Revealing Latent Self-Similarity in Cellular Automata via Recursive Gradient Profiling

Abstract

Cellular automata (CA), originally developed as computational models of natural processes, have become a central subject in the study of complex systems and generative visual forms. Among them, the Ulam-Warburton Cellular Automaton (UWCA) exhibits recursive growth and fractal-like characteristics in its spatial evolution. However, exact self-similar fractal structures are typically observable only at specific generations and remain visually obscured in conventional binary renderings. This study introduces a Recursive Gradient Profile Function (RGPF) that assigns grayscale values to newly activated cells according to their generation index, enabling latent self-similar structures to emerge cumulatively in spatial visualizations. Through this gradient-based mapping, recursive geometric patterns become perceptible across scales, revealing fractal properties that are not apparent in standard representations. We further extend this approach to UWCA variants with alternative neighborhood configurations, demonstrating that these rules also produce distinct yet consistently fractal visual patterns when visualized using recursive gradient profile. Beyond computational analysis, the resulting generative forms resonate with optical and cultural phenomena such as infinity mirrors, video feedback, and mise en abyme in European art history, as well as fractal motifs found in religious architecture. These visual correspondences suggest a broader connection between complexity science, computational visualization, and cultural art and design.