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Ben Reiniger wrote on Tue, Jun 25, 2013 03:22 AM UTC:In regard to dimensionality of chess variants, I think of it as the most convenient way to describe the rules. Any (most?) variants can be thought of purely combinatorially, with a set of locations and pieces occupying those locations, with a set of transitions between states. But often the transition rules are easiest to describe using some extra geometry. (Do FIDE rooks move orthogonally in 2d, or as in Eeeeeeeex as a 1 or 9 rider in 1d, or as some set of at most 14 locations that depend on the current locations of itself and the other pieces?)
Notice that it is possible to make the game which is the same as FIDE but has only one dimension. Therefore, how many dimensions does it really have? Does this help you to know how many dimension this game has? Probably not!
Ben Reiniger wrote on Mon, Jun 24, 2013 06:14 PM UTC:I'm not really sure if this qualifies as a 4D variant, but I've indexed it that way to be as inclusive as possible. If pieces couldn't turn at the knot, I would insist that it is 4D, but which is easier to explain here: four disjoint 1D boards with a glued square, or a subset of the 4D board with turning pieces? (Note that in the 4D interpretation, the Entity's teleport power is almost the same as a 4D bishop's move.)
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Continuous-space (non-discrete) games will require the correct number of dimensions. Geometry certainly does help to explain many things, and it is very convenient, as well as possibly supporting certain kind of mathematical generalizations; I am just saying that it is equivalent to the game with the different number of dimensions.
I myself would think this game described as one-dimensional with four boards sharing a square seems best, although doing it on a chessboard doesn't seem to work very well like that. But it could be described as 4D, if the rules for some of the pieces movements is corrected to make a bit more sense.