V’s and four E’s.
We almost have enough equations to avoid all future problems. But there are a few more that are so obvious you may have overlooked them. Suppose we change which vertex of the cube our triangle contains, and then do this again. We get back where we started! For example, first we go from the black one to the blue one:
and then we go back to black.
So, we have
This says switching vertices twice gets us back where we started. Similarly, we have
and
And now, although I haven’t proved it to you, we have a complete set of equations to give each triangle an unambiguous name… or more precisely, an unambiguous element of the ‘Coxeter group’ of the cube. Two different expressions, like EFE and FEF, give the same element of the Coxeter group if we can get from one to the other using our equations. For example, in the Coxeter group we have
We can do this stuff for other Platonic solids, too. The Coxeter group of the octahedron is secretly the same as that of the cube, since they’re dual. The only difference is that the names F and V get switched, because faces of the cube correspond to vertices of the octahedron, and vice versa! Similarly for the icosahedron and dodecahedron. So, I mainly have two puzzles for you:
Puzzle 1: Find equations defining the Coxeter group of the tetrahedron.
Puzzle 2: Find equations defining the Coxeter group of the dodecahedron.
If these seem hard, let’s reflect a bit more on what we did for the cube. For the cube we have
because of this picture:
Similarly, we have
because of this picture:
And finally, we have
because of this picture:
This should make it easy to solve the puzzles. We can also phrase the solutions in a different way:
Puzzle 3: Show that that Coxeter groups of the tetrahedron, cube and dodecahedron can be completely described by the equations
and
for some integers a, b, and c.
The story doesn’t stop here—far from it! Later we’ll meet Coxeter groups for the higher-dimensional analogues of Platonic solids, which are called regular polytopes. And we’ll use them to classify so-called uniform polytopes obtained by chopping vertices, edges, faces and so on off the regular ones. For example, the cuboctahedron:
can be gotten either by chopping the corners off a cube, or chopping the corners of an octahedron! We can classify such shapes using Coxeter diagrams, which are based on Coxeter groups. So, there’s no shortage of fun stuff to do… in fact, there’s way too much!
Actually, that’s the main problem with mathematics, once you start actually doing it. There’s just too much fun stuff.
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