Replication of spatial patterns with reversible and additive cellular automata

Abstract

In this article, the replication of arbitrary patterns by reversible and additive cellular automata is reported. The orbit of an 1D cellular automaton operating on $p$ symbols that is both additive and reversible is explicitly given in terms of coefficients that appear in the theory of Gegenbauer polynomials. It is shown that if $p$ is an odd prime, the pattern formed after $(p-1)/2$ time steps from any arbitrary initial condition (spatially confined to a region of side less than $p$) replicates after $p+(p-1)/2$ time steps in a way that resembles budding in biological systems.