Visualizing the Structure of Lenia Parameter Space

TL;DR

The paper addresses identifying and visualizing solitons in Lenia's high-dimensional parameter space. It formalizes Lenia dynamics with a single-channel setup and a Gaussian-like growth function, updating via $A^{t+\Delta t}(x) = [A^t + \Delta t G(K * A^t(x))]_0^1$ (with $\Delta t = 0.1$) and maps the μ–σ growth-parameter space across multiple kernels. An automatic classifier partitions initial-condition-driven dynamics into four dynamical classes, revealing soliton regions and uncovering new soliton families, with a qualitatively universal phase-space structure observed across kernels. The work enables interactive visualization of Lenia's parameter space and provides a framework for analytically characterizing transition regimes between stable and metastable dynamics.

Abstract

Continuous cellular automata are rocketing in popularity, yet developing a theoretical understanding of their behaviour remains a challenge. In the case of Lenia, a few fundamental open problems include determining what exactly constitutes a soliton, what is the overall structure of the parameter space, and where do the solitons occur in it. In this abstract, we present a new method to automatically classify Lenia systems into four qualitatively different dynamical classes. This allows us to detect moving solitons, and to provide an interactive visualization of Lenia's parameter space structure on our website https://lenia-explorer.vercel.app/. The results shed new light on the above-mentioned questions and lead to several observations: the existence of new soliton families for parameters where they were not believed to exist, or the universality of the phase space structure across various kernels.

Figures (4)

• Figure 1: a) All activity dies out. b) Activity expands to the whole array. c) All moving solitons get "unclassified".
• Figure 2: Traversing the space of initial configurations by increasing the area of noise in the shape of random polygons.
• Figure 3: x-axis: polygon sizes, y-axis: proportion of configurations in each phase. a) All configurations enter the stable phase. b) All configurations enter the metastable phase. c) A transition from stable to metastable phase as the patches of noise increase in size. d) A transition with solitons occuring around the transition region.
• Figure 4: (Left) "Phase space" of Lenia's dynamical classes for a fixed kernel while varying $\mu$ and $\sigma$. Dark orange region contains systems with emerging solitons, some of them showcased in the (middle). (Right) Analogous phase spaces for various kernel shapes depicted next to them with $0.1 \leq \mu \leq 0.5$ and $0.0 < \sigma \leq 0.1$.