Cellular Automaton
A grid of cells, each updated by a simple rule that looks only at its neighbors — discrete dynamics that generate astonishing complexity.
Cellular Automaton
A cellular automaton is a Dynamical System stripped to its barest discrete form: a grid of cells, each in one of a few states, all updated simultaneously by a single rule that looks only at a cell's immediate neighbors. Time ticks in steps, space is a lattice, and states are discrete — yet from this austere setup pours pattern, structure, and even universal computation. Cellular automata are the cleanest demonstration of the section's thesis: local rules, global complexity.
Because the update is a map applied over and over, cellular automata live squarely in the world of Flows and Maps — they are maps whose state space is an entire grid.
Elementary cellular automata
The simplest family lives on a one-dimensional row of cells, each either on or off. A cell's next state depends on just three cells: itself and its two neighbors. There are 2^3 = 8 possible neighborhoods, and a rule assigns an output to each — so there are exactly 2^8 = 256 such rules, numbered 0–255 by Wolfram's convention (read the eight outputs as a binary number).
Most are dull. A few are extraordinary. Rule 30 is the star: from a single on-cell it generates a stream of structure so disordered that it has been used as a random-number generator, yet it is produced by a rule you can state in one line.
Watch Rule 30 build itself
Each row is computed from the one above it and drawn beneath, top to bottom. The left edge marches in a clean periodic stripe while the right dissolves into apparent randomness — order and chaos from the same three-cell rule.
Classes of behavior
Survey all 256 rules and their long-term behavior sorts into four rough classes: settling to a uniform state, settling to stable or periodic stripes, producing chaotic noise, and — rarest and most interesting — generating localized structures that move and interact. That fourth class is where computation hides. The two-dimensional automaton Conway's Game of Life belongs to it, and its drifting "gliders" are the proof that a grid of bits can carry information across space.