Truncating from

o-3-•-3-o (octahedron)

gets you to

•-3-•-3-• (truncated octahedron).

So the numbers don’t really matter. In all cases, truncating from

o—•—o

gets you to

•—•—•

There seems to be a pattern here. Truncating from

•—o—-o

gets you to

•—•—o

which fills in to the right. And truncating from

o—o—•

gets you to

o—•—•

which fills in to the left. So it seems plausible that starting from the center fills in both sides.

OTOH, truncating from

•—o—•

doesn’t get you to

•—•—•

But the center would be double filled in. Extending your notation:

•—:—•

But there’s no reason to expect that to be a uniform polyhedron…

]]>I bet you thought this series had died. But it was only snoozing.

“That is not dead which can eternal lie, And with strange aeons even death may die…“

]]>So back in 1977 after botching a quals question on the simplicity of A

_{5}, I took some old APL output I’d never managed to draw, describing the Cayley graph of A_{5}. Lo and behold, it was the edges of this solid!I stayed up and made one, which I still have in my office.

I replied:

]]>Sorry about that qual question! But your comment is very interesting. The fancier solid called the truncated icosidodecahedron, with 120 vertices, can be seen as the Cayley graph of the group of all rotation and reflection symmetries of the dodecahedron: A

_{5}× Z/2. This is part of a more general pattern that I’ll explain soon in this series. But I hadn’t noticed that the rhombicosidodecahedron, with the 60 vertices, can be seen as the Cayley graph of the group of all rotation symmetries of the dodecahedron, namely A_{5}. I suspect this is part of a general pattern too! I’ll try to figure this out…

The Coxeter diagram with no dots blackened does not give a uniform polyhedron… or if it does, it’s one so degenerate I don’t want to bother with it.

Thanks for catching all these mistakes! I think I’ve fixed them all now.

]]>Wikipedia uses an *ad hoc* way of decorating Coxeter diagrams to denote the snub polyhedra—see here for how they fit the snub cube into the cube/octahedron series. But this doesn’t fit into the general theory of Coxeter groups and their ‘parabolic subgroups’, which is what the marked Coxeter diagrams is secretly all about.

Remember, in this game the left-hand dot is always the ‘vertex’ dot, the middle dot is the ‘edge’ dot, and the right-hand dot is the ‘face’ dot.

I’ve fixed these typos… thanks. I’ve always changed a few other things. I hope the article makes more sense now!

]]>