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๐ŸŒŒ Lenia โ€“ Continuous Cellular Automata

Lenia (Bert Chan, 2019) generalises Conway's Game of Life from discrete to continuous in every dimension: continuous space, continuous time, continuous state. The result: cell-like "creatures" that move, pulse, grow, and interact โ€“ emerging from nothing but a convolution kernel and a smooth growth function.

"Lenia is probably the most beautiful demonstration that complex life-like behaviour can arise from simple continuous rules."

๐Ÿง  How It Works

The system follows a single update equation:

A(t + dt) = clip[ A(t) + dt ยท G(K โˆ— A(t)) ]
Symbol Meaning
A Activity field โˆˆ [0, 1] โ€“ how "alive" each cell is
K Ring-shaped convolution kernel โ€“ "how alive is my neighbourhood?"
K โˆ— A Spatial convolution (computed via FFT)
G Growth function: G(u) = 2ยทexp(โˆ’(uโˆ’ฮผ)ยฒ/2ฯƒยฒ) โˆ’ 1
dt Time step (< 1 for smooth, continuous dynamics)
clip Clamp result to [0, 1]

Key insight: the growth function creates a Goldilocks zone. Too little neighbourhood activity โ†’ decay. Too much โ†’ also decay. Just right (u โ‰ˆ ฮผ) โ†’ growth. This is the same principle behind biological morphogenesis.

๐ŸŽจ Parameter Presets

Preset ฮผ ฯƒ R Pattern
Orbium (default) 0.15 0.017 13 Gliding, pulsing organisms
Geminium 0.14 0.014 10 Self-replicating cells
Smooth Life 0.30 0.050 15 Amoeba-like blobs

Change MU, SIGMA, and KERNEL_R at the top of the script.

๐Ÿ–ผ Visualisation

The window displays the activity field A as a heatmap with a custom colourmap (black โ†’ electric blue โ†’ white). The dynamics are mesmerising: structures appear, move, collide, and evolve.

Press ESC to exit.

๐Ÿ”— Connection to System Intelligence

Lenia challenges our categories:

  • Predictive Power (P): The "organisms" behave predictably once formed โ€“ they follow trajectories, respond to collisions
  • Regulation (R): Each creature maintains its form through the Goldilocks growth function โ€“ a kind of homeostasis
  • Adaptive Capacity (A): Creatures can deform, merge, and reform under perturbation

The philosophical question: are Lenia creatures alive? They satisfy several criteria of life (metabolism, self-maintenance, sensitivity to environment) โ€“ raising the question of where "life" begins.

๐Ÿ“š References

  • Chan, B. W.-C. (2019). Lenia โ€“ Biology of Artificial Life. Complex Systems, 28(3).
  • Chan, B. W.-C. (2020). Lenia and Expanded Universe. ALIFE 2020 Conference.

โ–ถ Run

cd simulation-models/lenia
python3 lenia.py

Requires: numpy, scipy, matplotlib

Experiment ideas

  • Try MU=0.14, SIGMA=0.014, KERNEL_R=10 for self-replicating cells
  • Increase GRID_SIZE to 512 for higher resolution (slower)
  • Modify the kernel shape in make_kernel() for entirely new physics