Emergent Patterning Phenomena in 2D Cellular Automata

Abstract

This article discusses mechanisms of pattern formation in 2D, self-replicating cellular automata (CAs). In particular, we present mechanisms for structure replication that provide insight into analogous processes in the biological world. After examining self-replicating structures and the way they reproduce, we consider their fractal properties and scale invariance. We explore the space of all possible mutations, showing that despite their apparent differences, many patterns produced by CAs are based on universal models of development and that mutations may lead either to stable or to unstable development dynamics. An example of this process for all possible one-step mutations of one specific CA is given. We have demonstrated that a self-replicating system can carry out many slightly different but related entities, realizing new different growth models. We infer that self-replicating systems exist in an intermediate regime between order and chaos, showing that these models degrade into chaotic configurations, passing through a series of transition stages. This process is quantified by measuring the Hamming distances between the pattern produced by the original self-replicator and those produced by mutated systems. The analysis shows that many different mechanisms may be involved in patterning phenomena. These include changes in the external or internal layers of the structure, substitution of elements, differential rates of growth in different parts of the structure, structural modifications, changes in the original model, the emergence of different structures governed by different CA rules, and changes in the self-replication process.