The Mathematics of a Flock

An elite and olympic-level synchronised swimmer typically trains up to 8 hours a day, 6 days a week.

Professional cheerleaders train approximately 2 to 3 hours a day, up to 5 times a week. 

Perfecting a single dance routine would take even the most experienced artists anywhere from 10 to 50 hours of focused rehearsal, and depending on the complexity of the performance, up to 100 hours or more.

The message here is clear: coordinated movement does not come easily to us. It takes time, repetition, correction and, depending on the context, an aspect of leadership. Even the most skilled human beings rehearse relentlessly before a group can move with anything resembling perfect synchrony.

This is the nature of human beings: social creatures who crave cohesion and unity, with inherently individualist tendencies.

Then there are starlings.

Thousands of them rise into the sky at once, turn almost simultaneously, compress into a dark ribbon, expand, split apart, and reform within seconds. There is no choreographer, no rehearsal, and no captain positioned at the front calling the next movement.

Yet somehow, the flock moves as though performing a well-rehearsed routine. How is this even possible?

Who Leads The Flock?

When hundreds, sometimes thousands, of birds turn almost simultaneously, the obvious assumption is that one must be leading. After all, this is how human coordination typically works. An orchestra has its conductor, football teams have their captain and coach, and dance troupes have their choreographer. Large groups rarely achieve precise, collective movement without somebody knowing and dictating what comes next.

Perhaps it’s no surprise, considering the content of this issue: murmuration appears to work differently. There is no permanent starling at the front deciding where the flock should go. Instead, decades of research into starling flight increasingly point towards a system of distributed coordination, where the movement of the whole emerges from interactions between individual birds.

One of the most influential starling murmuration studies was published in 2008 by researchers from the STARFLAG project. Using stereoscopic photography and computer-vision techniques, researchers successfully reconstructed the three-dimensional positions of individual starlings in 10 separate flocking events, some of which containing as many as 2,600 birds, above a roost in Rome.

What they discovered challenged a fairly intuitive assumption. At least on the surface, each individual starling does not appear to respond to every bird within a particular number of metres. Instead, its behaviour is best explained by its relationship with a relatively fixed number of surrounding birds: approximately six or seven neighbours.

Thousands of birds do not need to watch thousands of birds. Each one needs only a small window into what is happening around it.

This is what researchers have coined as a topological interaction. I’ll explain it this way: a starling does not need to keep track of every bird within a fixed physical radius. Instead, it appears to respond to a small, relatively constant number of neighbouring birds around it.

Imagine trying to monitor the movement of 100 people around you at once. If you’re anything like me, with a short attention span and trouble multitasking, even the thought of it can get pretty overwhelming.

Starlings avoid this problem entirely by focusing their attention and staying responsive to a handful of neighbours. This way, whether the flock compresses or spreads out, information is transferred swiftly and the movement remains fluid as each neighbour responds to their own.

However, where one mystery is solved, another emerges. Like a game of broken telephone, the original message is often barely recognisable once it's passed from person to person. So, if one bird turns, and another responds, and another responds to that one, shouldn't the signal eventually distort as it travels through the flock?

A 2014 study led by Alessandro Attanasi and Andrea Cavagna examined this exact question. Using three cameras, the researchers reconstructed the three-dimensional trajectories of individual starlings during collective turns and calculated the order in which the birds changed direction.

Their findings were remarkable. Information about a change in direction propagated across the flock with negligible attenuation. The signal could travel through the group rapidly without progressively fading away, helping the flock change direction while remaining cohesive. And most importantly, Attanasi and his team identified that despite all other observations that point to an absence of a leader, somebody still has to turn first.

A follow-up study published in 2015 examined spontaneous turns, including those that occurred without an obvious predator or external obstacle. By reconstructing the trajectories of every bird during a turning event, Attanasi and his team discovered that these changes tend to begin with individuals positioned at the elongated tips of the flock. Their deviation from the flock's existing direction then provoked responses from neighbouring birds, and the turn would spread.

One thing this study confirms is that there are initiators, but no specific leaders. The bird that happens to move first does not actually have a plan for the thousands behind it. It does not decide the shape the murmuration will eventually take, and might not even be aware of the larger formation it is helping to create.

One bird changes direction → Nearby birds react → Their neighbours follow → A ripple travels through the flock

From this chain of local interactions emerges something that, when viewed from the ground, looks uncannily like choreography.

The Mathematics Beneath the Movement

Before anything further, it’s important to clarify that the average cruising speed of starling flocks is approximately 10 to 12 metres per second (est. 36 to 43 km/h). That’s pretty fast, and you’re now more likely to understand when I say: moving together is one thing, but doing so without crashing into one another is a separate accomplishment.

Murmurations can stretch, compress and fold into itself at speed, sometimes packing enormous numbers of birds into what appears, from the ground, to be an impossibly tight space. Yet the formation remains remarkably coherent. How?

In 1987, computer scientist Craig Reynolds attempted to recreate flocking behaviour digitally. Instead of programming the path of every simulated bird individually, he created autonomous agents he called “boids” and gave them three basic behavioural tendencies.

1. Separation
   → Avoid getting too close to those around you.
2. Alignment
   → Adjust your speed and direction to resemble those around you.
3. Cohesion
   → Move towards the group rather than drifting away from it.

Nothing in the programme told the boids to form a sweeping arc, a wave or a swirling mass. In fact, there was no equation describing what the final flock should look like. Reynolds simply gave each simulated individual a few constraints governing how it should move. Once enough of them began moving together, recognisable flocking behaviour emerged on its own.

