A honeycomb of bucky-balls Here are two lovely honeycombs, the bitruncated {5,3,5} and bitruncated {3,5,3}. The first is a honeycomb of bucky-balls! (a.k.a. truncated icosahedra, which look like soccer balls.) Both are cell transitive , meaning every cell is exactly the same as every other. They are also vertex transitive , so every vertex is identical as well. Since they are vertex transitive, this also means they are examples of uniform honeycombs. I think they are edge transitive too! A truncated icosahedron is not edge transitive, so how can a honeycomb built of them suddenly be edge transitive? On a truncated icosahedron, some edges join two hexagons, and some join 1 hexagon and 1 pentagon, so you can't move one edge type to another as a symmetry. But on the honeycomb, all edges are connected to 2 hexagons and 1 pentagon. These two honeycombs are so close to being regular, but do fail in one respect. They are not face transitive because each has two kinds of faces. ...
The sad thing about spherical models of H3 is that the infinitely dense horizon obscures the large-scale centre! I like that you cut the {4,4,4} in half, presumably for this reason. I wonder about half-space models.
ReplyDeleteYep, that was one of the main reasons. The other nice thing about cutting in half is saving on printing costs. We should do half-ball models of these as well.
ReplyDeleteWe have played some with half-space models too, but haven't printed anything yet. Culling out the large parts in a nice way will be a challenge. I'll post some of the experimental pictures in a bit.