Continuous Game of Life: cell emergence and self-organization at the edge of growth
Abstract
Conway's Game of Life shows that simple rules can generate a rich diversity of emerging structures. This cellular automaton has been translated to continuous space by Rafler (2011) in a simulation called SmoothLife. The isotropic rule of this continuous Game of Life generates patterns whose beauty has attracted the attention of a growing community at the intersection of science and computer art. We study a minimal variant of this model, continuous in space and time, that generates cell-like patterns capable of self-replicating, gliding and disappearing. The phenomenology of these unit patterns is reported and related to homogeneous-state bifurcations, symmetry breaking, observed shape instabilities, finite-amplitude morphological changes, and a dilute-to-dense transition associated with cell proliferation. Its mapping onto a large reaction-diffusion system is interpreted in terms of homeostatic concentrations of morphogens, regulated by the nonlinear survival rule and generated through a cell-sourced cascade of auxiliary reactions. Introducing a global conservation law that limits resource availability causes the system to self-organize at this dilute-to-dense transition, which we call the edge of growth. A further exploration of parameter space reveals a variety of phases and the richness of life-like morphologies organized around this edge. Resemblance to biological processes such as division, motility, and death, together with a concise formulation and numerical implementation, makes the continuous Game of Life an appealing model system for investigating the emergence and self-organization of life-like patterns.
Related
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BibTeX
@misc{guillet2026continuous,
title = {{Continuous Game of Life: cell emergence and self-organization at the edge of growth}},
author = {Alexandre Guillet and Frank Jülicher},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.27402},
eprint = {2607.27402},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.27402},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Continuous Game of Life: cell emergence and self-organization at the edge of growth
AU - Alexandre Guillet
AU - Frank Jülicher
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2607.27402
UR - https://arxiv.org/abs/2607.27402
AB - Conway's Game of Life shows that simple rules can generate a rich diversity of emerging structures. This cellular automaton has been translated to continuous space by Rafler (2011) in a simulation called SmoothLife. The isotropic rule of this continuous Game of Life generates patterns whose beauty has attracted the attention of a growing community at the intersection of science and computer art. We study a minimal variant of this model, continuous in space and time, that generates cell-like patterns capable of self-replicating, gliding and disappearing. The phenomenology of these unit patterns is reported and related to homogeneous-state bifurcations, symmetry breaking, observed shape instabilities, finite-amplitude morphological changes, and a dilute-to-dense transition associated with cell proliferation. Its mapping onto a large reaction-diffusion system is interpreted in terms of homeostatic concentrations of morphogens, regulated by the nonlinear survival rule and generated through a cell-sourced cascade of auxiliary reactions. Introducing a global conservation law that limits resource availability causes the system to self-organize at this dilute-to-dense transition, which we call the edge of growth. A further exploration of parameter space reveals a variety of phases and the richness of life-like morphologies organized around this edge. Resemblance to biological processes such as division, motility, and death, together with a concise formulation and numerical implementation, makes the continuous Game of Life an appealing model system for investigating the emergence and self-organization of life-like patterns.
ER -