ExplorationsA crowd with no leader that falls into step

Fireflies That Keep Time

Heuristicsynchronisationcoupled oscillatorscollective behaviourintermediate

Each of 120 fireflies has its own natural rhythm and a random starting phase, the point in its cycle where it flashes. In Kuramoto’s 1975 model every firefly nudges its phase towards the others, harder when the coupling K is bigger. Below a critical coupling the flashes stay scattered and the order measure r stays near the level scattered phases give; above it a cluster locks together and r climbs towards 1. For rhythms spread in a bell curve of width 1, the limit of infinitely many fireflies puts that coupling at about 1.6; a run of 120 only blurs around it.

Current view: The flashing field

Watch the fireflies flash, see where each sits in its cycle, and follow r over time.

The flashing field · 120 fireflies · K 3.0 · spread 1seed 20,261,024

Running the fireflies, 0% done.

a firefly: a dot in the dark field, lit as its phase passes zero
a phase dot: the same firefly as a dot at its own angle on the circle
the mean arrow: the average of all the dots, whose length is r
r over time: the blue line, the length of the mean arrow at each moment
the scatter level: the dashed line at 1 over the square root of the number of fireflies

In the dark field a firefly lights up as its phase passes zero. On the phase circle every firefly is a dot at its own angle (the picture turns with the average rhythm, so a locked cluster stands still), and the arrow from the centre is the mean of all the dots: long when they agree, short when they are spread. Below, r is plotted against time with the scatter level 1/√N dashed, since N scattered phases typically give an r of about that size, not 0.

Running 120 fireflies.

Time: 48.0

How this page is laid out

Make a prediction

Change the assumption

What this assumes: Past the critical coupling 2/(π g(0)), the fireflies lock and r climbs towards 1.
What breaks it: Each firefly hears only its six nearest neighbours. The coupling is still 3, far past the marked 1.6, and r settles far below where the all-hearing field puts it.

Where it holds

Everyone hears everyone, with many fireflies and natural rhythms spread over a single-peaked bell curve: the case the formula is for.

The fireflies

Fireflies

Spread of natural rhythms

How far apart the fireflies’ own rates are, as the width of a bell curve. A wider spread asks for a stronger coupling.

Who each firefly hears

Each firefly is nudged by all the others, the case the critical coupling is a result about.

The coupling

Coupling K: 3.0

Marker: K_c = 1.60, the limit for infinitely many fireflies that all hear one another. A run of this size is a sample and blurs around it.

Step by step

Run, Step, Reset and Draw another field are under the figure. Run replays the 48 time units from the scattered start at this speed.

Speed

Inspect time 48.0 of 48

Moves how far along the run is shown. The whole run is fixed in advance by the seed and the settings.

Reproducible field

Advanced parameters

Random seedPositions, natural rhythms and starting phases all come from this integer through a generator that never touches the browser’s own randomness, and it is always written into the address bar.

Cases worth seeing

Try this

Choose the half-critical coupling and run it with 30 fireflies, then with 300. In which does r wander further from 0?

1/2

What is going on

Each explanation begins with a worked example and follows the same operation through intuition, formal statements and proofs. Later sections distinguish what is established from questions that remain open, so you can follow the level of detail useful to you.

Worked by handOne nudge between two fireflies, A at phase 0 and B a quarter of a cycle ahead at π/2, with natural rates 1.0 and 1.4 and coupling K = 2.

A: phase 0, own rate 1.0pull from B: (K/N) × sin(π/2 − 0) = (2/2) × 11B is a quarter of a cycle ahead, so it pulls A forward as hard as a pull can be. A pull from A on itself would be sin 0 = 0, so it adds nothing.
A: own rate 1.0 and a pull of 11.0 + 12.0A runs faster than it would alone, to catch up with B
B: phase π/2, own rate 1.41.4 + (2/2) × sin(0 − π/2) = 1.4 + 1 × (−1)0.4A is behind, so it pulls B back: B runs slower than it would alone
the gap between them, B minus Arate of B minus rate of A: 0.4 − 2.0−1.6Alone, B would pull ahead of A by 0.4 a time unit (1.4 − 1.0). Coupled, the gap shrinks at 1.6 a time unit, which is the whole of the story in miniature.

