Loop Patterns Formed by Cellular Automata

TL;DR

This paper introduces a cellular automata rule capable of generating stable loop patterns in a 2D grid, using tile-based templates and noise injection to evolve desired cyclic structures under fixed boundary conditions.

Contribution

It presents a novel CA rule and tile-based template matching method for reliably forming stable loop patterns, advancing pattern formation research.

Findings

CA rule successfully generates stable loop patterns

Templates enable local pattern matching for pattern control

Simulations confirm reliable evolution of loop structures

Abstract

A Cellular Automata (CA) rule is presented that can generate "loop patterns" in a 2D grid under fixed boundary conditions. A loop is a cyclically closed path represented by one-cells enclosed by zero-cells. A loop pattern can contain several loops that are not allowed to touch each other. The problem is solved by designing an appropriate set of tiles that can overlap and which are used in the CA rule. Templates are derived from the tiles which are used for local pattern matching. In order to drive the evolution to the desired patterns, noise is injected if the templates do not match or other constraints are not fulfilled. The general CA rule can be specialized by enabling certain conditions, and the characteristics of five rule variants are explained. Simulations illustrate that the CA rule can securely evolve stable loop patterns. The preliminary theoretical analysis of the obtained loop patterns raises many interesting research problems for the future -- several of them have been briefly discussed.