The n-Category Café

Tim wrote:

Why is the icosahedron a pure mathematical creation in contrast to the other four platonic solids?

I was quoting Benno Artmann on this because it’s an interesting notion, not because I necessarily agree with it.

I think his idea is that the Greeks could have seen things in nature shaped like regular tetrahedra, cubes, regular octahedra and roughly like regular dodecahedra. Or else ordinary folks could have invented some of these shapes without any mathematical thought. On the other hand, the icosahedron might have been discovered as part of the classification of regular polyhedra.

This classification theorem appears as a remark right near the end of Book XIII of Euclid’s Elements. It seems quite possible that Theaetetus was the first to prove this theorem, and that he discovered the icosahedron at about the same time.

If so, the icosahedron might be the first example of an ‘exceptional object’ in mathematics: roughly speaking, an object that’s not part of a systematic family of objects as its type, which is discovered while proving a classification theorem.

A more sophisticated example of an exceptional object is the simple Lie group $E_8$. And interestingly, the group $E_8$ is deeply connected to the icosahedron!

Is it because it does not have an “associated crystallographic point group” (meaning perfect crystals with it’s symmetry group do not exist) unlike the dodecahedron?

The regular dodecahedron doesn’t have an associated crystallographic point group either! The symmetries of the dodecahedron are the same as those of the icosahedron.

So what’s the difference?

Iron pyrite commonly forms ‘pyritohedra’, which look similar to regular dodecahedra, but are made of little cubic crystal cells, like this:

See:

• Steven Dutch, Building isometric crystals with unit cells.

The pyritohedron has 12 pentagonal faces, orthogonal to these vectors:

(2,1,0)   (2,-1,0)   (-2,1,0)   (-2,-1,0)
(1,0,2)   (-1,0,2)   (1,0,-2)   (-1,0,-2)
(0,2,1)   (0,2,-1)   (0,-2,1)   (0,-2,-1)

On the other hand, a regular dodecahedron has 12 pentagonal faces orthogonal to the following vectors, in which the number 2 has been replaced by the golden ratio Φ $=(\sqrt{5}+1)/2$:

(Φ,1,0)   (Φ,-1,0)   (-Φ,1,0)   (-Φ,-1,0)
(1,0,Φ)   (-1,0,Φ)   (1,0,-Φ)   (-1,0,-Φ)
(0,Φ,1)   (0,Φ,-1)   (0,-Φ,1)   (0,-Φ,-1)

These vectors are also the vertices of a regular icosahedron!

But if we use the number 2 instead of the number $\Phi$, we get the vertices of a so-called ‘pseudoicosahedron’:

Apparently iron pyrite can also form a pseudoicosahedron! But I think this happens very rarely. I’ve asked around to see a photo of such a crystal, but nobody has ever shown me one.

In short, it’s probably just some subtle property of iron pyrite that caused the Greeks in Sicily — where pyrite is abundant — to see almost-regular dodecahedra, but not almost-regular icosahedra.

I wrote a lot more about this in week241. It turns out the pyritohedron is just one of a sequence of ‘fool’s dodecahedra’ in which $\Phi$ is approximated better and better by ratios of Fibonacci numbers. And some of the better approximations do arise in nature!

Also, the connection between $E_8$ and the icosahedron is equally well a connection between $E_8$ and the dodecahedron. For more on that see the end of week241, and also the appendix to my talk on the number 5 — the appendix entitled Quaternions, the dodecahedron and $E_8$.

The cuboctahedron also has a bunch of equators.

I don’t know ‘why’ the octahedron is the only Platonic solid with equators. It’s an interesting question, but I don’t know how to get a handle on it.

(An easier question: the Poincaré dual of the octahedron is the cube. What’s special about the cube, corresponding to how the octahedron has equators?)