Though I’m shifting toward applied math, I find myself unable to quit explaining pure math to people—stuff that’s fun purely for its own sake. So, I’ve been posting about symmetry and the fourth dimension over on Google+. Now I want to take those posts, polish them up a bit, and combine them into blog articles.
The idea is to start with something very familiar and then take it a little further than most people have seen… without getting so technical that only people with PhDs understand what’s going on. I’m more interested in communicating with ordinary folks than in wowing the experts.
So, I’ll assume you know and love the five Platonic solids:
• The tetrahedron, with 4 triangular faces, 6 edges and 4 vertices:
• The cube, with 6 square faces, 12 edges and 8 vertices:
• The octahedron, with 8 triangular faces, 12 edges and 6 vertices:
• The dodecahedron, with 12 pentagonal faces, 30 edges and 20 vertices:
• The icosahedron, with 20 triangular faces, 30 edges and 12 vertices:
Starting from these, we’ll build the six Platonic solids that exist in 4 dimensions, and the various fancier shapes we can get from these by cutting off corners, edges and so on.
Luckily, a lot of heroic mathematicians and programmers have made pictures of these shapes freely available online. For example, the rotating Platonic solids above were made by Tom Ruen, who put them on Wiki Commons. It wouldn’t be so bad if all I did is show you lots of these pictures and explain them. But there are some underlying themes that make the story deeper, so I thought I’d reveal those now. As the series marches on, I’ll try to make it easy to ignore these themes or pay attention to them, depending on what you want.
One theme is the quaternions. This is a number system introduced by the famous mathematician William Rowan Hamilton back in 1843. A typical quaternion looks like this:
where are ordinary real numbers and are square roots of -1 that ‘anticommute':
As their name indicates, the quaternions are a 4-dimensional number system. We can use them to relate rotations in 3 dimensions to rotations in 4 dimensions… and this establishes links between 3d Platonic solids and 4d Platonic solids: special links that just don’t exist in higher dimensions.
For example, the dodecahedron has 60 rotational symmetries, and this fact gives a 4d Platonic solid—or as mathematicians say, a 4d regular polytope—with 120 dodecahedral faces. Getting to understand this in detail will be one of the high points of this series: it’s a really wonderful story!
Another theme is 5-fold symmetry. In 2 dimensions there’s an obvious polygon with 5-fold symmetry: the regular pentagon. In 3 dimensions we have a Platonic solid with pentagonal faces: the regular dodecahedron. In 4 dimensions, as I just mentioned, there’s a regular polytope with regular dodecahedra as faces. But then this pattern ends. There are no higher-dimensional regular polytopes with pentagons in them! Only squares and triangles show up.
But the biggest unifying theme is ‘finite reflection groups’. These show up as symmetry groups of Platonic solids and 4d regular polytopes. Technically, a finite reflection group is a finite group of transformations of n-dimensional Euclidean space that’s generated by reflections. Some examples in 3 dimensions will illustrate the idea: it’s not as scary as it might sound.
Take a regular dodecahedron, for example:
This has lots of rotations and reflections as symmetries—but a finite number of them. Each reflection corresponds to a mirror: a plane through the center of the dodecahedron. The reflection switches points on opposite sides of this mirror. We can get every symmetry by doing a bunch of these reflections, one after another: that’s what we mean by saying a group of symmetries is ‘generated by reflections’. So, the symmetry group of a dodecahedron is a finite reflection group.
But the fun starts when we take a sphere centered at the center of the dodecahedron, and slice it with all these mirrors. We get a picture like this, called a Coxeter complex:
The great circles in this picture are where the mirrors intersect the sphere.
You’ll notice there are 120 triangles in this picture: each of the 12 pentagonal faces of the dodecahedron has been subdivided into 10 right triangles. You should be able to see that if we pick one of these triangles, there’s a symmetry carrying it to any other. So, the symmetry group of the dodecahedron has 120 elements!
By the way: earlier I mentioned the 60 rotational symmetries of the dodecahedron; now I’m talking about 120 symmetries including rotations and reflections. There’s no contradiction. If we start by picking a black triangle, a rotation will take it to another black triangle, while a reflection will take it to be a blue one. There are 60 of each, for a total of 120.
We can play the same game starting with any other Platonic solid. If we start with the icosahedron, nothing really new happens. It has the same symmetry group, so we get the same Coxeter complex. Indeed, if you look carefully here:
you can see a bunch of equilateral triangles, each containing 6 right triangles. There are 20 of these equilateral triangles, and they’re the faces of an icosahedron:
The corners of the icosahedron are located at the centers of the faces of a dodecahedron, and vice versa. So we say these Platonic solids are dual to each other. Dual polyhedra, or more generally dual polytopes, have the same symmetry group and the same Coxeter complex.
But we get something different if we start with the cube:
This gives a Coxeter complex with 48 triangles, formed by subidividing each of the 6 square faces of the cube into 8 right triangles:
There’s a symmetry of the cube carrying any of these right triangles to any other, so its symmetry group has 48 elements.
The octahedron has the same symmetry group as the cube, because they’re duals. But we get something different if we start with the regular tetrahedron:
This gives a Coxeter complex with 24 triangles, formed by subidividing each of the 4 triangular faces of the tetrahedron into 6 right triangles:
There’s a symmetry of the tetrahedron carrying any of these right triangles to any other, so its symmetry group has 24 elements.
The tetrahedron is its own dual. So the Platonic solids only give us three finite reflection groups in 3 dimensions. There are also some some infinite sequences of less interesting ones, like the symmetry groups of the hosohedra, which are funny degenerate Platonic solids whose faces have just 2 sides… these faces need to be curved:
But in general, the possibilities are quite restricted. So, finite reflection groups are not only beautiful: they’re a bit rare. This makes them doubly precious. People have written books about them:
• James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge U. Press, 1992.
And as we’re beginning to see, the Coxeter complex is a vivid picture of the finite reflection group it comes from. We can already see that in 3 dimensions, it has one black triangle for each reflection in the group, and one blue triangle for each rotation. But it contains much more information than this, in neatly visible form… and it works well not just in 3 dimensions, but any dimension.
That’s enough for now: I want to keep these blog articles bite-sized, rather than letting them grow jaw-breakingly big. But if you’re hungry for more right now, try this:
• John Baez, Platonic solids in all dimensions.
I’ll also leave you with this:
Puzzle: How many great circles are there in these Coxeter complexes?
• The Coxeter complex of the tetrahedral finite reflection group, also known to mathematicians as A3:
• The Coxeter complex of the octahedral finite reflection group, also known as B3:
• The Coxeter complex of the icosahedral finite reflection group, also known as H3:
The answers will say how many reflections there are in the finite reflection groups we’ve met today.