Diagrammatic approach to cellular automata and the emergence of form with inner structure

We present a diagrammatic method to build up sophisticated cellular automata (CAs) as models of complex physical systems. The diagrams complement the mathematical approach to CA modeling, whose details are also presented here, and allows CAs in rule space to be classified according to their hierarchy of layers. Since the method is valid for any discrete operator and only depends on the alphabet size, the resulting conclusions, of general validity, apply to CAs in any dimension or order in time, arbitrary neighborhood ranges and topology. We provide several examples of the method, illustrating how it can be applied to the mathematical modeling of the emergence of systems composed of parts of different sizes. Specifically, we use the majority CA rule as a building block to construct more complex cellular automata in which separate domains (with substructures having different dynamical properties) are able to emerge out of disorder and coexist in a stable manner.