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.

What is a "class," and why only four of them?read in layers

Start here after running a rule for many generations, the picture tends to do one of four things: vanish into a single colour, fall into a stable stripe or repeat, keep producing static, or grow little objects that drift and bump into each other. The class label on each card tells you which family that rule lands in.

High school the four classes are named by what the long-run behaviour looks like, not by a measurement threshold. The same behaviour shows up across many rule numbers because reflection (left↔right) and colour-swap (black↔white) preserve the dynamics - so rules that look like mirror images of each other share a class.

Undergrad the boundary between Class III and IV is genuinely fuzzy and depends on the lattice width and boundary conditions. Open the lab with rules 41, 54, or 106 - their measured λ, entropy, and sensitivity sit in the no-man's land between chaotic and complex, and the class call flips depending on which scheme you use. The 256-rule classification problem is not algorithmically decidable in general; what we have is a stable taxonomy with disputed borderline cases.

Going deeper at least nine published classification schemes exist, and they disagree mostly on the III/IV boundary: Wuensche's basins-of-attraction and Z-parameter, Li-Packard refinements, communication-complexity (Bravi et al.), topological/surface-based, Lempel-Ziv compression, morphological-diversity, memory-induced. The honest position is that "Class IV" names a region of rule-space where interesting things happen, not a mathematically sharp category. Martínez (2013, arXiv:1306.5577) tabulates these schemes side-by-side; class labels follow that survey's consolidated table.

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 isread in layers

Start here each metric is one question with a numerical answer. Density: how much black? Entropy: how mixed-up? Correlation: does the future look like the past? Sensitivity: if I poke it, does the poke spread? The numbers let you compare two rules without arguing about what you saw.

High school Langton’s λ = count of 1s in the rule’s output table, divided by 8 — it’s a property of the rule, not the seed. Row entropy is Shannon’s H(p) = −p·log2 p − (1−p)·log2(1−p), where p is the fraction of black in the row; maxes at 1 for half-and-half. Row correlation (N→N+8) compares each row to the row 8 steps later, rescaled to [−1, +1]. Sensitivity flips one cell at the seed centre and tracks the cone of disagreement.

Undergrad the metrics that matter the most for distinguishing classes aren’t the single-row ones. Block entropy Hn (per-cell entropy rate over n-cell windows, n = 2 or 3 here) catches spatial structure that single-cell entropy misses. The perturbation lightcone is a one-trajectory analogue of the maximum Lyapunov exponent: linear growth (λmax > 0) is the discrete signature of sensitive dependence; saturation is Class II; sub-linear spread is what Class IV rules tend to show. Lattice width and boundary choice matter — narrow lattices truncate the lightcone and inflate correlation, so treat the numbers as evidence from this setup, not absolutes.

Going deeper the atlas’s composite 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, defined for ranking exploration here, not as standard published measures. They aren’t standard published measures, so don’t cite them. If you want a published axis to put 256 rules on, look at Wuensche’s input-entropy and Z-parameter (basin-of-attraction analysis), Lempel–Ziv compression ratio, topological entropy, or the communication-complexity measure of Bravi et al. Each highlights a different aspect of complexity, and none is canonical; the comparison table in Martínez (2013) shows where they agree and where they fight.

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, 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.

    Why don’t the class counts add to 88?read in layers

    Start here the four class chips count rules out of 256, not 88. The 88 number you might have heard counts something different: how many rules are essentially distinct once you ignore mirror-images and colour-swaps.

    High school reflect a rule left–right, or swap black↔white, and you get a rule that behaves the same way. Both operations together also count. Most rules have an orbit of size 4 under those symmetries, so 256 / 4 ≈ 64 — but rules that are their own mirror or colour-swap have smaller orbits, which is why the actual count is 88, not 64.

    Undergrad exact count via Burnside / Cauchy–Frobenius: with the order-4 group G = {e, r, c, rc} acting on the 256 rule tables, |orbits| = (|Fix(e)| + |Fix(r)| + |Fix(c)| + |Fix(rc)|) / |G| = (256 + 64 + 16 + 16) / 4 = 88. The fix counts come from how many output bits each symmetry leaves free: reflection identifies neighbourhood pairs like 001↔100 and 011↔110 (so 6 free output bits, 26 = 64 fixed rules); colour-complement pairs each input with its bit-flipped form and demands opposite outputs (4 free bits, 24 = 16); rc is similar.