The laboratory does exactly this for every firefly at once, many times a time unit, and flashes each one as its phase passes zero.

Left alone, the faster firefly B would pull steadily ahead of A. With the nudge, the gap between them shrinks instead. Nothing told either of them to agree: each only reacted to the other.

What you are seeing

A field of fireflies, each with its own beat, each listening to the rest

Every firefly in the field has a phase, which is just where it is in its flashing cycle, written as an angle round a circle. It flashes when that angle passes zero. Each one also has a natural rate of its own, a little faster or slower than its neighbours, and it starts at a random angle.

Each firefly is also nudged: it speeds up a little if the others are ahead of it and slows a little if they are behind, and the coupling K says how hard. With K at zero every firefly keeps its own time and the field twinkles at random. Raise K and, past a certain point, a cluster of fireflies begins flashing together. The arrow on the phase circle is the average of every firefly's angle, and its length, written r, is the measure of how together they are: near 0 for a scatter, near 1 when they all agree.

Why it starts to make sense

A nudge towards the crowd is a nudge the crowd gives back

Each firefly feels the pull of the average. If the crowd has any lean at all, the pull is a little stronger on the fireflies that sit near it, which holds them closer, which strengthens the lean. That is a feedback, and a feedback has a tipping point.

Set against it is the spread of natural rates. A firefly whose own rhythm is far from the average rate is not held by the pull and drifts past the cluster, like a boat that cannot make headway against the current. Only the fireflies near enough to the average rate can be caught. So there are two forces in a contest: the pull, which grows with K, and the spread, which tears the crowd apart.

When the spread is wide, a stronger coupling is needed before the pull wins. That is the whole content of the critical coupling below.

The key idea

Order is not built firefly by firefly: it switches on for the crowd

Nothing in the rule mentions a cluster. Each firefly obeys the same short rule and has no idea whether it is inside one. Yet there is a coupling below which the answer for the whole crowd is a scatter and, above it, a cluster that holds. The order is a property of the crowd that none of its members can see.

Two honest cautions come with it. The threshold is exact only in a limit, with infinitely many fireflies, and a run of a hundred and twenty blurs around it. And a model that explains how a crowd could lock is not a measurement of how any particular crowd does. The ledger at the foot of the page keeps both apart.

Formal statement

Kuramoto's model and its critical coupling

Write $θ_{i}$ for the phase of firefly $i$ and $ω_{i}$ for its natural rate. In the model, with coupling strength $K$ and $N$ fireflies that all hear one another,

\frac{d θ _{i}}{d t} = ω_{i} + \frac{K}{N} j = 1 \sum N sin (θ_{j} - θ_{i}) .

The sine is a pull towards whoever is ahead or behind, strongest at a quarter of a cycle apart and zero at no gap or at opposite sides. The order parameter is

r e^{i ψ} = \frac{1}{N} j = 1 \sum N e^{i θ_{j}},

and the equation can be rewritten exactly as $d θ_{i} / d t = ω_{i} + K r sin (ψ - θ_{i})$ , which is the form the laboratory integrates.

If the natural rates are drawn from a smooth, symmetric, single-peaked density $g$ , then in the limit of infinitely many fireflies the scattered state is stable for

K < K_{c} = \frac{2}{π g ( 0 )}

and above $K_{c}$ a synchronised cluster grows from it. For a bell curve of standard deviation $σ$ the peak is $g (0) = 1/ (σ 2 π)$ , so $K_{c} = σ 8/ π \approx 1.596 σ$ . Kuramoto found this in 1975, and a proof that the picture holds for the infinite model came much later.

