Belousov–Zhabotinsky Spirals
Cyclic rule with 14 states, threshold 1, range 1
Fourteen states chasing each other in a loop. Random noise organises itself into rotating spirals that compete for territory.
How it works
In plain English, before the notation
Think of rock-paper-scissors with fourteen hands arranged in a ring, where each one beats exactly the hand before it. A cell in state 5 becomes state 6 as soon as any neighbour is already in state 6. Start from noise and, within a few hundred steps, the noise is gone: the grid organises itself into rotating spirals, each pumping out rings of colour and competing with its neighbours for territory.
Watch order appear from noise
- Open Presets and load "Spiral Waves". Every cell starts at a random point in the cycle.
- Press Play. The first hundred steps look like static; after that spirals start to form.
- Once it has settled, pause, pick a colour from the state swatches in the dock, and paint a small dot inside a flat region. A new spiral usually forms around it.
Starting configurations
Loads straight into the simulatorTry any rule
The catalogue covers a few dozen rules. Here you can run any of the 262,144 two-state grid rules, or any of the 256 one-dimensional rules, including ones nobody has written up.
Well-known rules
Where it came from
Cyclic cellular automata were introduced by David Griffeath in the 1980s and studied in detail with Robert Fisch and Janko Gravner. Their work traced the progression a random start goes through — from noise, to droplets, to defects that seed spirals, to a final state dominated by whichever spirals rotate fastest.
The name refers to the Belousov–Zhabotinsky reaction, an oscillating chemical reaction first reported in the Soviet Union in the 1950s and initially disbelieved because it appeared to contradict expectations about how reactions settle to equilibrium. It produces spiral waves that look strikingly like this rule’s output.
The rule, precisely
What each cell looks at
The 4 cells sharing an edge — north, south, east and west (the von Neumann neighbourhood). With all 8 surrounding cells the same rule locks into a fine maze and never forms spirals.
What a cell can be
States 0 to 13, arranged in a cycle
The update
next = (k + 1) mod 14 if at least 1 neighbour is in state (k + 1) mod 14; otherwise k
A cyclic rule is defined by three numbers: how many states there are (14), how many neighbours in the next state are required to trigger a change (the threshold, 1), and how far the neighbourhood extends (the range, 1). Which cells count as neighbours matters too: this rule uses the four edge neighbours, as in Griffeath’s experiments.
Spirals, and why they win
The progression from random noise to organised spirals is reliable and happens in recognisable stages:
- Noise first: local patches synchronise into small regions of a single colour.
- Then defects: points where several colours meet in the wrong order, which cannot resolve locally.
- Then spirals: a defect that persists starts rotating and emits rings of colour outwards.
- Then competition: a spiral emitting rings faster overruns a slower one, so the final state is set by whichever spiral cores have the tightest rotation.
Self-organisation, not computation
This family is studied for how reliably it produces order from disorder, not for what can be built inside it.
Spiral waves in real media
Rotating spiral waves show up in several unrelated systems: the BZ reaction in a shallow dish, cAMP signalling in aggregating Dictyostelium slime mould, and — with clinical consequences — re-entrant electrical waves in heart tissue. What these share with the rule is the combination of local excitation and a recovery period, which is enough to make spirals the natural attractor.
Things to try
- The state swatches in the dock let you paint a specific colour. Painting the colour that comes next in the cycle into a flat region reliably starts a new spiral.
- Try the other cyclic rules for comparison: raising the threshold (Cyclic Turbulence) prevents clean spirals, and widening the range (Crystal Domains) slows everything down.
- Set Cell Scale to 2px. Spirals need space, and at 4px or higher only two or three will fit.
Frequently Asked Questions
References
The formal definition and the parameters that distinguish one cyclic rule from another.
The chemistry the rule is named after, and why the reaction was initially rejected.