August 28, 1995

This Week's Finds in Mathematical Physics (Week 62)

John Baez

Now I'd like to talk about a fascinating subject of importance in both mathematics and physics, the subject of "ADE classifications". Here A, D, and E aren't abbreviations for anything; they are just names for certain diagrams. But these diagrams show up all over the place when you start trying to classify beautiful and symmetrical things.

Let's start with something nice and simple: the Platonic solids. It's not terribly hard to classify all the regular polyhedra in 3-dimensional Euclidean space. Roughly, it goes like this. The faces could all be equilateral triangles. Obviously there need to be at least 3 faces meeting at each vertex to get a polyhedron. If there are exactly 3, you have a tetrahedron. If there are 4, you have an octahedron. If there are 5, you have an icosahedron. There can't be 6 or more, since when you have 6 they lie flat in the plane, and more is even worse. The faces could also be squares. If there are 3 squares meeting at each vertex you have a cube. There can't be 4 or more, since when you have 4 they lie flat in the plane. The faces could also be regular pentagons. If there are 3 pentagons meeting at each vertex you have a dodecahedron. There can't be 4 or more, since when you have 4 you already have more than 360 degree's worth of angles.

So, there we are: the 5 regular polyhedra are the tetrahedron, octahedron, icosahedron, cube, and dodecahedron! Of course, we haven't shown these solids actually exist. Sometimes people forget that you really need to check that all these possibilities are realized! But the Greeks did that a while back. This is perhaps the first example of an ADE classification.

This had such beauty that in his "Timaeus" dialog, Plato suggested that the 4 elements were made of these solids, not counting for the dodecahedron. Interestingly, Plato considered decomposing the faces of these solids into "elementary triangles", in order to explain how one element could turn into another. This is presumably why he left out the dodecahedron: one can't chop up a regular pentagon into 30-60-90 triangles. In a passage that's notoriously hard to translate, he suggested that the dodecahedron corresponding to some sort of "quintessence", or perhaps the zodiac. It's worth pointing out, also, that Plato explicitly says it's okay if someone comes up with a better scheme. He makes it clear that he is just trying to lay out an example of a mathematical scheme for explaining the elements, to get people interested.

Later, of course, Kepler suggested that the 5 Platonic solids corresponded to the orbits of the 5 planets:

As it turns out, Plato and Kepler were in the right ball-park, but not really right. Both the solar system and atoms are described pretty well by similar laws - the inverse-square force laws for gravity and electrostatics. And solving this problem (in either the classical or quantum case) does indeed require a deep understanding of rotations in 3-dimensional space. It's sort of amusing, however, that the Platonic solids have as their symmetries finite subgroups of the rotation group in 3 dimensions, while the study of quantum-mechanical atoms instead involves the theory of "representations" of this group, which are in some sense dual. The rotation group in n dimensions, by the way, is called SO(n). See "week61" for a bit more about it. For a grand tour of the inverse square law, both classical and quantum, read:

1) Victor Guillemin and Shlomo Sternberg, Variations on a Theme by Kepler, American Mathematical Society, Providence, Rhode Island, 1990.

You will see, among other things, that the real reason the inverse square force law problem is exactly solvable is that it has a hidden symmetry under SO(4), not just SO(3).

But I digress! Recall how I said that "obviously" a regular polyhedron has to have 3 faces meeting at each vertex? What would happen if you relaxed the definition a little bit, and let there be just 2 faces meeting at a vertex? Well, then any regular polygon could qualify as a regular polyhedron, I guess. Then we would have an infinite series of regular polyhedron with only two faces, together with 5 exceptions, the Platonic solids. That's actually typical of ADE-type classifications: often, when you are classifying really symmetrical things, you find some infinite series of "obvious" or "classical" cases, together with finitely many weird "exceptional" cases.

Before I get further into ADE classifications, let me note that the problem of why there are so many ADE classifications, and how they are all related, was explicitly raised by the famous mathematical physicist V. I. Arnol'd, in

2) Problems of Present Day Mathematics in Mathematical Developments Arising from Hilbert's Problems, ed. F. E. Browder, Proc. Symp. Pure Math. 28, American Mathematical Society, Providence, Rhode Island, 1976.

