Cyclic Cellular Automata: Spiral Waves from Noise

One rule, guaranteed order

Fill a grid with uniform random integers 0…K−1 and apply a single local rule: a cell advances to its next state if any neighbour already carries that successor state. Run it for a few hundred synchronous steps and something remarkable happens — the noise organises itself into rotating spiral waves that persist indefinitely. This is the cyclic cellular automaton (CCA), the simplest computational model of an excitable medium: a system where a local disturbance triggers a self-propagating wave that briefly resets before it can fire again.

The Greenberg-Hastings model

Formalised by Greenberg and Hastings in 1978, each cell carries a state s ∈ {0, 1, …, K−1}: state 0 is resting, and states 1…K−1 are successive excited/refractory stages that cycle. The synchronous update rule is:

s'(c) = (s(c)+1) mod K   if at least θ neighbours have state (s(c)+1) mod K
s'(c) = s(c)             otherwise
// neighbourhood: Moore (8 cells) or von Neumann (4 cells); threshold θ

With a von Neumann neighbourhood and K ≥ 3, Fisch, Gravner and Griffeath proved in 1991 that CCA started from random initial conditions form persistent spiral waves with probability 1 — the spirals are the dynamics' unique attractor, not a lucky outcome.

How a spiral is born

Spiral formation follows a precise three-step sequence. First, nucleation: a local patch of random noise happens to form a monotonically increasing sequence in one direction, and this "seed wave" expands outward as a target ring. Second, breakup: when a planar wave front meets a region already in a different phase, the front breaks and its free ends curl inward — the curling tips are topological defects where all K states meet at a point, and they become the spiral's rotating core. Third, annihilation: spirals of opposite chirality destroy each other on collision, so the survivors are the ones that opened enough territory to outlast their competitors. The rotation period is roughly T ≈ K steps and the wavelength is roughly λ ≈ K cells, so more states means slower, larger spirals.

Three numbers control everything

The qualitative behaviour of CCA is governed by just three parameters. Higher K (states) means a longer refractory period, larger spirals and slower rotation. Higher θ (threshold) makes cells harder to excite, giving fewer but larger spirals — θ = 1 is the most excitable, and above roughly K/4 the automaton stops forming spirals at all and freezes into a static state. A Moore neighbourhood (8 cells) propagates waves faster and produces more complex tips than a von Neumann one (4 cells). Gravner and Griffeath's 1993 analysis splits the (K, θ) parameter space into a dying phase, a chaotic transient phase, and a persistent phase where spirals form inevitably and never die.

The same mathematics as heart tissue

Spiral waves are not a curiosity of toy cellular automata — the same excitable-medium dynamics appears in the Belousov-Zhabotinsky chemical reaction, in slime-mould colonies signalling with cAMP pulses, and, more seriously, in cardiac tissue: a reentrant spiral wave of electrical activation in heart muscle is the mechanism behind ventricular fibrillation. Around each spiral tip the discrete state field winds by exactly ±2π, giving it a conserved topological charge of ±1 — mathematically identical to a vortex in a superfluid or a dislocation in a crystal lattice. That topological protection is why a lone spiral, once formed, essentially never disappears on its own: it takes an opposite-chirality partner to annihilate it.

Frequently asked questions

Try it live

Set K, θ and the neighbourhood type in Cyclic Cellular Automata and press randomise — watch order emerge from pure noise within a few hundred generations, entirely in your browser.