plenia.jl is a Particle Lenia implementation for Julia based on the work Particle Lenia and the energy-based formulation by Alexander Mordvintsev et al. (2022). The patterns and cell-like behavior shown in the video below are produced by an artificial "life" system that is called "Particle Lenia" -- based on the name Lenia Mathematical Lifeforms, which in itself is based on John Conway's Game of Life.

Particle Lenia's are systems of Ordinary Differential Equations (ODE) described by a collection of point particles $\boldsymbol{p_i}$ and the interaction with each other. Each particle is assigned an unique position $\boldsymbol{p_i} = [x_i, y_i]^\top$ together with an continuous energy field $E_i(\boldsymbol{x}) = R_i(\boldsymbol{x}) - G_i(\boldsymbol{x})$, being the repulsive and growth energy field, respectively. We refer to Mordvintsev et al. for the expressions of these fields. The dynamics for this system of particles arise from the gradient of the total energy $E = \sum E_i$, simply put, it reads as follows: $$\dot{\boldsymbol{p_i}} = -\nabla E(\boldsymbol{p_i}) $$

Despite its mathematical simplicity, the dynamics of such systems lead to interesting (stable) limit cycles, that are reminiscent to the behavior of microscopic organisms in water or single-cell organisms.

To run a Particle Lenia simulation, simply call:

julia main.jl