When you do this, it’s hard to help noticing a cool fact: if you take a Platonic solid and draw a dot in the center of each face, these dots are the vertices of another Platonic solid, called its dual. And if we do this again, we get back the same Platonic solid that we started with! These two solids have very similar Coxeter diagrams.
For example, starting with the cube, we get the octahedron:
Starting with the octahedron, we get back the cube:
These pictures were made by Alan Goodman, and if you go to his webpage you can see how all 5 Platonic solids work, and you can download his free book, which includes a good elementary introduction to group theory:
• Alan Goodman, Algebra: Abstract and Concrete, SemiSimple Press, Iowa City, 2012.
When we take the dual of a Platonic solid, or any other polyhedron, we replace:
• each vertex by a face,
• each edge by an edge,
• each face by a vertex.
So, it should not be surprising that in the Coxeter diagram, which records information about vertices, edges and faces, we just switch the letters V and F.
Here’s the story in detail.
The tetrahedron is its own dual, and its Coxeter diagram
doesn’t change when we switch the letters V and F. Remember, this diagram means that the tetrahedron has:
• 3 vertices and 3 edges around each face,
• 3 edges and 3 faces around each vertex.
The dual of the cube is the octahedron, and vice versa. The Coxeter diagram of the cube is:
because the cube has:
• 4 vertices and 4 edges around each face,
• 3 edges and 3 faces around each vertex.
On the other hand, the Coxeter diagram of the octahedron is:
because it has:
• 3 vertices and 3 edges around each face,
• 4 edges and 4 faces around each vertex.
If we switch the letters V and F in one of these Coxeter diagrams, we get the other one… drawn backwards, but that doesn’t count in this game.
The dual of the dodecahedron is the icosahedron, and vice versa. The Coxeter diagram of the dodecahedron is:
because it has:
• 5 vertices and 5 edges around each face,
• 3 edges and 3 faces around each vertex.
The Coxeter diagram of the icosahedron is:
because it has:
• 3 vertices and 3 edges around each face,
• 5 edges and 5 faces around each vertex.
Again, you can get from either of these two Coxeter diagrams to the other by switching V and F. That’s duality.
But now let’s think a bit about a deeper pattern lurking around here.
Puzzle. If we take the Coxeter diagrams we’ve just seen:
and strip off everything but the numbers, we get these ordered pairs:
Why do these pairs and only these pairs give Platonic solids? I’ve listed them in a cute way just for fun, but that’s not the point.
There could be a number of perfectly correct ways to tackle this puzzle. But I have one in mind, so maybe I should give you a couple of clues to nudge you toward my way of thinking—though I’d be happy to hear other ways, too!
First, what about this pair?
Well, last time we looked at the corresponding Coxeter diagram:
and we saw it doesn’t come from a Platonic solid. Instead, it comes from this tiling of the plane:
What I’m looking for is an equation or something like that, which holds only for the pairs of numbers that give Platonic solids. And it should work for some good reason, not by coincidence!
One more clue. If your equation, or whatever it is, allows extra solutions like (2,n) or (n,2), don’t be discouraged! There are weird degenerate Platonic solids called hosohedra, with just two vertices, like this:
You can’t make the faces flat, but you can still draw it on a sphere, and in some ways that’s more important. The Coxeter diagram for this guy is:
And each hosohedron has a dual, called a dihedron, with just two faces, like this:
The Coxeter diagram for this is:
So, if your answer to the puzzle allows for hosohedra and dihedra, it’s not actually bad. As you proceed deeper and deeper into this subject, you realize more and more that hosohedra and dihedra are important, even though they’re not polyhedra in the usual sense.
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