Just as a cube has 3 squares meeting at each vertex, a dodecahedron has 3 pentagons meeting at each vertex. So, instead of the Coxeter diagram for the cube:

everything today will be based on the Coxeter diagram for the dodecahedron:

The number 5 is much cooler than the number 4, which is, frankly, a bit square. So the shapes we get today look much more sophisticated, at least to my eyes. But the underlying math is very similar: we can use diagrams to keep track of these shapes as we did before.

Dodecahedron: •—5—o—3—o

First we have the dodecahedron, with all pentagons as faces:

I like this shape so much I gave a lecture about it, and you can see the slides here:

• John Baez, Tales of the dodecahedron: from Pythagoras through Plato to Poincaré.

Truncated dodecahedron: •—5—•—3—o

Then we get the truncated dodecahedron, with decagons (10-sided shapes) and triangles as faces:

Icosidodecahedron: o—3—•—3—o

Then, halfway through, we get the aptly named icosidodecahedron, with pentagons and triangles as faces:

Like that other ‘halfway through’ shape the cuboctahedron, every edge of the icosidodecahedron lies on a great circle’s worth of edges.

Truncated icosahedron: o—5—•—3—•

Then we get the truncated icosahedron, with pentagons and hexagons as faces:

This one is so beautiful that a whole sport was developed in its honor!

Icosahedron: o—5—o—3—•

And then finally we get the icosahedron, with triangles as faces:

Again, I like this one so much I gave a talk about it:

• John Baez, Who discovered the icosahedron?

I almost feel like telling you all the stuff that’s in these talks of mine… and if I turn these blog posts into a book, I’ll definitely want to include it all! But there’s a lot of it, and I’m feeling a bit lazy—so why not just go check it out?

Puzzle. Why did Thomas Heath, the great scholar of Greek mathematics, think that Geminus of Rhodes is responsible for the remark in Euclid’s Elements crediting Theatetus with discovering the icosahedron?