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.
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.