Last time in this series we completed the first phase of our quest: we got lots of polyhedra from Coxeter diagrams in a systematic way. But before we sail off into new seas, let’s review what we’ve done.
To spice things up, I’ll explain everything in a different way than before. If you don’t see how this new way relates to the old one, please ask! There’s a lot to say about this stuff that I’m not saying, so there are plenty of gaps left to fill.
And if you’re wondering what that thing is up there, read on.
Harold Scott MacDonald Coxeter, the ‘king of geometry’, developed a beautiful theory relating groups to highly symmetrical polyhedra and their higher-dimensional generalizations, called ‘polytopes’.
His book Regular Polytopes is required reading for anyone who wants to learn about these things. There’s a copy sitting by my bed, and there should be one by yours, too. It took him 24 years to write.
In his honor, finite groups generated by reflections in n-dimensional space are called Coxeter groups. They’re described by diagrams with n dots, called Coxeter diagrams. We’ve been looking at the 3d case, where there happen to be three such diagrams, with mysterious-sounding names:
I want to show you to use these diagrams. So let’s do an example, namely A3:
This diagram is telling us to draw a bunch of triangles on a sphere, like this:
This shape is called a Coxeter complex. As you can see, it contains lots of great circles. For each one we get a symmetry: the reflection across that circle, as if a mirror cut our Coxeter complex in half through that circle. So, the symmetries of the Coxeter complex form a finite group, generated by reflections. This is a Coxeter group!
But how do we get the Coxeter complex from the diagram? The dots in the diagram are secretly called V, E, and F:
Among other things, these letters are names for the edges of our favorite triangle in the Coxeter complex:
It doesn’t matter which is our favorite triangle, so I just picked one. Each edge of this triangle gives a symmetry: the reflection that reflects the Coxeter complex across that edge! We call these symmetries V, E and F.
The 3 on the line going between V and E in this diagram:
is a quick way of saying that
In other words, doing the reflections V, E, V, E, V, E gets us back where we started.
To see this, let’s call our favorite triangle 1. Reflecting this triangle across the great circle containing the edge V, we get a new triangle, which we’ll call V. Reflecting that across the great circle containing the edge E, we get a new triangle, which we’ll call VE. And so on:
By the time we get to VEVEVE = (VE)3, we’re back where we started! That’s a total of 6 reflections, so the V and E edges of each triangle must meet at a 60° angle.
Similarly, the 3 on the line going between E and V says that
so doing the reflections E, F, E, F, E, F also gets us back where we started:
On the other hand, there’s no line connecting the dots V and F in our Coxeter diagram:
But this is an abbreviation for a line with a 2 on it! You see, lines with 2 on them occur so often that Coxeter decided to save time by not drawing such lines. So, we have
and doing the reflections V, F, V, F also gets us back where started:
That’s a total of 4 reflections, so the V and F edges of each triangle must meet at a 90° angle.
So, I’ve sketched how the Coxeter group and the Coxeter complex arise from the Coxeter diagram. To be a bit more precise, the Coxeter group A3 has generators V, E, F obeying relations
since V, E, and F are reflections. If we draw a sphere with one great circle serving as the ‘mirror’ for each reflection in the Coxeter group, we get the Coxeter complex.
What makes Coxeter complexes special, compared to other ways of tiling a sphere with triangles? One thing is that they have exactly as many symmetries as triangles. If you pick any triangle and call it your favorite, there’s exactly one symmetry—that is, rotation and/or reflection—sending this triangle to any other.
So, the Coxeter complex is actually a picture of its own symmetry group!
The particular Coxeter complex we’ve been looking at has 24 triangles. So, its symmetry group has 24 elements. This group is called the tetrahedral finite reflection group, or A3 for short. There’s a lot to say about it, but not now! There’s another more urgent question.
How do we get polyhedra from our Coxeter complex?
The easiest one works like this. We take the Coxeter complex and create a polyhedron with a corner in the middle of each triangle, connecting two corners with an edge whenever their triangles touch:
The polyhedron looks better if we view it from a slightly different angle, and let a skilled artist like Tom Ruen do the drawing:
This polyhedron is called the Poincaré dual of the Coxeter complex.
There’s a notation for this particular polyhedron:
All the dots are black, because this is the fanciest, most interesting polyhedron that comes from our Coxeter complex. We get other polyhedra by blackening just some of the dots.
