[Paper Review] Circular Motion of Strings in Cellular Automata, and Other Surprises

Daniel Miller, Edward Fredkin|arXiv (Cornell University)|Jun 10, 2012

Cellular Automata and Applications1 references3 citations

TL;DR

This paper presents a novel two-state, three-dimensional, deterministic, and reversible cellular automaton that exhibits emergent complex dynamics, including approximately circular orbits of string-like configurations, wavelike undulations, and particle-like entities with half-life decay. The key contribution is demonstrating that such simple rules can produce behaviors reminiscent of physical systems, challenging assumptions about minimal complexity for emergent motion and decay patterns.

ABSTRACT

A two-state, three-dimensional, deterministic, reversible cellular automaton is shown to be capable of approximately circular orbits, wavelike undulations, and particle-like configurations that decay in accordance with a half-life law.

Motivation & Objective

• To investigate whether simple, deterministic, and reversible cellular automata can generate complex, lifelike dynamics.
• To explore the emergence of circular motion in string-like configurations within a 3D cellular automaton framework.
• To analyze wavelike undulations and particle-like decay behaviors in the same system.
• To demonstrate that minimal rule sets can produce behaviors analogous to physical systems, such as half-life decay.
• To provide a foundation for modeling physical phenomena using reversible, discrete systems.

Proposed method

• The authors define a two-state, three-dimensional cellular automaton with deterministic and reversible update rules.
• The system operates on a cubic lattice where each cell's state evolves based on its local neighborhood configuration.
• The update rules are designed to preserve global reversibility and conserve certain topological features of string-like structures.
• The dynamics are simulated over time to observe emergent patterns such as orbits, waves, and decaying particles.
• The behavior is analyzed through visual inspection and qualitative characterization of trajectories and decay profiles.
• A companion website is provided to host simulations and visualizations of the automaton's behavior.

Experimental results

Research questions

• RQ1 Can a simple, deterministic, and reversible cellular automaton produce stable, approximately circular orbits of string-like configurations?
• RQ2 What types of wavelike undulations emerge in the dynamics of such a system?
• RQ3 Do particle-like configurations in the automaton exhibit behavior consistent with half-life decay?
• RQ4 How do the topological and dynamical properties of the system relate to physical analogs?
• RQ5 What minimal rule complexity is sufficient to generate complex, lifelike behaviors in discrete, reversible systems?

Key findings

• The cellular automaton successfully generates string-like configurations that follow approximately circular trajectories over time.
• Wavelike undulations propagate through the system, indicating collective, coherent motion.
• Particle-like configurations are observed to decay over time with a behavior consistent with a half-life law.
• The system's dynamics are fully deterministic and reversible, preserving information at the cellular level.
• The emergence of complex behaviors from simple local rules suggests potential for modeling physical systems in discrete, reversible frameworks.
• The companion website provides interactive access to simulations, enabling further exploration of the automaton's behavior.