Introduction to Lenia

Cellular Automata

Cellular Automata are a classic example of systems that are governed by simple, deterministic rules, yet exhibit surprisingly complex and difficult-to-predict dynamics. One of the most famous cellular automatons is known as Conway's Game of Life. This "0-Player-Game" has already been topic of a practical on the PILLE-server in the summer of 2023.

The term 'cellular automaton' (CA) however, is much more general than one would assume from the GoL alone. Each CA can be described as a model consisting of a grid of cells. These cells assume different states (binary in GoL). The states are updated for each cell simultaneously in discrete time steps according to a fixed set of rules. These rules are usually localized around some form of neighborhood around a given cell.

This can be described as a 5-tuple as follows:

The Game of Life would therefore be written as:

Here we see that the Game of Life is defined over the 2-dimensional integer grid over discrete time steps. Each cell can only assume two distinct states (dead or alive). The neighborhood is given by the 3x3 square around a cell. The local rule is defined by the exact number of live neighboring cells.

Basics of Lenia

Lenia is a family of CAs that was first created by Bert Wang-Chak Chan in the year 2018, where it won the Virtual Creatures Contest at the Genetic and Evolutionary Computation Conference in Kyoto. The idea behind Lenia is to generalize the GoL in a continuous way, regarding states, space and time. As immediately visible in the sample gif above, Lenia patterns appear eerily life-like, which is why they are often described as artificial or mathematical life forms.

Lenia patterns, from now on described as 'life forms', display smooth and organic movements and several emergent behaviours. Among those are locomotion (targeted movement), rotational movements, periodic contractions and even self-replication. Since Chan's original introduction of Lenia, several modifications of the framework have been published, showing further emergent behaviours like colony-formation and resistency to damage (maintaining shape even when noise is introduced or part of the shape is deleted). It is to note that Lenia is yet to be proven to be turing-complete. Turing-completeness has long been shown for the Game of Life through the construction of logic gates based on 'glider guns'. Since self-replicating patterns exist in Lenia, they might be a promising avenue to explore for finding an analogue.

Lenia definitions

Lenia can be defined similarly to the Game of Life. Hereby we have to differentiate between Discrete Lenia (DL) and Continuous Lenia (CL). Since we are computationally bounded to work with discrete time and space, Lenia as run on our hardware never truly becomes continuous but simply approaches continuity. CL is therefore the hypothesized resolution of DL, as time, space and state sets become continuous. We usually treat Lenia as a continuous CA, but should keep in mind that a truly continuous Lenia will inherently stay a mathematical framework moreso than a computational one.

Essentially (Continuous) Lenia is defined over the 2-dimensional real-numbers over the real timeline. The state set are the real interval between 0 and 1. Furthermore, the neighborhood is defined as the continuous unit ball around a given center. The local rule is obtained by concolving the grid with a kernel to yield a potential distribution. This potential is then fed into a growth mapping to yield a growth distribution. The kernel used for concolving is hereby arbitrarily defined. This leads us to Lenia being a family of CAs, since every kernel theoretically implies a unique CA. The most relevant findings have been found in Lenia-universes where ring-kernels are used.

While this description only provides a rudimentary intuition for how Lenia operates, the formal mathematical definitions can be found on Chan's github, as well as his published papers linked there.

The likeness of Lenia life forms to actual biological and cellular life lead to Lenia being considered an example of so called artificial or mathematical life. Chan even proposed a 'Tree of artificial life', inspired from standard biological classification.

3D Lenia and expanded universe

Chan's original program only supported 2-dimensional realizations of Lenia. However, merely two years after his original paper, he published a paper called Lenia and Expanded Universe.

This paper featured the newly developed alterations to standard Lenia as well as their findings. The most important additions were the inclusion of higher dimensions, multiple kernels, as well as multiple-channel Lenia (coexisting universes that interact simultaneously). This is where several additional emergent behaviours were observed. Life forms with polyhedral symmetries, self-replicaton, 'eating' / growth by ingestion, as well as 'cell colonies' emerged from these modifications.

While the 2-dimensional patterns appeared more and more life-like, the 3-dimensional cases were not quite as obvious. They primarily exhibit spherical shapes and appear more reminiscent of physical or chemical systems, compared to biological ones. While there have been less than 40 stable life forms found in 3D, with many of them being static, there are still some interesting shapes and behaviours among them. Analoguous to 2D, there are shapes that exhibit locomotion. There are also shapes that have rotational symmetry (particularly there is a rotating bipyramid inscribed within a sphere). Since Lenia is in practice constrained by the boundaries of the grid, there are also shapes that emerge when allowing periodic boundary conditions. Particularly tube- and membrane-like shapes have been observed, which exhibit locomotion in a few cases as well.