Proof

What can be shown on the page, and what is cited

The peak arithmetic, shown. A bell curve of standard deviation $σ$ has density $g (x) = e^{- x^{2} /2 σ^{2}} / (σ 2 π)$ , so $g (0) = 1/ (σ 2 π)$ . Then $2/ (π g (0)) = 2 σ 2 π / π = σ 8/ π$ , which is the marker on the slider.

The scatter level, shown. Write $r^{2} = \frac{1}{N ^{2}} \sum_{j, k} cos (θ_{j} - θ_{k})$ . The $N$ terms with $j = k$ are each 1. For independent phases spread evenly round the circle every other term averages 0. So the mean of $r^{2}$ is exactly $N / N^{2} = 1/ N$ , and a scattered crowd has an $r$ of about $1/ N$ in typical size. It can come out smaller on a lucky draw, but it does not sit at 0 as a rule. That is why the dashed line on the plot is not at 0.

The threshold, cited. The step from the formula to "the scattered state is stable below $K_{c}$ and a cluster grows above it" is a real piece of analysis, and it is not reproduced here. Kuramoto (1975) obtained it by a self-consistency argument, Strogatz (2000) reviews it and what was and was not rigorous at the time, and Chiba (2015) gives a proof for the infinite model under conditions on $g$ that the paper states and this page does not. None of these texts was read in full for this page, which is why the sources carry a flag.

Limits and further questions

Where this stops, and what the real fireflies are doing

The marker on the slider is for infinitely many fireflies that all hear each other. A run of 120 is a sample: its realised rates have a spread of their own, it is run for a fixed 48 time units, and its r typically settles near the scatter level below the marker rather than at 0, and does not climb exactly at the marker. The sweep view shows the blur. The page does not say how quickly the blur narrows as the number of fireflies grows.

The neighbours setting is a different system, not a filtered version of the first. Each firefly feels only a local pull, and the marker is not an answer for it. The runs on the page show that local patches can lock without the whole field joining them, and they say nothing about whether a threshold exists there or where.

Real fireflies are not Kuramoto oscillators. Synchronous flashing has been observed in the firefly Pteroptyx malaccae in Thailand (Buck and Buck, 1968), and Photinus carolinus in the Great Smoky Mountains flashes in synchronised bursts (Moiseff and Copeland, 2010). Insects of this kind are often described as resetting their rhythm when they see a flash, which is a sharper nudge than the smooth sine here, so the model shows that locking without a leader is possible, not how any insect does it.

Two other systems are often drawn alongside it. Metronomes on a freely moving base drift into step through the base's small motion (Pantaleone, 2002). The London Millennium Bridge opened on 10 June 2000, swayed under its crowd and was closed on 12 June; Strogatz and colleagues (2005) propose that the walkers fell into step with the sway once the crowd passed a critical size. That is a proposed explanation, and the feedback runs through the bridge, which this page does not simulate.

What is actually established

Every statement on this page, with its status, its exact scope, and the date that status was last checked.

Proved

For natural rates drawn from a smooth, symmetric, single-peaked density g, with infinitely many oscillators that all hear one another, the scattered state is stable for coupling below K_c = 2/(π g(0)), and above K_c a synchronised cluster grows from it.

Scope: The limit of infinitely many oscillators, all-to-all coupling and a density meeting the smoothness conditions in the cited papers, which this page does not restate. Kuramoto (1975) obtained the formula from a self-consistency argument; the full bifurcation picture was proved much later (Chiba 2015). Cited, not proved here, and neither text was read in full. It says nothing about a finite run, which blurs around this value, or about fireflies that hear only some of the others.
Why: In the limit each firefly feels the same mean pull K r sin(ψ − θ), so the population can be described by a density of phases for each natural rate. Fireflies with a rate close enough to the average are held by the pull and the rest drift past it. The scattered state becomes unstable when the pull of the held fireflies can feed on itself, which happens exactly at 2/(π g(0)): the more fireflies near the average rate, the lower the threshold.
Status checked: 24 October 2026
Sources: Self-entrainment of a population of coupled non-linear oscillators citation unresolved; From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators citation unresolved; A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model citation unresolved

Proved

For natural rates drawn from a Gaussian of standard deviation σ, the peak is g(0) = 1/(σ√(2π)), so K_c = σ√(8/π), about 1.596 σ: 1.60 for σ = 1 and 3.19 for σ = 2.

