Section 1
Eight bits, 256 rules, 8 neighbours each
Each elementary CA rule is an 8-bit integer — the lookup table for the function f(left, center, right) → next state. The eight input patterns are 111, 110, 101, 100, 011, 010, 001, 000 (Wolfram numbering: bits 7 down to 0). Each bit independently specifies the output for one neighbourhood.
Flipping any single output bit produces a neighbouring rule. The 256 rules are the vertices of an 8-dimensional Boolean hypercube: 28 = 256 vertices, each with 8 edges, giving 8 · 27 = 1024 edges total. Every pair of neighbours is connected by a single bit-flip in each direction.
Wolfram's behavioural classes assign a qualitative label to each rule — Class I (uniform), Class II (periodic), Class III (chaotic), Class IV (complex). Those labels are not smooth over the hypercube. The walker below makes the jaggedness tangible.
How does the Wolfram numbering work?read in layers
Start here Each elementary CA rule is just a small lookup table: for each of the 8 possible 3-cell neighbourhoods, it says whether the centre cell lives or dies. Wolfram numbered the rules by reading those 8 outputs as a binary number. Rule 110, for example, outputs 01101110 in binary — that's 110 in decimal, hence "Rule 110".
High school The 8 input patterns are ordered from largest to smallest: 111 (7), 110 (6), …, 000 (0). Bit k of the rule number is the output when the neighbourhood encodes to decimal k. So rule R's output for neighbourhood k is (R >> k) & 1. Flipping bit k of R changes only the output for that one neighbourhood — nothing else.
Undergrad The bit-flip metric on {0,1}8 is the Hamming metric. The hypercube graph is the Cayley graph of ℤ28 under generators e0, …, e7. The 256 rules under the standard mirror and state-complement symmetries reduce to 88 inequivalent classes (Wolfram 1983). The symmetry group has order 4, so each equivalence class has size 1, 2, or 4. The hypercube's symmetry group is much larger (order 28 × 8! = 10,321,920) and does not preserve Wolfram classes — which is why walks in Hamming space can cross class boundaries so easily.
Going deeper The distribution of Wolfram classes across the hypercube is neither random nor clustered. Wuensche and Lesser (1992) mapped the "basin of attraction fields" of all 256 rules and found that the Class IV rules cluster near the boundary between the Class I/II and Class III regions in various parameterisations — Langton's λ parameter being the most studied. But λ is itself a smooth function on the hypercube (it counts 1-outputs), and the class boundaries it predicts are approximate. The empirical observation is that Hamming-distance-1 class transitions are common, even across the most dramatic class gaps.
Section 2
Walk the hypercube
The current rule is on the left. Its 8 neighbours — one per bit-flip — are on the right. Click any neighbour to make it the centre. The previous centre becomes one of its own 8 neighbours (every edge is bidirectional).
01101110
Class IV
Walk history
Section 3
Class doesn't change smoothly
Rule space is not smooth. Four rules make the point concretely. The neighbours listed are all exactly one bit-flip away.
Rule 110 Class IV
Cook's universal rule. Provably Turing-complete.
| Flip bit | Neighbour | Class | Character |
|---|---|---|---|
| 0 | Rule 111 | Class II | periodic |
| 1 | Rule 108 | Class II | periodic |
| 2 | Rule 106 | Class IV | complex |
| 3 | Rule 102 | Class III | chaotic |
| 4 | Rule 126 | Class III | chaotic |
| 5 | Rule 78 | Class II | periodic |
| 6 | Rule 46 | Class II | periodic |
| 7 | Rule 238 | Class I | uniform |
5 of Rule 110's 8 neighbours are Class I or II. Flip the right bit and universality collapses to a fixed point. Cook's universal rule is a knife-edge in the hypercube.
Rule 30 Class III
Wolfram's canonical chaotic rule; used in Mathematica as a pseudorandom number generator.
| Flip bit | Neighbour | Class | Character |
|---|---|---|---|
| 0 | Rule 31 | Class II | periodic |
| 1 | Rule 28 | Class II | periodic |
| 2 | Rule 26 | Class II | periodic |
| 3 | Rule 22 | Class III | chaotic |
| 4 | Rule 14 | Class II | periodic |
| 5 | Rule 62 | Class II | periodic |
| 6 | Rule 94 | Class II | periodic |
| 7 | Rule 158 | Class II | periodic |
7 of 8 neighbours are Class II. Rule 30's chaos is a single island — surrounded almost entirely by periodic rules.
Rule 90 Class III
The XOR rule. Additive, self-similar; produces the Sierpiński triangle from a single cell.
| Flip bit | Neighbour | Class | Character |
|---|---|---|---|
| 0 | Rule 91 | Class II | periodic |
| 1 | Rule 88 | Class II | periodic |
| 2 | Rule 94 | Class II | periodic |
| 3 | Rule 82 | Class II | periodic |
| 4 | Rule 74 | Class II | periodic |
| 5 | Rule 122 | Class III | chaotic |
| 6 | Rule 26 | Class II | periodic |
| 7 | Rule 218 | Class II | periodic |
The XOR rule's exact self-similar structure vanishes with any single bit-flip. 7 of 8 neighbours are periodic.
Rule 184 Class II
The traffic flow rule. Models single-lane highway density with a sharp phase transition.
| Flip bit | Neighbour | Class | Character |
|---|---|---|---|
| 0 | Rule 185 | Class II | periodic |
| 1 | Rule 186 | Class II | periodic |
| 2 | Rule 188 | Class II | periodic |
| 3 | Rule 176 | Class II | periodic |
| 4 | Rule 168 | Class I | uniform |
| 5 | Rule 152 | Class II | periodic |
| 6 | Rule 248 | Class I | uniform |
| 7 | Rule 56 | Class II | periodic |
Two of the 8 neighbours collapse immediately to Class I. The traffic rule's structure evaporates in two different directions.
The pattern holds across the whole hypercube. Class boundaries are not walls — they're more like faults: sharp, local, and pervasive. Walking the hypercube reveals this at every step.
Section 4
What the hypercube contains
The 256 vertices are not evenly distributed across classes. The vast majority collapse quickly. Genuinely interesting behaviour is rare.
192 of 256 rules are periodic. Class IV — the behaviour that supports localized interacting structures and, in the case of Rule 110, universal computation — covers only 14 rules.
Langton's edge-of-chaos framing is one attempt to explain why the 14 exist where they do: Class IV rules cluster near the boundary between the ordered and chaotic regions when the rules are parameterised by the fraction of 1-outputs (the λ parameter). That framing is productive but contested; not all researchers accept it as an explanation rather than a restatement. Measure the metrics directly on the home page lab.
Class labels follow Wolfram's qualitative classification expanded through mirror and state-complement equivalences per Martínez (2013). Rules 41, 54, 106, and their equivalents are sometimes listed as Class III in other sources. The classification here follows the Martínez survey table. The boundary is genuinely ambiguous at those rules.