Have $3$-dimensional cellular automata been studied at all? Recently, I have become interested in the Game of Life and other similar cellular automata. I notice how these cellular automata operate in a square tiling on $2$-space, and it seems as if similar cellular automata could operate on a cubic tiling in $3$-space. Have these $3$-dimensional cellular automata been studied much? Are any interesting specific examples known?

Of course the general theory of cellular automata is developed in any dimension (and more...), but there is a qualitative gap between dimension 1 and dimension 2: _i.e._ many results apply to dimension 2 and more, many others to dimension 1 only. This can partly explain why dimension 3 is less considered.

However there are sometimes specific 3D aspects, for instance:

1. the majority vote CA is hard to predict in 3D, but in 2D it's an open problem;
2. idem for the sandpile model: 3D hard, 2D unknown.



The intuitive explanation beyond this 2D/3D difference: embedding arbitrary information flows is easy in 3D, harder in 2D due to crossing problems. It is easier for a simple rule to induce a complex behavior in 3D than in 2D, and than in 1D even more.