Simple math explains how big crowds keep moving

As you walk through a busy airport, crosswalk, or train station, take a look at the crowd around you. Are people moving smoothly in organized lanes, or chaotically zigzagging through one another? 

A new study from the Massachusetts Institute of Technology (MIT) has revealed a mathematical tipping point that explains when crowds transition from orderly flow to disorganized congestion –  a discovery that could help design safer and more efficient public spaces.

MIT applied mathematics instructor Karol Bacik and an international team of researchers have pinpointed a precise factor that determines crowd behavior. The key lies in something called “angular spread” – a measure of how varied people’s walking directions are.

From organization to chaos

The analysis shows that when pedestrians cross paths from roughly opposite directions – like in a typical crosswalk – they tend to self-organize into lanes, flowing like vehicles in traffic.

But when the range of directions people walk in expands beyond an average angle of 13 degrees from a straight path, that orderly structure collapses into a disordered, inefficient tangle.

“This all is very commonsense,” Bacik said. “The question is whether we can tackle it precisely and mathematically, and where the transition is. Now we have a way to quantify when to expect lanes – this spontaneous, organized, safe flow – versus disordered, less efficient, potentially more dangerous flow.”

From social distancing to crowd dynamics

Bacik, originally trained in fluid dynamics and granular flow, first became interested in human crowd behavior during the COVID-19 pandemic, when he and collaborators began studying how people move while maintaining social distance. That research eventually evolved into a broader investigation of how people navigate crowds.

In 2023, Bacik and his colleagues explored “lane formation,” a phenomenon where pedestrians moving in opposite directions naturally form structured lanes. This happens, the team discovered, when small imbalances in how people choose to pass others – left versus right – cause paths to align and solidify.

Those early findings raised a new question: how much variation in walking direction can that organized system tolerate before it breaks down?

Treating a crowd like a fluid

To answer that, the researchers used mathematical equations typically used to describe the average behavior of fluid particles. By treating a crowd like a fluid, they could model global patterns without tracking each individual.

The team focused on a scenario common in urban life: people walking from one side of a space to the other, trying to avoid bumping into others headed in various directions. They adjusted parameters like the width of the space, the angle at which people walked, and how often individuals dodged one another.

The calculations showed that when people walk nearly straight across – within 13 degrees of a head-on path – lanes naturally form. But when the average path deviates beyond that, disorder takes over. The more chaotic the angles, the more inefficient the overall flow becomes.

Testing math in the real world

To see whether these predictions held up outside the equations, the team set up an experiment in a gymnasium. Volunteers wore paper hats marked with unique barcodes, which allowed an overhead camera to track their movements in detail.

Participants were instructed to walk from assigned starting points to specific destinations on the other side of the space, avoiding collisions.

The experiment was repeated many times, each time changing the configuration of start and end points to simulate different crowd flows. The results were clear: as the angular spread increased, the natural lanes broke down. Just like in the simulations, the tipping point was around 13 degrees.

The more disorganized the flow, the more time and effort it took for people to reach their destinations – suggesting real-world implications for managing foot traffic.

Designing smarter spaces

“We would like to analyze footage and compare that with our theory,” Bacik said. “And we can imagine that, for anyone designing a public space, if they want to have a safe and efficient pedestrian flow, our work could provide a simpler guideline, or some rules of thumb.”

These findings could help architects, urban planners, and event organizers design more navigable environments.

By keeping pedestrian angles within that 13-degree threshold – whether through signage, visual cues, or the layout of paths and barriers – spaces could be optimized to encourage natural, organized lanes of traffic.

What began as an effort to understand socially distanced movement has evolved into a deeper understanding of how people unconsciously self-organize – and when that organization fails.

With a blend of mathematics, experiments, and real-world insights, Bacik and his colleagues are helping to decode the flow of human motion, one crowd at a time.

The study is published in the journal Proceedings of the National Academy of Sciences,

—–

Like what you read? Subscribe to our newsletter for engaging articles, exclusive content, and the latest updates.

Check us out on EarthSnap, a free app brought to you by Eric Ralls and Earth.com.

—–