The fractal structure of cellular automata on Abelian groups

TL;DR

This paper investigates a new class of cellular automata on Abelian groups that, despite lacking common algebraic properties, still generate fractal spacetime diagrams, expanding understanding of fractal structures in automata.

Contribution

It introduces and analyzes a novel class of cellular automata on Abelian groups that produce fractal structures without relying on traditional algebraic properties.

Findings

Automata generate fractal spacetime diagrams despite weaker cell structures.

Fractal formation explained without ring or p-Fermat properties.

New insights into the diversity of automata capable of fractal behavior.

Abstract

It is well-known that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo 2 generates a Sierpinski triangle. Explaining the fractal structure of the spacetime diagrams of cellular automata is a much explored topic, but virtually all of the results revolve around a special class of automata, whose typical features include irreversibility, an alphabet with a ring structure, a global evolution that is a ring homomorphism, and a property known as (weakly) p-Fermat. The class of automata that we study in this article has none of these properties. Their cell structure is weaker, as it does not come with a multiplication, and they are far from being p-Fermat, even weakly. However, they do produce fractal spacetime diagrams, and we explain why and how.