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Chess Geometry I can see what you mean now. It is the old image of the hexagon with its diagonals drawn in that flips between representing transparent cubes facing in oppositte directions, as follows: -- /\/\ ---- \/\/ -- The Tetrahedral geometry corresponds to one Bishop binding of a cubic geometry. The asterisks on the 9x9x9 cubic board below indicate the plane corresponding to a hex board of side 5. If you colour the different cubic cells based on the four Unicorn bindings, you will find that this corresponds to colouring the hex cells based on the four Dabbaba bindings. Your projection will indeed project each of the unmarked cells onto one of the marked ones. Level 1 Level 2 Level 3 ----*---- -----*--- ------*-- ---*----- ----*---- -----*--- --*------ ---*----- ----*---- -*------- --*------ ---*----- *-------- -*------- --*------ --------- *-------- -*------- --------- --------- *-------- --------- --------- --------- --------- --------- --------- Level 4 Level 5 Level 6 -------*- --------* --------- ------*-- -------*- --------* -----*--- ------*-- -------*- ----*---- -----*--- ------*-- ---*----- ----*---- -----*--- --*------ ---*----- ----*---- -*------- --*------ ---*----- *-------- -*------- --*------ --------- *-------- -*------- Level 7 Level 8 Level 9 --------- --------- --------- --------- --------- --------- --------* --------- --------- -------*- --------* --------- ------*-- -------*- --------* -----*--- ------*-- -------*- ----*---- -----*--- ------*-- ---*----- ----*---- -----*--- --*------ ---*----- ----*----
I was reading over the Tetrahedral Chess page a while back, and decided to understand the statement that certain (skew) planes form hexagonal chessboards. I can confirm now that this is true, and Gilman's M&B even enumerates the number of such planes through each cell. Now note that the 4-coloring of Tetrahedral Chess lends all four of its colors to these hex-boards. I cannot recall finding a hex chess that uses a four-colored board. However, a longer period of time ago, I noticed that the hex board can be thought of as a certain quotient (that's a technical math term in this context) of a 3d cubic board along a single unicorn line. (For those not used to the math lingo, think of it as an optical illusion: you look at the cubic board so as to line up opposite corners, and all cells along your line of sight are treated as being equivalent.) This quotienting does some weird things with the pieces, but what about the colors? The only obvious coloring that could be maintained by this quotient is the unicorn's 4-binding! It probably comes as no shock that this coloring is the same as in Tetrahedral Chess (how many 4-colorings can there be of a hex board?) I think furthermore that some of the pieces in Tetrahedral chess, when restricted to one of the hex-planes, turn out to be very similar to the cubic pieces modulo the unicorn's diagonal. (I had worked some of this out, but don't have the notes handy.)
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