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.
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