Gliders and Ether in Rule 54

TL;DR

This paper analyzes the behavior of gliders and ether in elementary cellular automaton rule 54 using a new algebraic formalism called 'flexible time', providing detailed descriptions and formulas for their evolution.

Contribution

It introduces algebraic expressions for cellular automaton patterns in 'flexible time' and characterizes their behavior with general formulas, advancing understanding of rule 54 dynamics.

Findings

Derived algebraic expressions for cell groups and their evolution.

Described periodic patterns like ether and gliders efficiently.

Found general formulas characterizing pattern behavior.

Abstract

This is a study of the one-dimensional elementary cellular automaton rule 54 in the new formalism of "flexible time". We derive algebraic expressions for groups of several cells and their evolution in time. With them we can describe the behaviour of simple periodic patterns like the ether and gliders in an efficient way. We use that to look into their behaviour in detail and find general formulas that characterise them.