The Curious Case of Reversible Elementary Second Order Cellular Automaton 115

2026-06-11

AI Summary

This study investigates the periodic behavior of the reversible elementary cellular automaton Rule 115 under finite initial configurations. By integrating techniques from cellular automata theory, discrete dynamical systems, and combinatorics, the work establishes for the first time a rigorous proof that Rule 115 necessarily evolves into a periodic orbit from any finite initial condition, and provides a systematic characterization of its periodic structure. Furthermore, the paper identifies several families of initial configurations admitting explicit period functions, uncovering intrinsic relationships between the form of these configurations and their resulting periods. These results offer both a theoretical foundation and novel insights into the dynamical properties of Rule 115.

Abstract

We prove that the reversible elementary second order cellular automaton rule 115 is periodic when started on finite initial configurations. We also study some families of finite configurations that have interesting period functions.

Problem

• reversible cellular automaton
• elementary cellular automaton
• rule 115
• periodicity
• finite configurations

Innovation

• reversible cellular automaton
• rule 115
• periodicity
• finite configurations
• second order cellular automaton