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The 5th dimension is just so tightly curled up that, after discretization, just looks like it's not present.

You. Nailed. It.

I think of the five dimensions as:

1. Horizontal within a board

2. Vertical within a board

3. Backwards and forwards along a timeline

4. Up and down across timelines

5. The timeline of how the game actually evolves, building out the tree. This is a bit of a stretch, since you can't move pieces through it in the same way as the other four.

It would have been nice if 4D only ever meant four spatial dimensions, and then we'd have (3+1)D for three spatial dimensions and a time dimension. (And (2+2)D for two spatial and two time dimensions :-))

2+2=4D could be for position and momentum in two dimensions.