Earlier in this series, I described how to build these other polyhedra using the concept of ‘flag’. But there’s another way, using the Coxeter complex, which I’ll sketch here.
Suppose we blacken just some of the dots in our Coxeter diagram. Then our favorite triangle belongs to a bunch of triangles, all related by reflections corresponding to the dots we left white. And indeed, all the triangles can be grouped into bunches that all look the same… and there’s a polyhedron that has a corner in the middle of each bunch of triangles.
For example, suppose we leave the E dot white:
Then our favorite triangle belongs to a bunch of triangles—in this case, just a pair!—that are related by reflections along the E edge of our favorite triangle. If we group our triangles into pairs like this, we get a polyhedron with a corner in the middle of each pair:
Again, it looks better if we let Tom Ruen draw it:
I should do more examples, but I think I’ll wrap up by describing the procedure in more highbrow language. Skip the next paragraph if you don’t know group theory, and move on to the complete list of examples!
If we call the Coxeter group G, blackening all the dots of the Coxeter diagram gives a polyhedron with one corner for each element of G. But if we don’t blacken all of them, the reflections corresponding to the white dots generate a subgroup B of G. Then we get a polyhedron with one corner for each element of G/B. Last time I said each way of blackening some dots describes some sort of ‘flag’. In these terms, B is the subgroup fixing your favorite flag of this sort, and G/B is the set of all flags of this sort. Each of those flags corresponds to what I’m calling a ‘bunch of triangles’ in my current story.
But now let’s see what we get from all this! We get three families of polyhedra, and these are almost all the ‘Archimedean solids’ in 3 dimensions.
The A3 family: o—3—o—3—o
This family of polyhedra can all be gotten from the tetrahedron by chopping off vertices, edges or faces. They’re associated to the Coxeter complex we’ve just been looking at:
It’s built from triangles whose interior angles are and . These numbers come from taking π and dividing it by the numbers on the edges of the Coxeter diagram: 3, 3, and the invisible edge labelled 2.
As mentioned, this Coxeter complex has 24 triangles, and its symmetry group, with 24 elements, is called the tetrahedral finite reflection group, or A3.
Here are all the polyhedra in this family. The list has some repeats, because this Coxeter diagram is its own mirror image!
The B3 family: o—3—o—4—o
This family of polyhedra can all be gotten by taking the cube or the octahedron and chopping off vertices, edges or faces. They come from the Coxeter complex whose triangles have interior angles and :
This Coxeter complex has 48 triangles, and its symmetry group is a Coxeter group with 48 elements, called the octahedral finite reflection group, or B3.
Here are all the polyhedra in this family. We’ve seen some of these already in the A3 family, and there’s a reason for that: the group B3 happens to contain A3 as a subgroup! It’s twice as big.
The H3 family: o—3—o—5—o
This family of polyhedra can all be gotten by taking the dodecahedron or icosahedron and chopping off vertices, edges and faces. They come from the Coxeter complex built from triangles whose interior angles are and :
This Coxeter complex has 120 triangles, and its symmetry group is a Coxeter group with 120 elements, called the icosahedral finite reflection group, or H3.
Here are all the polyhedra coming from this Coxeter complex. These are my favorites, because they look the most fancy, and I have a fondness for the quirky charm of 5-fold symmetry:
The picture at the start of this post was taken by my friend Allen Knutson:
It’s the H3 Coxeter complex, and for some reason it’s in the music library at Cornell University, where Allen teaches math.
The picture of H. S. M. Coxeter is from the cover of a book about him:
• Siobhan Roberts, King of Infinite Space, Walker & Company, 2006.
As usual, the pretty pictures of solids with brass balls at the vertices were made by Tom Ruen using Robert Webb’s Stella software. Tom Ruen also drew the Coxeter complexes; I’m to blame for the crude extra stuff added on.
You can see the previous episodes of this series here:
• Part 1: Platonic solids and Coxeter complexes.
• Part 2: Coxeter groups.
• Part 3: Coxeter diagrams.
• Part 4: duals of Platonic solids.
• Part 5: Interpolating between a Platonic solid and its dual, and how to describe this using Coxeter diagrams. Example: the cube/octahedron family.
• Part 6: Interpolating between a Platonic solid and its dual. Example: the dodecahedron/icosahedron family.
• Part 7: Interpolating between a Platonic solid and its dual. Example: the tetrahedron family.
• Part 8: The missing solids, coming from Coxeter diagrams with both ends blackened.