Math for the masses: Why pedestrians unconsciously form lines

Equations explain how people spontaneously follow lanes to avoid bumping into each other, according to a recent study published in ‘Science’

Tokyo Shibuya
Pedestrians cross the Shibuya intersection in Tokyo, one of the busiest in the world.Nikada (Getty Images)

Anyone who has watched crowds of people forming restroom lines at a concert or hurrying to catch buses and trains has probably noticed that they bump into each other less than you might think. The crowd usually moves in an orderly fashion in both directions. A recent article in Science explains the mathematics behind the order amid the chaos.

The long-recognized phenomenon is known as “spontaneous lane formation,” which is what happens when people in a crowd unconsciously create multiple lanes (approximately two bodies wide) to walk in both directions. Walkers will join a lane moving in the right direction and follow others going the same way. When someone walking against the flow moves in, the lane itself will push them out and into the one going in the other direction. All this happens in an orderly fashion, with no one organizing it.

Mathematician Karol Bacik from the University of Bath (U.K.) coauthored the paper in Science and explains the study’s findings: “It’s a new way of describing that movement using differential equations. The new mathematical approach enabled us to explain the phenomenon of lane formation, but also generated new hypotheses that we tested through experiments.”

Bacik and co-author Tim Rogers, a colleague from the University of Bath, formulated several differential equations and conducted an experiment to test their effectiveness. “We gathered a group of volunteers in a sports facility and asked them to perform a series of simple tasks to simulate real-life pedestrian traffic. To track their movements, we gave them red and blue hats with bar codes on top. We did not make any suggestions about forming lanes, but they did this anyway spontaneously,” said Rogers.

The researchers recorded the group’s movements from above and made some surprising discoveries, says Rogers: “The lanes are not always straight. They can curve if some people are heading toward a door, or the lanes can slant in one direction when people are told, for example, that they have to pass on the right. But pedestrians don’t realize they are forming these patterns. They are only aware they are dodging people going in the opposite direction.”

The study found that curved lanes are parabola-shaped when one side is headed toward a narrow exit, but are elliptical if both exits are narrow. This finding that the lanes are not always straight was previously unknown.

While it’s interesting to learn that spontaneous lane formation can be described by elegant mathematical equations, there is a practical application for this knowledge. Knowing how crowd flow develops in confined spaces is crucial to preventing injury and death in emergency situations. Architects and urban planners are well aware of this need. In 2022, over 150 people died trapped in an alley in Seoul (South Korea) during a Halloween celebration, and other human stampedes around the world have also killed and injured many people.

Muslims gather around the Kaaba at the grand mosque in Mecca (Saudi Arabia) during Ramadan.
Muslims gather around the Kaaba at the grand mosque in Mecca (Saudi Arabia) during Ramadan. FAISAL Al NASSER (REUTERS)

The study by Barik and Rogers focused only on bidirectional travel, but other researchers found problems can arise when more than two flows exist (when pedestrians are headed in more than two directions). When this happens, collisions occur more frequently, lanes fade away and stable flow patterns disappear. Because of this, experts have been researching how to promote the persistence of spontaneous lane formation — how to make people stay in their lanes.

One heavily studied case was the stampede in Mina, a city near Mecca (Saudi Arabia), during the Hajj, the annual Muslim pilgrimage to Mecca. On September 24, 2015, a stampede happened when two groups of pilgrims traveling in opposite directions rain into each other on a narrow street. The ensuing crush killed 2,426 people, a disaster that has been studied extensively to find ways of propagating safe and spontaneous lane formation in crowds.

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