Cellular automata · every rule, every class

Elementary CA Zoo

Every Wolfram elementary cellular automaton, grown from a single living cell, classified, and turned into a lab for Langton's edge-of-chaos lens.

An elementary cellular automaton is a one-dimensional row of black/white cells evolving one generation at a time. Each new cell looks only at its left neighbour, itself, and its right neighbour above it. There are eight possible neighbourhoods, each choosing black or white, so there are 2⁸ = 256 rules. Wolfram numbering reads those eight output bits as a binary number, ordered from 111 down to 000.

Wolfram classes: Class I rules quickly collapse to a uniform fixed point; Class II rules settle into stable or periodic structure; Class III rules stay chaotic and random-looking; Class IV rules support localized structures that interact. Edge of chaos comes from Chris Langton's λ framing; Wolfram Class IV is related, not identical. Class labels here follow the Martínez (2013) survey table expanded across mirror/complement equivalents; Rules 41, 54, 106, and their equivalents are sometimes classified as Class III elsewhere.

Need to find your footing first? Read in layers.

If this is your first time: picture a long row of squares, each painted black or white. A rule is a little decision table that says “if these three squares above me look like this, I become black; otherwise I become white.” Apply that decision at every position, and you get a new row underneath the first. Do it again and again, stacking rows downward. Each of the 256 rules tells a different visual story — some collapse to nothing, some settle into stripes, some look like static, and a rare few grow little moving shapes. Click any preview below and watch what happens.

If you’ve seen binary before: a rule looks at three cells — left, centre, right — each black (1) or white (0). That’s 2³ = 8 possible inputs. The rule picks an output (0 or 1) for each input, so a complete rule is 8 bits. Reading those 8 output bits as a binary number from input 111 down to 000 gives Wolfram’s rule number, 0–255. The explorer’s Bits readout shows that decoding live for whichever rule you load.

If you’re comfortable with discrete math: these are one-dimensional, two-state, three-neighbour cellular automata on ℤ/nℤ with periodic boundary conditions. The symmetry group generated by spatial reflection and 0/1 colour-complement has order 4 and partitions the 256 rules into 88 equivalence classes — which is what Wolfram’s qualitative I–IV classification is actually defined on. The lab measures standard descriptive statistics for the resulting space–time orbit: Langton’s active-fraction parameter λ, Shannon row entropy, block entropies Hₙ, lagged Hamming-based row correlation, and a one-cell perturbation lightcone (a discrete analogue of a Lyapunov probe).

If you want the deeper story: rule 110 was proven Turing-complete by Matthew Cook, “Universality in Elementary Cellular Automata,” Complex Systems 15 (2004) 1–40 — making it the simplest universal cellular automaton currently known. Class IV behaviour is often discussed alongside Langton’s critical-λ hypothesis, but the link between “edge of chaos” and computational capacity is genuinely contested — see Mitchell, Hraber & Crutchfield, “Revisiting the Edge of Chaos: Evolving Cellular Automata to Perform Computations,” Complex Systems 7 (1993) 89–130. For a survey of competing classifications (Wuensche basins-of-attraction, Li–Packard refinements, communication-complexity, topological, compression-based, morphological diversity, memory-induced), see G. J. Martínez, “A Note on Elementary Cellular Automata Classification,” arXiv:1306.5577 (2013), Journal of Cellular Automata 8(3–4):233–259.

Guided tour

Why this rule matters

Four stops, four ideas: collapse, randomness, additive symmetry, and localized computation near the ordered/chaotic boundary. Each one runs live with marked features. Watch first; explore after.

Annotation layers

What the metrics reveal

Connect each mark to a measurement

The tour marks what your eye should catch. The lab measures the same behaviour, so a pattern can move from “I see it” to “the rule exposes it in the numbers.”

Collapse / order
Low row entropy and high row correlation: the rule forgets the seed and keeps repeating itself. Measure collapse.
Noise / random-looking behaviour
High entropy, lively density, and low correlation: rows stay mixed and stop resembling their recent past. Measure Rule 30.
Triangles / additive symmetry
Block entropy stays structured while correlation follows repeated algebraic texture. Measure Rule 90.
Localized packets / gliders
Sensitivity spreads as tracks instead of a blast; entropy and correlation sit between frozen order and noise. Measure Rule 110.

Explorer

Rule 30

GIF export uses the current rule, seed, width, and generation count.

Hand-set seed row: click cells below, then choose “hand-set row”.

Lab

Langton λ lab

Turn a rule into measurements you can see. Langton's λ counts how many of the eight neighbourhoods produce a live cell; Langton used λ to frame phase transitions and computation near the edge of chaos. Density and entropy show whether rows collapse, repeat, or keep information mixed. Block entropy measures short-cell patterns, normalized per cell so H2 and H3 stay comparable. Row correlation checks how much a future row still resembles its past, and sensitivity flips one starting cell to follow the lightcone of disagreement.

