Triangular Automata: The 256 Elementary Cellular Automata of the 2D Plane

TL;DR

The paper addresses the dynamics of binary states on a triangular lattice by introducing Elementary Triangular Automata (ETA), the 2D counterpart to Wolfram's Elementary Cellular Automata. It develops a graph-theoretic evolution framework using an adjacency matrix and a rule index to encode the 256 ETA rules, and it analyzes a broad range of behaviors from beauty and fractals to chaos and self reproduction. Key contributions include formalizing ETA, demonstrating qualitative phenomena across the rule space, linking a subset of ETA to ECAs, and providing practical tools such as an interactive website and a Mathematica package for exploration. The work establishes ETA as a fundamental, minimal 2D CA model on a triangular grid, enabling systematic, resource efficient exploration of complex dynamics and potential extensions to broader triangular automata.

Abstract

Cellular automata on the triangular grid are here referred to as Triangular Automata (TA). This paper focuses on the simplest class of TA, called Elementary Triangular Automata (ETA). They are argued to be the two-dimensional counterpart of Wolfram's Elementary Cellular Automata. Conceptual and computational tools for their study are presented, along with an initial analysis. This paper is accompanied by a website where the results can be explored interactively. The source code is available in the form of a Mathematica package.

Figures (32)

• Figure 1: The triangular grid
• Figure 2: All possible local configurations
• Figure 3: Structure of the triangular grid
• Figure 4: Rule $210=1101\,0010_2$
• Figure 5: The beauty of ETA