Simple Block-Substitution Rule Exhibits Interesting Patterns

Abstract

The goal is to find all patterns (space-time evolutions) that can be generated by a simple binary 1D block substitution rule (BCA), the Twin-Toggle Rule. The 1D space is partitioned into even or odd cell pairs. The even and odd partition are alternated in time. If the two bits in a pair are equal, then they are substituted by their logical inverses. Firstly, all possible initial configurations are reduced to a set of representatives taking into account 0/1-inversion, cyclic shift and mirroring. Secondly, the BCA was simulated and all patterns were compared to each other for similarities (black/white-inversion, shift, rotation, mirroring). Only a small amount of them was stored, representing all possible patterns. Most of the patterns are single patterns (the pixel structure is the same for black and white pixels). For $N = 8, 12$ cells interesting dual patterns were discovered, which show a different structure for black and white pixels.