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With the addition of the 'pentagonal' boards, this article now covers five of the most familiar of the nineteen uniform and dual-uniform tilings. Several of the others -- Floret pentagonal comes to mind -- can be addressed using the AVERSE system, so coverage is likely much better than that.

I'd like to suggest the quadrilateral tilings on this list for further study. Quadrilateral tilings tend to suggest obvious interpretations of FIDE pieces, but when the quadrilaterals are not rectangles, those pieces may behave oddly. On the deltoidal trihexagonal tiling, the obvious Rook travels in a twelve-step circle.

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 see several principal radial directions, each generally with its own SOLL:

- Wazir-wise
- Ferz-wise
- Viceroy-wise
- Rumbaba-wise

Their unique characteristic, setting them apart from other directions, appears to be that as moved in space they do not move through cells other than the one they're leaving and the one they're entering (though they may pass along edges, as Viceroy, or faces, as Rumbaba).

In the rhombic dodecahedral board, I observed an interesting direction. (I tried reading Oddly Oblique to see if you'd named it there, but I couldn't tell; your construction of the pieces for a rhombic dodecahedral board was difficult to follow without more knowledge than I have of your cubic pieces and of your reducing process.) Its SOLL is 6, analogous to your cubic Sexton, and it can be found by making three mutually perpendicular Ferz steps (not possible in cubic) or equivalently a Ferz step and a perpendicular Dabbabah step. (I am guessing it is the one that Oddly Oblique calls a Sexton?)

The interesting property is that it, too, does not pass through any other cell. It passes along the edges of six other cells (three a Wazir step away, three a Viceroy step away) but the first cell it enters is the destination.

I've illustrated it here. The starting square is red; green is a Wazir step, while blue is a Ferz step. Yellow is a Viceroy step. The direction in question is included in purple.

Only distantly related, but also worthy of note, the family of shapes that brought us cubic and rhombic dodecahedral has a third member. The truncated octahedral tiling is as yet unexplored in chess variants, to my knowledge. It is found by removing three-quarters of the cells of a cubic; if we consider the chess pieces as residing on points instead of in cells, it is a body-centered cubic crystal system (as compared to a simple cubic, a hexagonal, or rhombic dodecahedral's face-centered cubic). I don't think its properties are as interesting as those of the other varieties, but it could be worth investigating.

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