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Ben Reiniger wrote on Tue, Jan 31, 2012 06:44 PM UTC:
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.)

Charles Gilman wrote on Thu, Feb 2, 2012 06:54 AM UTC:
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
---------       ---------       ---------
---------       ---------       ---------
--------*       ---------       ---------
-------*-       --------*       ---------
------*--       -------*-       --------*
-----*---       ------*--       -------*-
----*----       -----*---       ------*--
---*-----       ----*----       -----*---
--*------       ---*-----       ----*----

Charles Gilman wrote on Fri, Feb 3, 2012 07:00 AM UTC:
Actually I can see more of it now that you've thought of collapsing the
board along a Unicorn line. It is not hard to see that, given that 1:1:0
and 1:-1:0 are at right angles, so are 1:1:1 and 1:-1:0 - the principle on
which Man and Beast 09's Hyperrhino alternator is built. However, it is
equally clearly at right angles to 1:0:-1 and 0:1:-1 - so the three axes of
the Bishop at right angle to any axis of the Unicorn correspond to the
three axes of the Rook on the hex board.

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