CA Rule-Space Classifier: Langton's λ Parameter

Rule-Space Parameters

λ center (Langton parameter) 0.35

Neighborhood radius r 2 (32-entry table)

Initial condition

Speed 8 gen/s

Generation Time-Series

Generation

Steps / sec

0.0

The Six Rules (by measured λ)

How it works

Each of the six panels runs an independent k = 2-state, radius-r cellular automaton. A cell's next state comes from a lookup table indexed by its (2r+1)-cell neighborhood:

idx = Σ state(i+j)·2^(j+r),  j = −r..r
state(i, t+1) = table[idx]      (table[0] ≡ 0, quiescent)

Langton's λ parameter (Langton, 1990, "Computation at the edge of chaos") is the fraction of table entries mapped to the non-quiescent state: λ = (#entries = 1) / tableSize. Each panel's table is drawn at random subject to a target λ, then run forward — this is exactly the random-table experiment Langton used to map Wolfram's four qualitative behaviour classes onto one continuous control parameter:

λ ≈ 0 — almost every neighborhood decays to 0: Class I, the lattice freezes to a fixed point.
small λ — isolated blinkers and stripes survive: Class II, periodic/nested cycles.
λ near a critical λ_c — gliders, long-lived structures, long transients: Class IV, the "edge of chaos" where the automaton is conjectured capable of universal computation (Rule 110 is the canonical elementary example).
large λ — almost every update flips: Class III, spatiotemporal chaos.

The class label under each panel is not hand-assigned — it is measured live from the Hamming distance between consecutive rows (the fraction of cells that flip each step). A near-zero, low-variance flip rate reads as Class I; a low-variance moderate flip rate reads as Class II; a high, steady flip rate reads as Class III; a moderate flip rate with high variance — long quiet stretches punctuated by bursts of change — reads as Class IV, exactly the signature complexity researchers use to locate the edge of chaos empirically.

Active cells are pushed forward in the panel (relief/extrusion) purely so the pattern is readable in 3D from an angle — the underlying automaton itself is one-dimensional, evolving row by row exactly like the classic 2D Wolfram diagrams.

quiescent cell (state 0) active cell (state 1, colour = λ)

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Cellular Automaton Rule Classifier — Langton's λ Parameter