This is known as emergence: when relatively simple components interact in ways that produce something far more complex at a larger scale.

I’d urge you not to mistake Reynolds' simulation for a literal instruction manual inside a starling's brain. Real birds are biological creatures responding to vision, aerodynamics, environmental conditions and one another in ways far more complicated than three lines of computer logic. But that doesn’t mean we toss his experiment out entirely. Reynolds’ model demonstrates something more fundamental: complexity does not always require complex instructions. A few constraints, repeated again and again across a large enough system, can produce forms that were never explicitly designed into any individual part of it.

Wait, there’s more.

Studies of real starlings have uncovered something even stranger. In 2010, researchers measured the three-dimensional positions and velocities of birds in flocks ranging from 122 to 4,268 individuals. They found that fluctuations in the birds' movement displayed what physicists call scale-free correlations. The distance across which their movements were correlated did not remain fixed. Instead, it increased with the size of the flock itself.

A larger murmuration did not simply become a collection of smaller, independently behaving regions. The mathematics continued to connect the flock across its scale. Researchers went on to argue that this resembles what physicists call a critical system: a system poised in a state where it can remain ordered while still being extraordinarily sensitive to change.

No Bird Sees the Whole

Sometimes beautiful things are best appreciated from a distance. Observe woven tapestry too closely, and the design is lost to a haphazard array of individual threads crossing over one another, disappearing beneath others and reappearing somewhere else. Lean back a little, and those same seemingly disconnected threads resolve into a complete visual masterpiece.

Similarly, consider the coral reefs that stretch across an enormous expanse of ocean, creating a landscape of ridges, branches, crevices, and entire habitats. Yet much of its physical foundation is built by tiny coral polyps, many measuring only one to three millimetres across.

Thousands of birds in harmonious flight can adorn the sky with the same kind of beauty, even if they cannot see it for themselves. With our feet firmly planted on the ground, we watch sweeping ribbons, dark waves and fluid forms take shape overhead.

All the while, the artists themselves, each individual starling, have no access to the view we are admiring.

They are inside the masterpiece.

In a 2014 study examining the visual information available to birds within large flocks, researchers pointed out that it would be unrealistic for an individual bird to recognise and track the position and orientation of any significant proportion of the flock. From within a dense murmuration, many of the other birds would simply appear as moving silhouettes against the sky.

Just as we cannot see ourselves as others do, a starling cannot step outside the flock to witness the shape it is helping to create. It knows only the view from within: the birds nearest to it, the movement unfolding around it, and the small adjustments required from one moment to the next.

The whole exists beyond its perspective.

There is no bird's-eye view of the bird's-eye view

And yet researchers measuring real starling flocks have found an astonishing degree of collective order. In one analysis of 24 flocks containing between 122 and 4,268 birds, the average measure of directional alignment was 0.96 on a scale where 1 represents near-perfect alignment. Thousands of individually moving animals were, at the level of the group, behaving as an extraordinarily ordered whole.

If you can, sit with that for a moment. A pattern exists, and individuals help create it without ever needing to comprehend the whole. A murmuration, then, is more than a mysterious display of collective behaviour. Much of its intrigue lies in a simple matter of perspective: what appears fragmented from within can look remarkably ordered from afar.

The Order We Cannot See

The most beautiful sight we have ever seen, the most memorable birthday, the funniest joke, and countless other moments are little more than blips in the span of our existence. Every day, we move through a collection of moments that will never return. Some are routine, while others resemble those that came before. But none are ever exactly the same.

Perhaps that is one reason a starling murmuration can feel so captivating, even to those of us who have never witnessed one in person. Its mechanics reflect a logic that is not entirely foreign to us. We, too, live from within.

Rarely with access to the full picture, decisions are made, plans fall apart or come together, dreams are delayed, new friendships emerge while others fade away. One event always follows another, and from where we stand, the connections between them are not always obvious.

We’re quick to see the thread, but not the tapestry. And without any means to verify, we are even quicker to decide what the threads represent and what the whole’s worth.

We call one moment fortunate and another disastrous. We label an interruption pointless, a delay inconvenient, an outcome unfair. Sometimes those judgements may prove accurate. But other times, we simply do not yet possess enough of the picture to know.

A starling inside a murmuration faces a similar limitation of perspective. It cannot see the enormous pattern unfolding around it. It knows only the few movements immediately relevant to its own flight. Were it capable of judging the flock solely from that view, perhaps the experience would appear fragmented: birds moving here, openings appearing there, one sudden turn followed by another.

From the ground, our perspective is much wider. The movements connect and shape emerges.

What seemed like thousands of isolated adjustments to the birds, becomes one continuous form for us to witness.

It is through this spectacular display of nature, we are reminded that not being able to see an order is not the same as there being no order at all.

The thread of our lives looks tangled until we step back. The polyp looks insignificant until we see the reef. The starling sees only movement until we look up and see the murmuration. And human beings, for all that we know, remain creatures standing somewhere inside the picture.

Have They Not Seen the Birds Above Them?

Al-Mulk (67:19)

أَوَلَمْ يَرَوْا۟ إِلَى ٱلطَّيْرِ فَوْقَهُمْ صَـٰٓفَّـٰتٍۢ وَيَقْبِضْنَ ۚ مَا يُمْسِكُهُنَّ إِلَّا ٱلرَّحْمَـٰنُ ۚ إِنَّهُۥ بِكُلِّ شَىْءٍۭ بَصِيرٌ ١٩

Have they not seen the birds above them, spreading and folding their wings? None holds them up except the Most Compassionate. Indeed, He is All-Seeing of everything.

— Dr. Mustafa Khattab, The Clear Quran