Cellular automata.

A cellular automaton is a grid of cells, each in some state, updated in steps. At every step each cell looks at its neighborhood (itself and the cells around it) and takes a new state from a rule. Every cell uses the same rule, and every cell updates at once.

The point of them is how much comes from how little. A rule that fits in a sentence can produce stable patterns, endless randomness, moving structures that collide and interact, or, in some rules, any computation a computer can do. John von Neumann and Stanisław Ulam used them in the 1940s to ask how a machine could build a copy of itself; John Conway’s Game of Life (1970) made them famous; Stephen Wolfram studied the simplest ones systematically in the 1980s. They are now used as models of growth, physics, chemistry and computation, and as a test bed for the question this site keeps returning to: how do local interactions add up to behavior nobody wrote down?

The simplest kind: one row of cells, each either 0 or 1, and each cell’s next state set by itself and its two neighbors. Three cells have 8 possible patterns, and a rule says 0 or 1 for each, so there are exactly 2⁸ = 256 rules. A rule’s number is its eight answers read as a binary number. Drawing each step as a new row below the last turns a run into a picture of its history.

Wolfram's four classes

Runs die out or freeze (class 1), settle into stable or repeating patterns (2), look random (3), or form structures that move and interact (4).

Rule 30

Starts from one cell and looks random: its center column was used as a random number generator in Mathematica, though every cell is determined by the rule.

Rule 110

Class 4: its moving structures can be arranged to carry out any computation (Matthew Cook's proof, 2004).

Damage spreading

Flip one cell and watch how far the difference travels: a measure of how sensitive a rule is, related to chaos.

Mixed rules

Different rules in different parts of the row. Where they meet, structures can be blocked, changed or passed through.

Background: elementary cellular automata, Rule 110, non-uniform cellular automata.

Lenia (Bert Chan, 2019) makes everything continuous: states are numbers between 0 and 1, the neighborhood is a smooth ring-shaped weighting, and time steps are small. The result looks less like a grid and more like living tissue: gliding, rotating and dividing “creatures” that hold their shape. Flow-Lenia (Plantec and colleagues, 2023) adds conservation of mass, so material moves rather than appearing and vanishing, and creatures can’t simply fade away; their parameters can travel with their matter.

Sources: Chan, “Lenia: Biology of Artificial Life”; Plantec et al., “Flow-Lenia”.

In a neural cellular automaton the rule is a small neural network, the same in every cell, and it is learned rather than written. Each cell carries a handful of numbers: some visible (its color or tissue type) and some hidden, which only the rule reads. Training adjusts the network until a single starting cell grows into a target shape, holds it, and regrows it when part is cut away (Mordvintsev and colleagues, 2020). Because nothing in training says how to do this, the hidden channels end up holding whatever information growth and repair need: a place to look for something like a developmental plan.

Sources: Mordvintsev et al., “Growing Neural Cellular Automata”; Sudhakaran et al., growing 3D artefacts.

• Rule Observatory: all 256 elementary rules, run and traced cell by cell.
• A Single Difference: damage spreading, measured over many runs.
• Worlds at the Boundary: walls of one rule inside another that let one pattern through.
• Anthrobot Lenia: Flow-Lenia creatures with cilia pushing along their rims.
• Growing Anthrobots and the Rejuvenation Field: a 3D neural cellular automaton that grows, rolls and heals.