Ezra Bradford wrote on Tue, Jul 15, 2008 07:28 PM UTC:
Yes, you've correctly identified the truncated octahedron with the desired property. Its hexagons are regular. (If you truncate all the way down to triangles, you have the cuboctahedron, dual to the rhombic dodecahedron.)
I agree that the truncated octahedral system seems to complicate matters unnecessarily. I suspect there's an advantage hiding in there somewhere but I haven't found it. I would tend to consider the hexagonal direction the single step, and the square direction a funny sort of diagonal - but then the board has the likely undesirable property that no two of its eight main axes are perpendicular.
Hmmm. That is an interesting geometric property. Note also that in the coloring of a board for this partial knight, a cell adjoins each of the four other colors exactly once on the orthogonals, and once on the diagonals.
PS for 3-d visualization I suggest vZome. http://www.vorthmann.org/zome/
It's very good at the cubic and rhombic dodecahedral systems, I find. (It's not much good at hex-prism just yet, but I'm still talking to Scott Vorthmann about that.)
I agree that the truncated octahedral system seems to complicate matters unnecessarily. I suspect there's an advantage hiding in there somewhere but I haven't found it. I would tend to consider the hexagonal direction the single step, and the square direction a funny sort of diagonal - but then the board has the likely undesirable property that no two of its eight main axes are perpendicular.
Hmmm. That is an interesting geometric property. Note also that in the coloring of a board for this partial knight, a cell adjoins each of the four other colors exactly once on the orthogonals, and once on the diagonals.
PS for 3-d visualization I suggest vZome. http://www.vorthmann.org/zome/ It's very good at the cubic and rhombic dodecahedral systems, I find. (It's not much good at hex-prism just yet, but I'm still talking to Scott Vorthmann about that.)