2D CA Rule-Space Classifier: Langton's λ on a Moore Lattice

Rule-Space Parameters

λ center (Langton parameter) 0.35

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, genuinely two-dimensional k = 2-state outer-totalistic cellular automaton on its own toroidal Moore lattice (8-neighbor, like Conway's Life universe — not a flattened 1D rule). A cell's next state comes from a lookup table indexed by its own current state and the count of live neighbors among the 8 surrounding cells:

n = Σ state(neighbor),  8 Moore neighbors
idx = ownState·9 + n            (0 ≤ idx ≤ 17)
state(x,y,t+1) = table[idx]      (table[0] ≡ 0, quiescent)

Langton's λ parameter (Langton, 1990, "Computation at the edge of chaos") generalizes directly to this 2D outer-totalistic family: λ is the fraction of table entries mapped to the non-quiescent state. Each panel's 18-entry table is drawn at random subject to a target λ, then run forward on a real 2D grid — the same random-table experiment Langton used, but on an actual 2D lattice instead of a 1D row extruded to look three-dimensional:

λ ≈ 0 — almost every neighborhood decays to 0: Class I, the lattice freezes to a fixed point.
small λ — isolated blinkers and still lifes survive: Class II, periodic/nested cycles.
λ near a critical λ_c — gliders, long-lived structures, long transients: Class IV, the "edge of chaos" (Conway's Life itself sits near here).
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 fraction of cells that flip state each generation (a 2D Hamming distance / cell count) across a rolling window: 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 — quiet stretches punctuated by bursts of change — reads as Class IV.

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

2D Cellular Automaton Rule Classifier — Langton's λ on a Moore Lattice