The Growth of One-cell-thick Patterns within Cellular Automata
John M. Campbell
Applications of one-dimensional and nonlinear cellular automata were pioneered by Wolfram. This motivates new research investigations based on growth properties related to such automata and to how such properties relate to higher-dimensional cellular automata. In this direction, a naturally occurring instance of a one-cell-thick configuration associated with a two-dimensional cellular automaton is given by the pattern consisting of a given number of consecutive 𝙾𝙽 cells, arranged horizontally or vertically, in the plane associated with the Life cellular automaton due to Conway. We investigate the problem of determining the growth of such one-cell-thick configurations, inspired by the work of Sloane on the enumeration of 𝙾𝙽 cells within one-dimensional cellular automata. We introduce and prove, using Wolfram’s elementary cellular automaton classified as Rule 22, a formula for the number of live cells arising from our one-cell-thick patterns, subject to a constraint related to the Sierpiński triangle fractal.
Keywords: cellular automaton, Conway’s game of life, rule 22, discrete dynamical system, Sierpi´nski triangle, integer sequence, rule 90, Gould’s sequence
MSC: 68Q80
DOI: 10.32908/jca.v18.300326