Local Structure Theory: Calculation on Hexagonal Arrays, Interaction of Rule and Lattice

We apply local structure theory calculations 7 to the study of cellular automata on the two-dimensional hexagonal lattice. A particular hexagonal lattice rule denoted (3422) is considered in detail. This rule has many features in common with Conway's Life. The local struc-1 ture theory captures many of the statistical properties of this rule; this supports hypotheses raised by a study of Life itself 6. As in Life, the state of a cell under (3422) depends only on the state of the cell itself and the sum of states in its neighborhood at the previous time step. This property implies that evolution rules which operate in the same way can be studied on diierent lattices. The diierences between the behavior of these rules on diierent lattices are dramatic. The mean eld theory cannot reeect these diierences. However, a generalization of the mean eld theory, the local structure theory, does account for the rule/lattice interaction.