Scope: The arithmetic of putting the Gaussian peak into the formula of the previous claim, so it inherits that claim’s limit and conditions. The laboratory’s marker is this number for the chosen spread, and a unit test checks it against the formula 2/(π g(0)) with g(0) computed directly.
Why: Substituting g(0) = 1/(σ√(2π)) into 2/(π g(0)) gives 2σ√(2π)/π, which simplifies to σ√(8/π).
Status checked: 24 October 2026

Proved

If N phases are independent and uniformly spread round the circle, the mean of r² is exactly 1/N, so a scattered crowd typically has an r of about 1/√N, not 0.

Scope: Independent uniform phases, which is how every run here starts, for any N. It is a statement about the mean of r², not a bound on any one draw. Later in a run the phases are no longer independent.
Why: r² is the sum over all pairs of the cosine of the gap between them, divided by N². The N terms of a firefly with itself are each 1. For independent uniform phases every other pair averages 0, which leaves N/N² = 1/N.
Status checked: 24 October 2026

Observed in this run

The run this page opens on, 120 fireflies with natural rates of spread 1, everyone hearing everyone and K = 3, seed 20261024, starts with r = 0.07 and settles at r = 0.90, the mean over its last 12 time units. At K = 0.8 the same fireflies settle at 0.17.

Scope: That one seed, that run length of 48 time units, and those couplings. Another seed gives different numbers of the same kind. This is a record of what was computed, not evidence for the theorem above.
Status checked: 24 October 2026

Exhaustively checked

Over every seed from 1 to 100, with 120 fireflies of natural-rate spread 1 and everyone hearing everyone, r settles between 0.05 and 0.20 at K = 0.8 and between 0.87 and 0.96 at K = 3.

Scope: The 100 seeds from 1 to 100, those two couplings, 120 fireflies and a run of 48 time units, with r read as its mean over the last 12. It says nothing about other seeds, other couplings or other numbers of fireflies, and a fixed-step floating-point integration is not exact arithmetic.
Computation (approximate): Seeds 1 to 100, K = 0.8 and K = 3, N = 120, spread 1, all-to-all coupling, fourth-order Runge-Kutta with step 0.05 to time 48, double-precision arithmetic. Algorithm kuramoto-rk4-0.05.
Status checked: 24 October 2026

Observed in this run

For the opening seed with 120 fireflies and spread 1, where r settles climbs through 0.32 at K = 1.25, 0.45 at K = 1.5, 0.60 at K = 1.75 and 0.68 at K = 2: a steep rise that begins before the marked 1.6, not a step at it.

Scope: One seed, 120 fireflies, run lengths of 48 time units. The many-fireflies limit has an onset at exactly the marked value; a finite run blurs it, and the page makes no claim about how fast the blur narrows as N grows.
Status checked: 24 October 2026

Observed in this run

With each firefly hearing only its six nearest, 120 fireflies, spread 1 and K = 3, the opening seed settles at r = 0.39, and over seeds 1 to 100 the settled r ranges from 0.17 to 0.77, against 0.87 to 0.96 when everyone hears everyone.

Scope: These 101 runs of 48 time units, with the six nearest neighbours of each firefly fixed by its position in a random scatter. The page does not state a critical coupling for this system, does not claim local coupling can never lock the whole field, and does not say what happens for longer runs.
Status checked: 24 October 2026

Observed in this run

Fireflies that flash together have been observed: Pteroptyx malaccae in Thailand (Buck and Buck, 1968), and Photinus carolinus in the Great Smoky Mountains, which flashes in synchronised bursts (Moiseff and Copeland, 2010).