What each metric actually is

Langton’s λ — plainly: count how many of the 8 input patterns make a black cell, divide by 8. λ = 0 means the rule never produces black; λ = 1 means it always does. Pure property of the rule itself; doesn’t depend on the seed. Reference: C. G. Langton, “Computation at the edge of chaos: Phase transitions and emergent computation,” Physica D 42 (1990) 12–37.

Row entropy: per-row Shannon entropy H(p) = −p·log₂ p − (1−p)·log₂(1−p), where p is the fraction of black cells in the row. Maxes at 1 when the row is exactly half-and-half; drops to 0 when the row is all one colour. The sparkline is this value over generations.

Block entropy Hₙ: slide a window of n adjacent cells across the row, count how often each of the 2ⁿ possible windows appears, take the Shannon entropy of that distribution, then divide by n. The per-cell normalisation is what makes H₂ and H₃ directly comparable here.

Row correlation (N → N+8): for every row, compare it cell-by-cell with the row 8 generations later, count matches, and rescale Hamming agreement to [−1, 1]. Near +1: the future is essentially a repeat. Near 0: the row has forgotten its past. Below 0: there’s alternating/inverting structure.

Sensitivity & spread: evolve the seed once, then flip one cell at the centre of the seed and evolve again. The pink delta canvas highlights every cell where the two evolutions disagree. The lightcone you see is the discrete-CA analogue of a Lyapunov probe; spread rate is the average growth of the disagreement’s width per generation.

About the composite scores: the atlas’s complexity (≈ 0.72·entropy-score + 0.28·decorrelation) and sensitivity (≈ 0.62·max-changed/width + 0.38·spread-rate) are weighted combinations of the metrics above. They’re defined here for ranking and exploration on this site; they’re not standard published measures, so don’t cite them.

Try for random-looking irreducibility, for Class IV gliders and computation, and for clean nested structure. Narrow widths and wraparound boundaries can change the numbers, so treat lab metrics as evidence from the current setup.

Langton λ —

Fraction of neighbourhood outputs that create a live cell.

Mean row density —
Mean row entropy —
Sensitivity lightcone —
2-bit block entropy —

Read adjacent pairs. Higher values mean more short local patterns stay in play.

3-bit block entropy —

Read neighbourhood-sized triples. Rich rules use more of the tiny pattern alphabet.

Row N → N+8 correlation —

Compare rows eight steps apart. Near 1 repeats, near 0 forgets, negative inverts.

Spread rate —

Track how fast one flipped cell widens its future disagreement cone.

Metric lens

Choose what to look for next

Pick one measurement and the lab highlights the matching evidence: the score card, the sparkline, or the canvas where the behaviour becomes visible.

Follow

Sensitivity shows where one flipped cell matters

Watch the magenta delta canvas: narrow tracks mean structure carries the change; a broad burst means chaos; a quick fade means the rule forgets.

    All-rule measurement map

    λ × complexity × sensitivity

    Scan every rule in one field. The horizontal axis is Langton's λ, the vertical axis is a weighted entropy/decorrelation score, and colour marks a weighted one-cell perturbation score. “Complexity” and “sensitivity” are composite rankings for exploration here, not standard canonical measures. Click a ranked rule to open its lab view.

    Ranked rules in the current measurement pass
    RuleClassλComplexitySensitivitySpread

    How to read it

    Class I rules erase information; metrics dive toward zero or one. Class II rules make tidy repetitions. Class III rules keep high entropy and spread small changes fast. Class IV rules are the interesting middle in Wolfram's classification: block entropy stays alive, row correlation neither freezes nor vanishes completely, and changes travel as structured packets. Langton's edge-of-chaos idea is a nearby lens, not a synonym.

    Why rule 110 and rule 30 matter

    Rule 30 is a compact demonstration of computational irreducibility: the only honest way to know row 100 is to run the intervening rows. Rule 110 is famous because its moving local structures can perform universal computation. The lab view makes both claims visible: not as slogans, but as density, entropy, and perturbation growth.

    Original seed

    Perturbed seed

    Changed cells highlighted

    Magenta cells are positions where the one-cell perturbation changed the future. A narrow, persistent cone hints at mobile structure; a full blast suggests chaos; a quick fade means the rule forgets.

    Comparison view

    Two rules, one seed

    Pin two rules and watch them evolve side-by-side from the same starting row. Use to share the pair.

    Rule 30

    Rule 110

    Gallery

    Click any preview to load it into the explorer, or filter by Wolfram class.