Self-Sustaining Spiral Waves — Cyclic Cellular Automaton

Automaton Rule

States K 12

Excitation threshold T 1

Neighborhood radius 1

Speed (gen/s) 12

Live Readouts

Generation

Active cells

State entropy (order ↔ chaos)

0.00 bits

How it works

This is a cyclic cellular automaton (Fisher / Greenberg–Hastings model): every cell on a toroidal grid holds an integer state s ∈ {0 … K−1} and cycles forward through a fixed rock‑paper‑scissors order, one step ahead "eats" the state one step behind it:

next(s) = (s + 1) mod K
cell advances s → next(s) at generation t+1
  if  #{neighbors in state next(s)}  ≥  T
  else the cell stays at s

Neighbors are counted in a Moore neighborhood of the chosen radius (8 cells at radius 1, 24 at radius 2). There is no leader, no blueprint and no external driver — every cell obeys the same local rule using only what its neighbors are doing right now.

K (states) — length of the cycle a cell chases through. Small K gives blocky, fast-churning domains; large K gives slow, wide spiral arms.
T (threshold) — how many "predator" neighbors are needed to trigger advance. T = 1 is the classic regime that reliably nucleates rotating spiral cores from pure noise.
Radius — how far a cell can sense. Radius 2 smooths and speeds up wave fronts.

From uniformly random noise, local three-cell cycles occasionally close on themselves and become rotating spiral demons: topological defects that continuously re-emit the very wavefronts that keep them spinning. Each arm is destroyed by the state chasing it and instantly re-created by the state it is chasing — the spiral is transcendent in the literal sense of the word: no single cell or generation persists, yet the pattern as a whole sustains itself indefinitely. This is the same class of self-organizing excitable-medium dynamics behind Belousov–Zhabotinsky spiral chemistry and re-entrant electrical waves in cardiac tissue.

State s = 0 → K−1 (cyclic hue)

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