Explore all 256 elementary cellular automaton rules discovered by Stephen Wolfram. Watch Rule 30 produce chaos from order, Rule 90 draw Sierpinski triangles, Rule 110 compute (it's Turing-complete!) and Rule 184 model traffic flow.
Each cell looks at itself and its two neighbours, then applies a simple rule to determine the next state. 256 possible rules create an incredible diversity of behaviour.
Select a rule number (0-255) or pick a preset. Watch the automaton evolve row by row. Compare how tiny rule changes create vastly different patterns.
Wolfram's Rule 110 was proven Turing-complete in 2004 — meaning this tiny, one-dimensional system can compute anything a modern computer can, given enough time and space.
Wolfram elementary cellular automata are one-dimensional systems where a row of binary cells evolves step by step according to a simple rule: each cell's next state is determined solely by its current state and those of its two immediate neighbours. Despite this extreme simplicity, the 256 possible rules — numbered 0 to 255 — produce a stunning variety of behaviours ranging from static patterns and periodic oscillations to fractal self-similarity and apparent randomness. Watching the automaton grow row by row reveals how complexity can emerge from the simplest possible rules.
Stephen Wolfram systematically studied all 256 elementary rules in the 1980s and published his findings in the landmark 2002 book "A New Kind of Science," proposing that simple computational rules underlie much of the complexity seen in nature. These automata have since been used to model phenomena as diverse as seashell pigmentation patterns, traffic flow on highways, cryptographic pseudorandom number generation, and theoretical models of computation.
A cellular automaton is a grid of cells, each in one of a finite number of states, that evolves over discrete time steps. In a 1D elementary cellular automaton, each cell is either alive (1) or dead (0), and the next state of every cell is determined by a fixed rule applied to that cell and its two neighbours. Repeating this process generation after generation, row by row, builds up a visual pattern from top to bottom on the screen.
Type any rule number from 0 to 255 into the input box, or click one of the preset buttons (Rule 30, 90, 110, 184) to jump to famous examples. You can also toggle individual bits in the 8-cell rule table to fine-tune the rule. Adjust cell size to see finer or coarser detail, set the speed to control how many generations advance per frame, and choose between starting from a single live cell, random noise, or a symmetric random row. Press Pause to freeze the evolution and Reset to restart.
Rule 90 computes the XOR of a cell's two neighbours, ignoring the cell itself. Starting from a single live cell, this XOR operation is mathematically equivalent to computing binomial coefficients modulo 2 — exactly the operation that generates Pascal's triangle mod 2, which is the Sierpinski triangle fractal. The self-similar triangular pattern emerges inevitably from this arithmetic, not from any special design, demonstrating how fractal geometry can arise from elementary operations.
A system is Turing-complete if it can simulate any computation that a general-purpose computer can perform, given sufficient time and memory. Matthew Cook proved in 2004 (with Wolfram's permission to publish) that Rule 110 is Turing-complete by showing it can simulate a specific type of universal machine called a cyclic tag system. This means Rule 110 — a one-dimensional line of binary cells updated by a table of just 8 bits — is in principle capable of running any algorithm, making it one of the simplest known universal computational substrates.
Rule 184 maps directly onto the asymmetric simple exclusion process (ASEP), a standard model of traffic flow. Each live cell represents a vehicle, and the rule moves vehicles forward by one position if the cell ahead is empty, otherwise they stay put. This naturally reproduces real traffic phenomena: at low density, vehicles flow freely at maximum speed; above a critical density, a jam propagates backward as a wave even while individual cars occasionally move forward. Rule 184 is used in physics and transportation science as the simplest discrete traffic model that captures this jam-wave behaviour.
Rule 30 is not random — it is entirely deterministic — but the pattern it generates from a single live cell is provably chaotic and passes standard statistical tests for randomness. Wolfram used the centre column of Rule 30 as a pseudorandom number generator in Mathematica for many years, and it has been studied as a source of cryptographic randomness. The pattern is so sensitive to initial conditions and so visually unpredictable that predicting future states without running the rule step by step appears computationally irreducible: there is no known shortcut faster than actually simulating it.
One-dimensional cellular automata were first studied in the early 1980s by Stephen Wolfram, who performed a systematic computer-aided survey of all 256 elementary rules and published his results in a series of papers starting in 1983. Wolfram classified the rules into four behavioural classes — fixed points, periodic patterns, chaotic patterns, and complex (Class IV) behaviour — a taxonomy that remains influential in complexity science today. His later book "A New Kind of Science" (2002) extended these ideas into a grand theory proposing that simple rules are the foundation of all natural complexity.
Conway's Game of Life is the most famous two-dimensional cellular automaton, using a 3x3 neighbourhood instead of a 1x3 strip; it also achieves Turing-completeness and produces gliders, oscillators, and spaceships. The Sierpinski Triangle simulation on this site shares its fractal structure with Rule 90. Reaction-diffusion systems like the Gray-Scott model extend the idea of local rules governing pattern formation into continuous chemistry. Larger-neighbourhood and totalistic automata — where only the count of live neighbours matters — include the family containing Brian's Brain and Wireworld, which can emulate digital logic circuits.
Cellular automata underpin several practical technologies. Rule 30 and related automata have been used as pseudorandom number generators in computer algebra systems. CA-based models power fluid dynamics simulations via lattice-Boltzmann methods, which approximate the Navier-Stokes equations on a grid of local collision rules and are used in aerodynamics and microfluidics research. In hardware, CA shift registers appear in stream ciphers and error-correcting codes. Researchers have also used CA to design self-replicating robotic systems and programmable matter, inspired by von Neumann's theoretical self-reproducing automaton from the 1950s.
Current research explores reversible cellular automata, where every configuration has a unique predecessor — relevant for quantum computing and thermodynamically efficient computation. There is active work on continuous cellular automata (SmoothLife and Lenia), which generalize discrete rules to real-valued states and produce lifelike organisms that move and reproduce. Researchers are also investigating CA as models of spacetime at the Planck scale, asking whether the universe itself might be a cellular automaton. In machine learning, convolutional neural networks are structurally analogous to CA, and Neural Cellular Automata have been trained to grow target images and self-repair, opening a new area combining deep learning with the CA paradigm.