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. ...
I found it rather hard to predict how the topology of the board affects play. Hexagonal boards in the form of an 10*10 rhombus were interesting, some peices proved more or less useful, and needed some work to make them function.
ReplyDeleteWhat about hyperbolic go?
ReplyDeleteYou'd probably have to play go in the style of reversi, where you can only build onto an existing structure.
ReplyDeleteEduard Baumann, check out Jenn3D for Go played on spherical polytopes! See the very bottom of this page:
ReplyDeletehttp://www.math.cmu.edu/~fho/jenn/
wendy krieger, I believe you that it's difficult, based on my experience with my "global chess" board (spherical chess).
ReplyDeletehttp://www.pa-network.com/global-chess/indexf.html
That's quite hard to play, and I'm sure some of the hyperbolic topologies have to be even more mind-bending.