This lists a lot of important math problems, following up on Hilbert's famous turn-of-the-century listing of problems. Problem VIII in this book is the "ubiquity of ADE classifications". Arnol'd lists the following examples:

Don't worry if you don't know what those are except for the first one! I'll try to explain some of them. Later I'll also explain two new ones that came out of string theory:

Perhaps the best single place to start learning about ADE classifications is:

3) M. Hazewinkel, W. Hesselink, D. Siermsa, and F. D. Veldkamp, The ubiquity of Coxeter-Dynkin diagrams (an introduction to the ADE problem), Niew. Arch. Wisk., 25 (1977), 257-307. Also available at http://repos.project.cwi.nl:8888/cwi_repository/docs/I/10/10039A.pdf or http://math.ucr.edu/home/baez/hazewinkel_et_al.pdf

Okay, so what the heck is an ADE classification, after all? It's probably good to start by looking at "finite reflection groups." Say we are in n-dimensional Euclidean space. Then given any unit vector v, there is a reflection that takes v to -v, and doesn't do anything to the vectors orthogonal to v. Let's call this a "reflection through v". A finite reflection group is a finite group of transformations of Euclidean space such that every element is a product of reflections. For example, the group of symmetries of an equilateral n-gon is a finite reflection group. (This is a useful exercise if you don't see it right off the bat.)

Note that if we do two reflections, we get a rotation. In particular, suppose we have vectors v and w at an angle A from each other, and let r and s be the reflections through v and w, respectively. Then rs is a rotation by the angle 2A. Draw a picture and check it! This means that if A = π / n, then (rs)n is a rotation by the angle 2π, which is the same as no rotation at all, so (rs)n = 1. On the other hand, if A is not a rational number times π, we never have (rs)n = 1, so r and s can not both be in some finite reflection group.

With a little more work, we can convince ourselves that any finite reflection group is captured by a "Coxeter diagram". The idea is that the group is generated by reflections through unit vectors that are all at angles of π/n from each other. To keep track of things, we draw a dot for each one of these vectors. Suppose two of the vectors are at an angle π/n from each other. If n = 2, we don't bother drawing a line between the two dots. Otherwise, we draw a line between them, and label it with the number n. Typically, if n = 3 people don't bother writing the number; they just draw that line. That's what I'll do. (People also sometimes draw n - 2 lines instead of writing the number n, but I can't do that here.)

Algebraically speaking, if someone hands us a Coxeter diagram like

      7
o---o---o

we get a group having one generator for each dot, and with one relation r2 = 1 for each generator r (since that's what reflections do), and one relation of the form (rs)n = 1 for each line connecting dots, or (rs)2 = 1 if there is no line connecting two dots. It turns out that if a Coxeter diagram yields a finite group this way, it's a finite reflection group.

However, not every diagram we draw yields a finite group! Here are all the possible Coxeter diagrams giving finite groups. They have names. First there is An, which has n dots like this:

o---o---o---o

For example, the group of symmetries of the equilateral triangle is A2. The two dots can correspond to the reflections r and s through two of the altitudes of the triangle, which are at an angle of π/3 from each other. Thus they satisfy (rs)3 = 1. More generally, An corresponds to the group of symmetries of an n-dimensional simplex - which is just the group of permutations of the n+1 vertices.

Then there is Bn, which has n dots, where n > 1:

          4
o---o---o---o

It has just one edge labelled with a 4. Bn turns out to be the group of symmetries of a hypercube or hyperoctahedron in n dimensions.

Then there is Dn, where n > 3:

              o
             /
o---o---o---o
             \
              o

Then there are E6, E7, and E8:

      o
      |
o--o--o--o--o

      o
      |
o--o--o--o--o--o

      o
      | 
o--o--o--o--o--o---o

Interestingly, this series does not go on. That's what I meant about "classical" versus "exceptional" structures.

Then there is F4:

      4
o---o---o---o

Then there's G2:

  6
o---o

and H3 and H4:

  5
o---o---o

  5
o---o---o---o

H3 is the group of symmetries of the dodecahedron or icosahedron. H4 is the group of symmetries of a regular solid in 4 dimensions which I talked about in "week20". This regular solid is also called the "unit icosians" - it has 120 vertices, and is a close relative of the icosahedron and dodecahedron. One amazing thing is that it itself is a group in a very natural way. There are no "hypericosahedra" or "hyperdodecahedra" in dimensions greater than 4, which is related to the fact that the H series quits at this point.

Finally, there is another infinite series, Im:

  m
o---o

This corresponds to the symmetry group of the 2m-gon in the plane, and people usually require m = 5 or m > 6, so as to not count twice some Coxeter diagrams that we've already run into.

THAT'S ALL.

So, we have an "ABDEFGHI classification" of finite reflection groups. (In some future week I had better say what happened to "C".) Note that the symmetry groups of the Platonic solids and some of their higher-dimensional relatives fit in nicely into this classification, so that's one sense in which the Greeks' discovery of these solids counts as the first "ADE classification". But there is at least one another, deeper, way to fit the Platonic solids themselves into an ADE classification. I'll try to say more about this in future weeks.

You may still be wondering what's so special about A, D, and E. I'll have to get to that, too.


© 1995 John Baez
baez@math.removethis.ucr.andthis.edu