isotropy in chess piece movement is elegant, but because armies start on opposite sides and pawns can't move backwards, forward and backward movement are inherently different in chess. shogi has this reflected in piece types, for example, the gold general which can advance diagonally but not retreat diagonally. we enumerate possible fairy chess pieces which are left-right symmetric but not necessarily forward-backward symmetric. previously on anisotropy.
compounds of deconstructed wazir, ferz, rook, bishop, dabbaba, strong dabbabarider, knight, and alfil. product of two C-shaped regions around a piece.
inner C: 3^5 = 243
outer C (jumping pieces): 3^3 * 2^6 = 1728
dabbabarider, the only jumping rider we choose to include, grew the outer C from 2^9 = 512.
one can also think of it as symmetries as inducing equivalence classes among directions.
there are now 15 variations of knight (not including immobile stone) because of its 4 possible amounts of forward or backward movement. this includes crab but not pinwheel knight.
total is 3^8 * 2^6 = 419904, a huge increase from isotropic 72. this includes piece types which can only move backward and piece types confined to their starting rank or file. perhaps cull these, because they seem practically (nearly) useless unless the game has drops like shogi.
raise that to the Nth power to investigate N-piece endgames. there is a question of what pawns may promote to (future post mpyfvglu). finding interesting positions and mechanics will be art.
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