Spatiotemporal Dynamics of Conway's Game of Life in the Entropy--Complexity Plane

Abstract

We investigate the dynamics of Conway's Game of Life (GoL) through the lens of Ordinal Pattern analysis. Each GoL generation is transformed into a one-dimensional time series, from which ordinal patterns are extracted to characterize its underlying dynamics. The resulting ordinal distributions are then represented in the Entropy--Statistical Complexity ($H\times C$) plane, enabling a quantitative analysis of the system's evolution. We evaluate different encoding strategies, including patch sizes and linearization methods, and assess their impact on the resulting $H\times C$ trajectories. Finally, we characterize the emergent dynamics in terms of convergence behavior, trajectory patterns, and statistical complexity, showing how the proposed entropy--complexity framework provides insights into the dynamics of cellular automata.