Structural Properties of Additive Nano/Microcellular Automata

Abstract

A detailed structural analysis of 2D patterns generated by one-dimensional M-state additive cellular automata (ACA), where M is a prime ( p) to a non-negative power (t), is performed by using the transition matrix (TM), the barycentric fixed-mass (BFM) method, and the small-angle scattering (SAS) technique. The BFM method shows that, for M-state ACA, the subsets of particular states are multifractals. The fractal dimensions of ACA of a given transition rule are shown to be equal at constant p and arbitrary t, and there is a prime p at which the fractal dimensions of different ACA are equal. The SAS technique is shown to be able to differentiate between patterns of such ACA. In addition, SAS can also provide complementary information about the overall size, number of elements, and number of rows generated by ACA. In the case of monofractals, SAS can recover also the scaling factor, which is shown to be, for the M = p t -state ACA, the inverse of p. The above findings on the 2-and 3-state Rule 90 and Rule 150 are illustrated as idealized models with possible applications toward assembly of nanoparticles into programmable patterns or in modeling biological systems such as fibroblast aggregation, the growth of bacterial colonies, neuronal maps, and virus spreading.