Single Comment

Jared, I believe the cells of the board shown here are topologically
connected in the same way as the rhombic dodecahedron tiling you mention.
Only the topological form of the board is relevant to play, so I wouldn't
think that the translated rules would be enlightening ... if I'm
visualizing correctly what you have in mind, I think it would be far
harder to understand what the game is about. The trouble is that in any
diagram I can imagine, you can only see a cross-section of each level,
which prevents the full geometric form of the 3D cell from being seen. If
you have 3D raytracer software you might be able to demonstrate it. I'd
be interested in seeing that too. The ideal thing would be a virtual
reality board, that players would see by donning those goggles that
present stereoscopic 3D images that you can see all sides of by moving
your head. When those become commonplace I predict a lot of wonderful 3D
games will get implemented on them. I still haven't seen that technology,
but I hope someday to use them to play Renju on a 'tetrahedral' board of
order 13 or so.

Charles, I'm reading your post for about the tenth time and am starting
to figure out what you're talking about. You say 'square roots' but I
believe you mean 'squares.' The base 36 business was confusing to me but
you're really just doing it for compactness, so you can indicate each
distance (or its square root) by a single character. And your use of
'coprime' doesn't seem to match the meaning I understand by that word.
But I'm interested to see that the cells to which a knight at your origin
can move are all labelled as distance sqrt(3) from the origin - well, that
would make sense, just as a FIDE knight's moves are all sqrt(5) in
length. Okay, I'm starting to follow your arithmetic - and I'm
surprised, I wouldn't have guessed that the centers of cells in a rhombic
dodecahedral grid would have distances whose squares are integers - though
now that you point it out, I don't see why not.

I'm not sure how playable your proposals for Unicorns and Nightriders
would be on this grid -- it seems to me that to give them sufficient scope
to practice their powers the board would have to be considerably larger
and so have a huge number of cells, and a IMO game whose board has too
many cells becomes too complicated to be interesting, because the moves
have so many consequences no human player can foresee them; hence, it
turns into a game of chance rather than skill. However, many people
disagree with me, and I would be glad to see other game developers try
their hand at this grid. If you're inspired, go for it!