Block Cellular Automaton - Neighborhoods

Neighborhoods

The simplest partitioning scheme is probably the Margolus neighborhood, named after Norman Margolus, who first studied block cellular automata using this neighborhood structure. In the Margolus neighborhood, the lattice is divided into 2-cell blocks (or 2 × 2 squares in two dimensions, or 2 × 2 × 2 cubes in three dimensions, etc.) which are shifted by one cell (along each dimension) on alternate timesteps.

A closely technique due to K. Morita and M. Harao consists in partitioning each cell into a finite number of parts, each part being devoted to some neighbor. The evolution proceeds by exchanging the corresponding parts between neighbors and then applying on each cell a purely local transformation depending only on the state of the cell (and not on the states of its neighbors). With such a construction scheme, the cellular automaton is guaranteed to be reversible if the local transformation is itself a bijection. This technique may be viewed as a block cellular automaton on a finer lattice of cells, formed by the parts of each larger cell; the blocks of this finer lattice alternate between the sets of parts within a single large cell and the sets of parts in neighboring cells that share parts with each other.