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There are a few more types of move in this geometry. For example, whereas wave paths turn in alternating directions like the path of a Crooked Bishop on a square board, there is also a possible path where the turn is repeated inthe same direction analogous to the Rose path. Regarding 2d and 3d, David Cannon has underestimated existing boards. A hex board can be seen as a subset of a cubic-cell 3d board, in a way analogous to the e.g. white squares of a square-cell board as a smaller square-cell board at 45° to the original.