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

TL;DR

The paper addresses latent self-similarity in the Ulam-Warburton cellular automaton (UWCA) where fractal structure is not evident in traditional binary renderings. It introduces the Recursive Gradient Profiling Function (RGPF), grouping generations between powers of two and assigning grayscale values to newly born cells to reveal cumulative self-similarity. Applied to UWCA and multiple neighborhood variants, the method yields fractal patterns with a fractal dimension around $D=2.6827$ and normalized error $E_{norm}≈0.112\%$, indicating a robust scale-invariant structure. The authors further connect these computational visuals to optical and cultural phenomena, highlighting potential cross-domain relevance for art, architecture, and design.

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.

Figures (10)

• Figure 1: (1) and (2): Elementary 1D CA Rule 30 exhibits a pattern resembling the natural shell of Conus textile; (3) and (4): Elementary 1D CA rule 30 and 2D Moore neighborhood outer totalistic CA rule 510 both exhibit Sierpiński triangle-like patterns. These CA patterns are visualized as space-time diagrams. (Image (2) adapted from shell)
• Figure 2: Visualization of UWCA from generations $n = 1$ to $5$. The initial state ($n = 1$) begins with a single live cell. Within each generation, dead cells evaluate their von Neumann neighborhood (up, down, left, right). If exactly one neighbor is alive (green), the cell becomes alive in the next generation. Cells with more than one live neighbors (blue) indicate invalid transitions.
• Figure 3: Top-left: UWCA at generation $n=125$; Top-middle: Using image processing to extract only the boundary contours of the pattern; Top-right: Edge feature extracted with edge detection method. Bottom-left: At generation $n=32$. Bottom-right: Edge feature extracted with edge detection method Kawaharada.
• Figure 4: Top: Mapping of generation index ($n=1$ to $256$) to grayscale intensity values based on the Recursive Gradient Profile Function (RGPF), as defined in Equation \ref{['eq:grayscale']}. Bottom: Number of newly born live cells at each generation, computed using the UWCA update rule in Equation \ref{['eq:UWCA']}. Red lines indicate when $n=2^k$. At these generations, live cells form square patterns.
• Figure 5: Left: Shows the UWCA multiplying recursive gradient profile, and output up to generation 256; The red squares refer to Figure \ref{['fig_uwca_rgpf']} red lines, indicating generations where $n = 2^k$ and square outlines emerge. Right: Displays the corresponding log–log plot, from which a fractal dimension of $D = 2.6827$ is computed. Fit error $E = 0.0063$ quantifies the average vertical distance between the data points and the regression line. The error distance lines representing the vertical distance from each point to the fitted line, visualizing the fitting quality.