Exploration of Genotype-Phenotype Relationships in Nontotalistic Cellular Automata Using HexCode, a Novel Rule Encoding System

The spacetime evolution of a cellular automaton (CA) is determined by its rule table. Whereas a rule table is analogous to the DNA of the CA, that is, its genotype, the resulting spacetime evolution is equivalent to its phenotype. Conventional rule codes carry no easily recognizable features that would allow establishing a relationship between the rule genotype and the rule phenotype. This is particularly relevant for totalistic cellular automata (CAs) with radii and dimensions greater than one where the rule space is rapidly expanding. Here, a universal rule code, called HexCode, has been created that enables a hitherto unachievable systematic and exhaustive exploration of geometric features of CA phenotypes based on their genotypes in the large rule spaces of totalistic CAs with radii and dimensions greater than one.