OctHex 146

This was my firt attempt at a variant in the xyrixa geometry as used in, and named after an obscure variant with the same board as, Mark Thompson's Tetrahedral Chess. On examining Tetrahedral Chess I identified the cell shape as that resulting from removing cubic cells of one Bishop binding and expanding a sixth of each neighbouring cell of the other binding into the void. Another contributor put a name to this shape: rhombic dodecahedron. This gives the board 12 orthogonal and 6 diagonal directions, the reverse of the cubic board. Likewise it gives a vast 24 Unicorn directions but only 8 Sexton (root 6 leaper) ones. Numbers of Knight and Camel directions are unchanged from cubic as both are 24 anyway.

My reason for choosing a board in the shape of an octahedron is twofold. Firstly, this shape complements Mr. Thompson's tetrahedron - neither can tesselate on its own, but the two can together, in a ratio averging twice as many tetrahedra as octahedra, as Maurirts Escher's lithograph Flatworms illustrates. Secondly it helps visualise the two kinds of plane in this geometry, for like the hex-prism one it contains both square and hex planes. As with the board in the shape of a tetrahedron, each edge of the board is an orthogonal, and each face a 21-cell hex plane, but additionally each of the three symmetric bisectors is a 36-cell square plane.

It is worth contrasting this variant with Michael Ward's Octahedral Chess, from which it differs in several ways. Firstly, the dimensions are 11x11x11 if each dimension is square board, as against Octahedral's 8x8x9. Yet the number of cells is far smaller, at 146 as against 340, and the number of true ranks, which are hex planes, is 6. The key to this is that each Bishop binding, of which there are four, can be viewed as a cubic-cell board. The three based on the three pairs of vertices have 1+5+13+13+5+1=38 cells each, while the remaining one has 4+12+12+4=32 cells. This adds up to 38+38+38+32=146 cells.

Setup

The single-cell levels are the top and bottom, with the 6x6 level in the middle. Each player's symmetric pieces are on a single hex plane, and each player's Pawns are also on a single hex plane.

Pieces

	The ROOK moves along the orthogonals. Its shortest move is to any of up to 12 equal nearest cells: the 4 obvious ones on the same level, the 4 closest to directly above it on the next level up, and the 4 closest to directly below it on the next level down. Having taken one step in such a direction it can continue that move in the same direction until it captures an enemy or is blocked by an ally. It always moves within one square plane and two hex planes.
	The BISHOP moves along the standard diagonals. Its shortest move is to any of up to 6 cells: the 4 obvious ones on the same level, and the first directly above and below it. Having taken one step in such a direction it can continue that move in the same direction until it captures an enemy or is blocked by an ally. It always moves within two square planes, and never within a hex one. It can reach alternate cells of alternate levels, averaging to a quarter of all cells. Three Bishops aside start on the board, one on each vertex and one off-board.
	The UNICORN moves along the hex diagonals. Its shortest move is to any of up to 24 cells: the 8 on each adjacent level immediately surrounding the 4 one-step Rook destinations, and the 4 on each level beyond those immediately surrounding the one-step Bishop directions. Having taken one step in such a direction it can continue that move in the same direction until it captures an enemy or is blocked by an ally. It always moves within two hex planes, and never within a square one. Surprisingly, a Unicorn can reach the entire board - in contrast to being bound to a quarter of a cubic-cell board and a third of one hex plane on a hex-prism one! On the other hand the small size of the hex planes limits Unicorns to two steps on this board.
	The KNIGHT simultaneously moves two steps along any orthogonal and one along the orthogonal at right angles to that one. It cannot move any further. It has up to 24 destinations - 8 on the same level, 4 on each adjacent one, and 4 each on the 3rd above and below. It always moves within one square plane, and never within a hex one.

b  r  b            u u          r u r             b
 BNRNB   ------   -----   R--R   ---   --   B
 NBRBN   -NUUN-  u-RUR-u  -NN-  u-B-u  --
rRR*RRr  -URRU-   -UBU-   -NN-   ---
 NBRBN   -URRU-  u-RUR-u  R--R  r u r
 BNRNB   -NUUN-   -----
b     b  ------    u u

	The QUEEN is the compound of Rook and Bishop. It always moves through a square plane, and either a second one or two hex planes.
	The DUCHESS is the compound of Rook and Unicorn. It always moves through two hex planes, and sometimes one square one.
	The MARSHAL is the compound of Rook and Knight. It always moves through one square plane, and sometimes two hex ones.
	The GOVERNOR is the compound of Bishop and Unicorn. It always moves through two square or two hex planes, and never one or more of each.
	The CARDINAL is the compound of Bishop and Knight. It always moves through one or two square planes, and never through a hex plane.
	The CAVALCADE is the compound of Unicorn and Knight. It always moves through two square planes or one hex plane, and never one or more of each.

The PAWN moves one step in any of 3 orthogonal directions except when capturing, when it moves one step in any of 3 diagonal ones. In both cases the 3 directions are those taking it away from its own camp toward the enemy one. A guide to these directions is that a Pawn's move from its starting cell is never to any other piece's starting cell.

Rules

The first time each player moves a non-Bishop compound, the fourth Bishop fills the vacated cell, as these are the only back-rank cells on the non-vertex binding.

A Pawn reaching the starting cell on a symmetric enemy piece is promoted to any array piece.

Victory is by capturing all enemy... Rooks and Rook compounds, or Bishops and Bishop compounds, or Unicorns and Unicorn compounds, or Knights and Knight compounds. This is modelled on my recent variants Notchess and Nichtschach, as there was no room for a King or Emperor in the array.