Three Dimensions
For a three-dimensional grid, the only regular polytope that fills the whole space is the cube, while the only regular polytopes with a sufficiently large symmetry group are the dodecahedron and icosahedron (without the second constraint the model will suffer the same drawbacks as the HPP model). To make a model that tackles three dimensions therefore requires an increase in the number of dimensions, such as in the 1986 model by D'Humières, Lallemand and Frisch, which employed a face-centered hypercube model.
Read more about this topic: Lattice Gas Automaton
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