Consider a multidimensional lattice that is thick, perhaps infinite in extent, in only 2 of its dimensions. The other dimensions it is only a few, perhaps 3, cells thick. Hypercube honeycomb is the obvious lattice, but it need not be.
The goal is to implement cellular automata in hardware. We need at least 3 dimensions to allow signals to easily cross, but thick 3 dimensions or more thwart implementing it on a mostly two-dimensional microchip. Thin dimensions greater than 2 do not require too many wire crossings.
Having multiple dimensions is an alternative to allowing each cell to have many states. I think it allows complexity to be allocated more finely in the places it is needed, rather than requiring every identical cell to support many states, which current multicore computers are like.
Curiously, this is the same structure as the universe as posited by string theory, except with thin dimensions 4 through about 11, and the thin dimensions wrap around to get periodic boundary conditions.
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