Suppose are trying to classify all the Platonic solids. We’re looking for ways to tile the surface of a sphere with regular -gons, with meeting at each vertex. Suppose there are a total of vertices, edges, and faces. Since the Euler characteristic of the sphere is 2, we have

Since each face has edges but 2 faces meet along each edge, we have

Since each vertex has edges meeting it but each edge meets 2 vertices, we also have

Putting these equations together we get

or

Of course this implies the inequality we’ve already seen:

A priori the equation is stronger than the inequality, but it just happens to be equivalent, at least when

Whenever a sum of two reciprocals of two numbers like this exceeds 1/2, it exceeds 1/2 by a reciprocal of a number like this! And this number is the number of edges of your Platonic solid…

… or hosohedron, or dihedron.

For example

gives the cube, but in fact

so the cube has 12 edges! Simple but pretty stuff.

The square tiling is self-dual; the other two are dual to each other.

These tilings correspond to the solutions of

where are positive integers.

That’s weird.

Yes, the modularity theorem is weirdly self-referential. I’ve always been puzzled why everyone explaining this stuff doesn’t mention that.

It’s even more amusing that this self-referential result is what it took to prove something seemingly ‘concrete’ like Fermat’s Last Theorem. As it happened, Gerard Frey suggested that a counterexample to Fermat’s Last Theorem

would give an elliptic curve

that couldn’t be covered by a modular one. This was later proved by Jean-Pierre Serre and Kenneth Ribet. And that made it clear what had do be done to prove Fermat’s Last Theorem: prove the modularity theorem, or at least enough of it to cover this case.

Modular curves can be seen as parametrizing families of elliptic curves. [..] Sometimes an elliptic curve can be covered by a modular curve using a so-called ‘branched cover’. There’s a big theorem called the modularity theorem which says (very roughly) that all elliptic curves defined by equations using just rational numbers can be covered in this way by modular curves.

That’s weird: a modular curve, equipped with some additional structure defining the ‘branched quotient’ and this torsion subgroup iso, becomes itself a *point* in a modular curve! (In general, I suppose that this second curve is in general different from the first.)

I guess you’ve seen John McKay’s terse blast of information on this subject. It mainly serves to make you want to learn more.

Since Klein’s quartic curve is a curled-up piece of the hyperbolic plane, it’s called a ‘modular curve’. Modular curves can be seen as parametrizing families of elliptic curves (roughly speaking, tori) equipped with extra structure.

This is a nice big story already. However, it then takes a weird turn and gets much more intense.

There’s another weirder relation between modular curves and elliptic curves. Sometimes an elliptic curve can be covered by a modular curve using a so-called ‘branched cover’. There’s a big theorem called the modularity theorem which says (very roughly) that all elliptic curves defined by equations using just rational numbers can be covered in this way by modular curves.

And this theorem implies Fermat’s Last Theorem!

So, as usual in number theory, the flashy easy-to-explain result follows as a corollary from something that’s harder to explain, but ultimately more interesting and more connected to geometry and symmetry.

The proof of the modularity theorem is not easy: Andrew Wiles and a grad student of his proved enough of it to get Fermat’s Last Theorem, and other good mathematicians finished off the job.

Instead of reading about that, it’s a lot less stressful to start with:

• Tim Silverman, Pictures of Modular Curves (Part I), n-Category Caf&eeacute;, 10 October 2006.

and read all 11 parts. You’ll get a deeper understanding of creatures like

as shown here:

But, understanding which Coxeter diagrams give which things isn’t simply a matter of taking sums of reciprocals. We’ll see, I guess!