The B36/S125 "2x2" Life-Like Cellular Automaton

TL;DR

The paper explores the 2x2 cellular automaton, revealing its patterns, theoretical limits on spaceship speeds, and a family of oscillators with periods related to powers of two, expanding understanding of this Life-like rule.

Contribution

It provides a comprehensive analysis of the 2x2 automaton, including pattern descriptions, speed limit proofs, and the discovery of oscillators with specific periods, which are novel insights.

Findings

Diagonal spaceship speed limit c/3

Orthogonal spaceship speed limit c/2

Existence of oscillators with period 2^m(2^k - 1) for integers m,k ≥ 1

Abstract

The B36/S125 (or "2x2") cellular automaton is one that takes place on a 2D square lattice much like Conway's Game of Life. Although it exhibits high-level behaviour that is similar to Life, such as chaotic but eventually stable evolution and the existence of a natural diagonal glider, the individual objects that the rule contains generally look very different from their Life counterparts. In this article, a history of notable discoveries in the 2x2 rule is provided, and the fundamental patterns of the automaton are described. Some theoretical results are derived along the way, including a proof that the speed limits for diagonal and orthogonal spaceships in this rule are c/3 and c/2, respectively. A Margolus block cellular automaton that 2x2 emulates is investigated, and in particular a family of oscillators made up entirely of 2 x 2 blocks are analyzed and used to show that there exist oscillators with period 2^m(2^k - 1) for any integers m,k \geq 1.