Other geometries [for cellular automata]
Systems like cellular automata can readily be set up on any geometrical structure in which a limited number of types of cells can be identified, with every cell of a given type having a similar neighborhood.
In the simplest case, the cells are all identical, and are laid out in the same orientation in a repetitive array. The centers of the cells form a lattice, with coordinates that are integer multiples of some set of basis vectors. The possible complete symmetries of such lattices are much studied in crystallography. But for the purpose of nearest-neighbor cellular automaton rules, what matters is not detailed geometry, but merely what cells are adjacent to a given cell. This can be determined by looking at the Voronoi region (see page 987) for each point in the lattice. In any given dimension, this region (variously known as a Dirichlet domain or Wigner-Seitz cell, and dual to the primitive cell, first Brillouin zone or Wulff shape) has a limited number of possible overall shapes. The most symmetrical versions of these shapes in 2D are the square (4 neighbors) and hexagon (6) and in 3D (as found by Evgraf Fedorov in 1885) the cube (6), hexagonal prism (8), rhombic dodecahedron (12) (e.g. face-centered cubic crystals), rhombo-hexagonal or elongated dodecahedron (12) and truncated octahedron or tetradecahedron (14) (e.g. body-centered cubic crystals), as shown below. (In 4D, 8, 16 and 24 nearest neighbors are possible; in higher dimensions possibilities have been investigated in connection with sphere packing.) (Compare pages 1029 and 986.)
In general, there is no need for individual cells in a cellular automaton to have the same orientation. A triangular lattice is one example where they do not. And indeed, any tiling of congruent figures can readily be used to make a cellular automaton, as illustrated by the pentagonal example below. (Outer totalistic codes specify rules; the first rule makes a particular cell black when any of its five neighbors are black and has code 4094. Note that even though individual cells are pentagonal, large-scale cellular automaton patterns usually have 2-, 4- or 8-fold symmetry.)
There is even no need for the tiling to be repetitive; the picture below shows a cellular automaton on a nested Penrose tiling (see page 932). This tiling has two different shapes of tile, but here both are treated the same by the cellular automaton rule, which is given by an outer totalistic code number. The first example is code 254, which makes a particular cell become black when any of its three neighbors are black. (Large-scale cellular automaton patterns here can have 5-fold symmetry.) (See also page 1027.)