Rule Observatory.

A single row of cells, each either on (1) or off (0), updated again and again by one rule. The big picture is the row’s whole history: each horizontal line is one moment, and time runs down the canvas, so a pattern you see is a record of how the row changed. Space runs across.

• The space-time canvas keeps the latest 512 generations. Bright cells are 1, dark cells 0. In the “Luminous” display, color changes with the row only for presentation; “Pixel exact” shows plain binary pixels.
• The rule table (“Eight neighborhoods. One rule.”) shows the whole rule: for each of the eight possible neighborhoods, the state the middle cell takes next. It also lists the rules that behave the same way when mirrored or with 0 and 1 swapped.
• The initial row is the starting state, which you can paint.
• The inspector shows, for a selected cell, the neighborhood above it and the rule output that produced it.

Rule 30 / 90 / 110

Quick picks for three famous rules: chaotic-looking, fractal and complex.

Rule

Any of the 256 rules. Changing it starts a new run.

Boundary

Periodic wraps the row into a ring, so the right edge neighbors the left. Fixed-zero treats cells beyond the edges as always 0.

Cells across

The row's width, 65 to 1,025 cells.

Steps per second

Playback speed.

Display

Luminous (smooth, colored by row) or Pixel exact (binary, no smoothing).

Play / Step / Restart

Run continuously, advance one generation, or go back to generation zero.

Zoom

Enlarge the canvas; scroll to pan around a large history.

Rule table outputs

Tap an output (0 or 1) to flip it. That changes the rule number and restarts the run: the quickest way to see what one decision does.

Initial row

Single cell puts one 1 in the middle; Seeded random fills the row at the chosen density from a seed; Clear row empties it. Paint directly on the strip, or set a cell by index.

Inspect a local update

Choose a generation and position (or click a cell on the canvas) to see why it has its value.

Copy share link / Export / Import replay

Save the experiment, including any edits, as a link or a file and restore it later.

Try this: start Rule 30 from a single cell and look at the center column, which looks random although nothing random is involved. Then try Rule 90 from a single cell (a Sierpiński triangle), and Rule 110 from a random row, where small moving structures collide and interact.

Elementary cellular automata

An elementary cellular automaton is the simplest kind: one row, two states, and a rule that looks only at a cell and its two neighbors. Three cells give 2³ = 8 possible neighborhoods, and the rule assigns 0 or 1 to each, so there are 2⁸ = 256 rules. All cells update at once, from the previous row.

Rule numbers

Stephen Wolfram’s numbering lists the neighborhoods from 111 down to 000 and reads the eight outputs as a binary number. Rule 30 is 00011110 in binary: neighborhoods 100, 011, 010 and 001 produce 1, the rest 0. Mirroring a rule left to right, or swapping the roles of 0 and 1, gives rules that behave identically up to relabeling, so the 256 rules fall into 88 distinct families.

Four kinds of behavior

Wolfram sorted the rules informally into four classes: patterns that die out or freeze (class 1), settle into stable or repeating structures (class 2), look random (class 3, such as Rule 30), or produce long-lived structures that move and interact (class 4, such as Rule 110). Rule 110 has been proved capable of universal computation: in principle, with the right starting row, it can carry out any calculation a computer can.

The classes are a description, not a test. Whether a rule looks random depends on its starting row and its boundary, which is why the page lets you change both.

The Observatory is an instrument rather than an experiment with a single result. What it demonstrates is the core observation of the field: rules with eight bits of description produce behavior ranging from nothing to apparent randomness to computation. Rule 30’s center column was used for years as a random number generator in Mathematica, although it is fully determined by its rule and starting cell.

The inspector makes a second point concrete: however complex a pattern looks, every cell is explained by three cells above it and one entry in the rule table. The complexity is in how those local decisions compound, not in any single one.

• These are mathematical models, not simulations of anything physical or biological.
• A pattern’s appearance doesn’t prove it is chaotic or computationally universal; those are properties that need proofs or careful measurement.
• A finite row eventually repeats: with N cells there are at most 2^N states. The repetition can take far longer than you will watch.