Scope: Reported by the cited field studies; the page did not make these observations and has not read the papers in full. It claims the synchrony, not that these fireflies obey the Kuramoto equations.
Status checked: 24 October 2026
Sources: Mechanism of rhythmic synchronous flashing of fireflies citation unresolved; Firefly synchrony: a behavioral strategy to minimize visual clutter citation unresolved

Heuristic

The Kuramoto model shows how a crowd of different rhythms can lock without a leader; it does not establish that real fireflies, metronomes or walkers follow its equations.

Scope: Real fireflies are often described as resetting their rhythm when they see a flash, which is a sharper kind of nudge than the smooth sine used here. The model is a clean caricature of the idea of coupled rhythms, not a measurement of any insect.
Status checked: 24 October 2026
Sources: Mechanism of rhythmic synchronous flashing of fireflies citation unresolved; From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators citation unresolved

Observed in this run

Metronomes standing on a freely moving base drift into step, because the base’s small motion couples them.

Scope: Reported by Pantaleone (2002) for his bench experiments, generally in step and sometimes in opposite step under special conditions; not reproduced by this page, and not read here beyond its record. The paper describes the setup as a mechanical realisation of the Kuramoto model.
Status checked: 24 October 2026
Sources: Synchronization of metronomes citation unresolved

Observed in this run

The London Millennium Bridge opened on 10 June 2000, swayed sideways under its crowd, and was closed on 12 June 2000 pending investigation.

Scope: The engineers’ account as recorded in citing literature; the paper was not read in full. Crowd sizes are not stated here because no source for them was confirmed.
Status checked: 24 October 2026
Sources: The London Millennium Footbridge citation unresolved

Heuristic

Strogatz and colleagues (2005) propose that the sway fed back on the walkers, who fell into step once the crowd passed a critical size, a synchronisation of the kind the Kuramoto model describes.

Scope: A proposed explanation and a model, published in Nature, not a measurement of the opening day and not a claim that the walkers obey the fixed-coupling Kuramoto equations. The feedback runs through the bridge, which this page does not simulate.
Status checked: 24 October 2026
Sources: Crowd synchrony on the Millennium Bridge citation unresolved

Sources

Review notes show which bibliographic details Mathomaly has checked and which remain unresolved. Checking a publication record does not independently verify its proof.

