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Check out Janggi (Korean Chess), our featured variant for December, 2024.
Check out Janggi (Korean Chess), our featured variant for December, 2024.
I see several principal radial directions, each generally with its own SOLL:
- Wazir-wise
- Ferz-wise
- Viceroy-wise
- Rumbaba-wise
You observe each of these, and treat on them at length (though as it appears in only one geometry, #4 gets a separate article).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.