A conjecture about the size of a particular cellular automaton : the perfectly growing crystal
Hugues Dreyssé, Roland Riedinger · Journal de physique · 1987
In order to optimize a computer implementation of the recursion method, (initially proposed by Heine, Haydock and Kelly), we build a cluster from an initial seed of points by adjoining the new sites obtained from the actual cluster by translation of a given set of vectors (the generators), on a lattice. This cluster is organized in shells and a conjecture is given about its size. This conjecture is checked on lattices with inequivalent sites and seems to remain valid this case. The conjecture about the size seems to be related to a self-similarity of the growing cluster. We relate the construction of these clusters to other topics : cellular automata, animals, growth of a crystal and combinatories. We show also how the growing process may be used to generate cluster with grain boundaries.