Yoshiki Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics 39 (Springer), 1975. Link
The model and the critical coupling 2/(π g(0)), derived by a self-consistency argument in the limit of infinitely many oscillators. Cited for the formula, not read here.
Citation details remain unresolved after review by Mathomaly.
Bibliographic review: 2026-10-24
AI-assisted bibliographic audit. Evidence type: unresolved.
The Springer DOI and the volume (Lecture Notes in Physics 39, pages 420 to 422) were seen in search results; the text was not read, so the formula is attributed to it on trust.
- Review evidence 1 for Self-entrainment of a population of coupled non-linear oscillators
Steven H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Physica D 143, 2000.
A review of the model and of what was known about its onset of synchronisation 25 years on, including how K_c is obtained and what was and was not rigorous at the time. Pages 1 to 20.
Citation details remain unresolved after review by Mathomaly.
Bibliographic review: 2026-10-24
AI-assisted bibliographic audit. Evidence type: unresolved.
A copy of the paper was located by search; its text was not read for the statements this page attributes to it. The DOI resolves to the publisher record for the paper.
- Review evidence 1 for From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators
Hayato Chiba, A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model, Ergodic Theory and Dynamical Systems 35 (3), 2015. Link
Proves, for the infinite-oscillator model, that below the critical coupling the incoherent state is stable and above it a synchronised state bifurcates from it. Pages 762 to 834. The conditions on the frequency density are in the paper and are not restated here.
Citation details remain unresolved after review by Mathomaly.
Bibliographic review: 2026-10-24
AI-assisted bibliographic audit. Evidence type: unresolved.
The publisher listing gives journal, volume, issue, pages and DOI; the proof was not read, and the abstract was seen only in summary.
- Review evidence 1 for A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model
John Buck, Elisabeth Buck, Mechanism of rhythmic synchronous flashing of fireflies, Science 159 (3821), 1968.
Field observations of synchronous flashing in Pteroptyx malaccae in Thailand, with the suggestion that the fireflies adjust the timing of their own flashes in response to others. Pages 1319 to 1327. The page does not claim the fireflies obey the Kuramoto equations.
Citation details remain unresolved after review by Mathomaly.
Bibliographic review: 2026-10-24
AI-assisted bibliographic audit. Evidence type: unresolved.
Volume, issue and pages were seen in citing literature, and the PubMed number in a library record; the paper itself was not read.
- Review evidence 1 for Mechanism of rhythmic synchronous flashing of fireflies
Andrew Moiseff, Jonathan Copeland, Firefly synchrony: a behavioral strategy to minimize visual clutter, Science 329, 2010.
Photinus carolinus, studied in the Great Smoky Mountains, flashes in synchronised bursts; the paper tested females with programmed lights and reports that they answer synchronised flashes far more often than unsynchronised ones. Page numbers not confirmed.
Citation details remain unresolved after review by Mathomaly.
Bibliographic review: 2026-10-24
AI-assisted bibliographic audit. Evidence type: unresolved.
Title, authors, year and the findings were seen in press coverage of the study; the journal page could not be retrieved and the paper was not read.
- Review evidence 1 for Firefly synchrony: a behavioral strategy to minimize visual clutter
James Pantaleone, Synchronization of metronomes, American Journal of Physics 70 (10), 2002. Link
Metronomes on a freely moving base couple through the base’s small motion and, generally, settle in step. Pages 992 to 1000. The paper itself describes the setup as a mechanical realisation of the Kuramoto model.
Citation details remain unresolved after review by Mathomaly.
Bibliographic review: 2026-10-24
AI-assisted bibliographic audit. Evidence type: unresolved.
Journal, volume, pages and DOI were seen in search results and a copy was located; the text was not read for the statements attributed to it.
- Review evidence 1 for Synchronization of metronomes
Steven H. Strogatz, Daniel M. Abrams, Allan McRobie, Bruno Eckhardt, Edward Ott, Crowd synchrony on the Millennium Bridge, Nature 438, 2005. Link
Proposes that the wobble on the bridge’s opening was a feedback between the deck’s sway and the walkers’ footsteps, which fell into step once the crowd passed a critical size, using the mathematics of synchronising oscillators. Pages 43 to 44. A proposed explanation and a model, not a measurement of the opening day.
Citation details remain unresolved after review by Mathomaly.
Bibliographic review: 2026-10-24
AI-assisted bibliographic audit. Evidence type: unresolved.
Volume, pages, date (3 November 2005) and DOI were seen in a bibliographic record; the article text could not be read from the copy located.
- Review evidence 1 for Crowd synchrony on the Millennium Bridge
Peter Dallard, A. J. Fitzpatrick, A. Flint and co-authors, The London Millennium Footbridge, The Structural Engineer 79 (22), 2001.
The engineers’ account: the bridge opened on 10 June 2000, showed lateral vibration under a crowd, and was closed on 12 June 2000 pending investigation. Cited for those dates only. Pages 17 to 33 as given in the reference list of Strogatz and co-authors (2005); the full author list is not confirmed here.
Citation details remain unresolved after review by Mathomaly.
Bibliographic review: 2026-10-24
AI-assisted bibliographic audit. Evidence type: unresolved.
Venue, volume and the opening and closing dates were seen in a citing letter and a listing; the paper was not read and the full author list was not checked.
- Review evidence 1 for The London Millennium Footbridge

Connected by how they work, not by sharing a topic label.

Interactive laboratory

Make a prediction

Change the assumption

The fireflies

The coupling

Step by step

Reproducible field

Cases worth seeing

Try this

What is going on

What you are seeing

Why it starts to make sense

The key idea

Formal statement

Proof

Limits and further questions

What is actually established

Sources

Related by mechanism