This explorable illustrates a type of continuous cellular automata called Lenia. It was derived from the famous discrete cellular automata, Conway's Game of Life, by changing its rules in a couple of ways. The cells are no longer just dead or alive, but can take one of an infinite number of states. The local rule involves a much larger neighborhood, and the states are updated differentially.

Unlike the patterns found in the Game of Life, the emergent patterns or "creatures" discovered in Lenia look and behave more like microorganisms as observed under a microscope. This may be due to the intrinsic statistical, stochastic, fuzzy-logic-like nature of continuous cellular automata that shares with biological systems. These continuous patterns exhibit self-organization and self-repair, as well as more advanced phenomena like structural symmetry, self-replication, polymorphism, and colony formation.

Press Play, try out the controls, and keep on reading ....

This is how it works

Cellular automata can become continuous in terms of states, space or time, in Lenia we make them all. Inside an n-dimensional lattice, each site has a continuous state variable with real value between 0 and 1, representing the level of activity. The states are visualized using a color gradient as seen in the display panel.

Each site \(a_{ij}\) in the grid \(\mathbf{A}\) interacts with an arbitrary large circular area of neighbors, and updates in a differential fashion akin to the Euler method, making space and time continuous. The local rule goes like this:

1. First take a weighted sum of neighbor sites with a weight distribution \(\mathbf{K}\) (called kernel),
2. then get the growth value using a function \(G\) (called growth mapping),
3. finally add a small portion \(dt\) of the growth to the state \(a_{ij}\) itself,
4. don't forget to clip the value back to the interval \([0, 1]\).

Update the whole grid synchronously and repeatedly. The algorithm can be written in formula (\([\;]_0^1\) is the clip function):

\[ \mathbf{A}^{t+dt} = \big[ \mathbf{A}^t + dt \cdot G(\mathbf{K} * \mathbf{A}^t) \big]_0^1 \]

The weighted sum is mathematically a convolution. It can be interpreted as the "perception" of the world as being perceived by the "sensors" (kernel) of the creature. After a decision making process (growth mapping), the creature takes the appropriate "actions" (growth values) back to the world. This process mimics a sensori-motor feedback loop that may contribute to the self-organization of patterns.

In more advanced rules called extended Lenia, there can be multiple channels \(\mathbf{A}_c\), multiple kernels and growth mappings \(\mathbf{K}_k, \mathbf{G}_k\). The algorithm is similar, but the results are more surprising.

Try this

Select a pattern from the list, and observe its dynamic structure and behavior. Select again to randomly spawn 1-3 creatures. Gently move the space, time, and states sliders to control the degree of smoothness of the system: to the left more discrete and coarse-grained, to the right more continuous and fine-grained. Move the mu and sigma sliders to adjust parameters of the growth mapping. You may discover a new species with the right parameter values!

While playing with the sliders, switch to different views to observe the effects of parameter changes. The world, weighted sum, and growth views visualize each stage of cellular automata calculation, the kernel view shows the kernel and the growth mapping.

Model species

Like in biology, we can investigate how self-organizing patterns behave through a few "model species".

• Orb - It is the simplest pattern that possess individuality, meaning it can maintain a "personal space" and survive collisions. Can you feel the attractive & repulsive forces between them? You may also witness 2 of them fuse into a new species.
• Chain - It is a linear combination of smaller components. Try evolve shorter chains by adjusting the mu and sigma sliders.
• Star - Many species have radial symmetry. They are usually stationary (like real starfish) while rotating or oscillating.
• Spinner - Asymmetric patterns may be spinning or zig-zagging.
• Shape-shifter - Some edge case patterns have chaotic behaviors, like metamorphosis where the pattern keeps changing mode randomly. It is like it cannot decide which species to become, and in fact, it exists at the crossroads of various species (see this parameter map, y-axis is mu, x-axis is sigma).
• Self-replicator - In extended Lenia where the system has multiple kernels and multiple channels (shown in RGB colors), patterns have much more complex structures and behaviors. For example, this cloud-like creature can self-replicate.
• Mother ship - This pattern emits smaller patterns, analogous to "glider guns" or "rakes" in the Game of Life.
• Amazing cells - Some cell-like creatures are polymorphic - able to switch among various phenotypes, while engaging in complex interactions. How many phenotypes and combos can you find?
• Bizarre cells - They exhibit a kind of strange behavior that looks like communication (?), and they sometimes form colonies.
• The last pattern is Gosper's glider gun in the Game of Life. This is to demonstrate that discrete cellular automata is a special case of continuous cellular automata, and thus can be simulated inside Lenia (using a step kernel).

References

• Chan, B. W.-C. (2019). Lenia: Biology of Artificial Life. Complex Systems, 28 (3), 251–286.
• Chan, B. W.-C. (2020). Lenia and Expanded Universe. Artificial Life Conference Proceedings, (32), 221–229.
• More resources on Lenia: https://chakazul.github.io/lenia.html