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Because normal 2D chess is played in 3D, the added two dimensions make up 5D.
Or consider this: In 2D chess, you pick up the pieces, (use the THIRD dimension) to move them around. Otherwise, the pieces would need to phase through each other. In these 4D, you would need a 5th dimension to move these pieces around in. So I guess if you tried to play this game in full 4D, you would need 5D to make it happen!
1) length
2) height
3) depth (not represented)
4) time
5) abstract (in this case, parallel timeline)
Read.
My view is that it's really Quantum Chess, but that's not the best name for a videogame.
In Soviet Russia, 5th dimension is YOU!