Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2

Abstract

A binary difference pattern (BDP) is a pattern obtained by covering an equilateral triangular grid by black and white circles in a dense hexagonal packing under a simple symmetric local matching rule. It is a subpattern in a specific graphical representation of the orbit of a cellular automaton that generates Pascal's triangle modulo 2. Analytic conditions for certain types of geometric symmetry of these patterns are derived. These allow us to find all symmetric solutions and the cardinalities of the different symmetry classes. In the analysis, a central role is played by the so-called Pascal matrix — a square matrix that contains Pascal's triangle modulo 2 (up to a certain size) — and by certain groups of geometric transformations of this matrix, featuring remarkable product properties for the Pascal matrix.