Balanced simplices

Abstract

An additive cellular automaton is a linear map on the set of infinite multidimensional arrays of elements in a finite cyclic group Z/mZ. In this paper, we consider simplices appearing in the orbits generated from arithmetic arrays by additive cellular automata. We prove that they are a source of balanced simplices, that are simplices containing all the elements of Z/mZ with the same multiplicity. For any additive cellular automaton of dimension 1 or higher, the existence of infinitely many balanced simplices of Z/mZ appearing in such orbits is shown, and this, for an infinite number of values m. The special case of the Pascal cellular automata, the cellular automata generating the Pascal simplices, that are a generalization of the Pascal triangle into arbitrary dimension, is studied in detail.