Two-dimensional cellular automata—Deterministic models of growth

Cellular automata are simple discrete deterministic rules that can however produce different, from simple to very complex, dynamics, and it would be useful to have a criterion to classify their behaviour. Here, we argue that the investigation of surface growth as described by the cellular automata provides a quantitative method to classify them. To this aim, the growth behaviour of cellular automata describing pure growth has been analysed. The automata fall into three classes: a first class is formed by the rules where the surface width saturates, and includes also rules that display Family-Vicsek scaling. A second class is constituted by the rules where the surface width grows indefinitely, which we call the dendritic-growth class. Finally, some rules belong to the non-growing class. A quantitative analysis shows a finer sub-division in clusters, some of which are close to known models of growth, while others do not have any counterpart in the literature. This work demonstrates the capabilities of deterministic cellular automata to describe a large variety of growth regimes, and suggests that their growth behaviour may be also used as an effective tool for their classification.