Automaton on a sphere
A Conway-class cellular automaton on a latitude–longitude mesh wrapped to a closed surface. Drag to rotate; click the front face to seed.
Run
Initial condition
Rule (B/S)
Experiment
Export saves the seed, dimensions, rule and the exact packed grid, so an experiment reloads identically.
What the metrics mean
consensusLangton's λ (Langton, "Computation at the edge of chaos," Physica D 42, 1990, 12–37) is a property of the rule table — the fraction of transitions mapping to a non-quiescent state — used as a swept control parameter. This app instead reports activity Δ: the fraction of cells that changed state this generation. That is a dynamic observable (a normalized Hamming distance between successive states), not Langton's λ. It is named accordingly to avoid the conflation present in the first prototype.
Why the phase space is colored by block entropy
my noteThe binary entropy H(p) = -p·log2 p - (1-p)·log2(1-p) is a deterministic function of the live fraction p. Coloring a plot by it when p is already an axis adds no information. The color channel here is a spatial 2×2 block entropy: tally the 16 possible 2×2 occupancy patterns across the lattice and take their normalized entropy. Two states with identical density but different clustering get different block entropy, so the color separates "same density, different organization" — the axis the live fraction cannot show.
Why a lat/lon mesh, not a geodesic sphere
consensusA hexagon-based sphere tiling forces exactly 12 pentagons (Euler's formula; the fullerene/Goldberg result), giving cells 5 or 6 neighbours. my noteConway's B3/S23 is defined for the 8-neighbour Moore template, so it does not transfer to a geodesic mesh without re-deriving the rule. The lat/lon mesh preserves the 8-neighbour structure. Its cost is real: cells near the poles cover far less area and their angular neighbour spacing is non-uniform, so pole regions are over-weighted in longitude. That distortion is a property of this projection, not a bug to be patched away.
On "edge of chaos" framing
live debateLangton (1990) argued complex behaviour sits at intermediate λ. Mitchell, Crutchfield and Hraber ("Revisiting the Edge of Chaos," Complex Systems 7, 1993, 89–130, and the 1994 re-examination in the SFI Complexity volume) evolved cellular automata and found λ does not cleanly predict computational capability. The descriptive backdrop is Wolfram's four classes ("Universality and complexity in cellular automata," Physica D 10, 1984, 1–35). Treat the activity / density / entropy traces as exploratory description, not evidence for a particular complexity claim.
Pole handling
correctionAn earlier review draft suggested the reflect-and-shift pole wrap double-counts a neighbour. On inspection it does not for any longitude count ≥ 6: crossing a pole maps to longitude + 180°, and the three across-pole cells stay distinct from the same-row neighbours. The residual issue is distortion (above), not a discrete miscount.