Why do spirals appear instead of just expanding rings?

Spirals are seeded wherever a wavefront is broken — an open tip cannot annihilate against a partner, so it curls around itself indefinitely. Random initial conditions give the wavefronts plenty of free ends, so spirals usually win over the slower target patterns.

What does the refractory period correspond to chemically?

In the real BZ reaction, an oxidised patch of catalyst (e.g., ferroin → ferriin) cannot be re-excited until reductant is replenished. The integer countdown in the automaton plays the same role: a cell that has just fired is locked out for several frames before it can respond again.

Is this a true reaction–diffusion model?

No. The real chemistry is captured by Oregonator-style PDEs with diffusion. The cellular automaton is a coarse cartoon: it keeps the three essential ingredients — local excitation threshold, refractoriness, and propagation — and is fast enough to render full-screen patterns in real time.

Belousov–Zhabotinsky (Excitable)

Greenberg–Hastings excitable cellular automaton: target waves and rotating spirals as a qualitative BZ analogue.

About this model

A discrete Greenberg–Hastings cellular automaton on a toroidal grid stands in for the Belousov–Zhabotinsky reaction. Each cell is in one of three states — quiescent (0), excited (1), or refractory (2…N−1). A quiescent cell becomes excited when enough of its eight Moore neighbours are excited; an excited cell then runs through a fixed refractory countdown before becoming sensitive again. The model is qualitative, not chemically calibrated, but it reproduces the hallmark behaviour of an excitable medium: target waves expanding from local pacemakers and counter-rotating spiral waves around free wave-tips.

Who it's for: Intro nonlinear chemistry, complex systems, and computational physics; an accessible bridge between reaction–diffusion PDEs and cellular automata.

Key terms

Belousov–Zhabotinsky
excitable medium
Greenberg–Hastings
spiral waves
target patterns
refractory period
cellular automaton

How it works

Belousov–Zhabotinsky-style target waves and rotating spirals from a tiny excitable cellular automaton — a qualitative, lattice version of the bromate–malonic acid clock.

Key equations

state(0) → 1 if Σ[neigh = 1] ≥ k

state(s) → (s+1) mod n for s ≥ 1

Frequently asked questions

Why do spirals appear instead of just expanding rings?: Spirals are seeded wherever a wavefront is broken — an open tip cannot annihilate against a partner, so it curls around itself indefinitely. Random initial conditions give the wavefronts plenty of free ends, so spirals usually win over the slower target patterns.
What does the refractory period correspond to chemically?: In the real BZ reaction, an oxidised patch of catalyst (e.g., ferroin → ferriin) cannot be re-excited until reductant is replenished. The integer countdown in the automaton plays the same role: a cell that has just fired is locked out for several frames before it can respond again.
Is this a true reaction–diffusion model?: No. The real chemistry is captured by Oregonator-style PDEs with diffusion. The cellular automaton is a coarse cartoon: it keeps the three essential ingredients — local excitation threshold, refractoriness, and propagation — and is fast enough to render full-screen patterns in real time.

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Belousov–Zhabotinsky (Excitable)

About this model

How it works

Key equations

Frequently asked questions

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About this model

How it works

Key equations

Frequently asked questions