Anyone who has ever observed a crowd in the entrance area to the toilets of a concert or at rush hour in a subway or train station has noticed that people moving in two directions within this crowd hardly collide, less than you might thinks. The crowd moves surprisingly orderly and in both directions. A recent article published in the journal Science explains the mathematics that exist in this order in chaos.
The phenomenon, which has been known for a long time, is called “spontaneous track formation”. It was already known that when people move in two directions in a crowd, they unconsciously create multiple paths in each direction. These lanes are approximately two bodies wide. People who move join one of the tracks going in the desired direction and move with it. When someone traveling in a different direction changes lanes, it is the lane itself that ejects them into the correct lane. And the movement goes on in an orderly fashion, without anyone organizing it.
Mathematician Karol Bacik, a researcher at the British University of Bath and one of the authors of the study, explains what they found: “It’s a new way of describing this motion using differential equations. With the new mathematical approach, we were not only able to explain the phenomenon of track formation, but also to generate new hypotheses that we tested experimentally.”
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Bacik and his partner Tim Rogers, also researchers at the University of Bath, decided to run an experiment to see if the differential equations they had formulated worked. “We gathered a group of volunteers at a sports center and asked them to complete a series of simple tasks to simulate real-life pedestrian traffic. To track their movements, we gave them hats of two different colors (red and blue) with barcodes printed on the top. We didn’t imply in any way that they would form lanes, and yet they formed them spontaneously,” Rogers elaborates.
The researchers recorded the movement of their crowd from above. And what they found was surprising, according to Rogers: “The lanes aren’t always straight. They can be curved when some people are heading towards a door, or the lines can be skewed when people are told to turn right, for example. But pedestrians have no idea they are forming these structures, they are only aware that they are dodging those walking in a different direction.
According to Barik and Rogers’ studies, curved lanes have a parabolic shape when one part heads for a narrow exit, or an elliptical shape when both exits are narrow. This possibility that the lines do not always move in a straight line is a finding from work by mathematicians at the University of Bath and was previously unknown.
The importance of this research lies not only in the surprise at the elegance of the mathematical description of a fact that we all at some point unconsciously share. In order to avoid avalanches and crushing, it is important to know how the crowds develop in a confined space. Architecture and urban planning know it very well. In 2022, more than 150 people trapped in an alley in Seoul, South Korea, died while celebrating Halloween, and other avalanches have previously occurred in crowds in different countries around the world.
Muslims gather around the Kaaba at the Grand Mosque FAISAL Al NASSER (Portal) during Ramadan in Mecca, Saudi Arabia
The Barik and Rogers study only focused on two-way travel, and this was because it is known that in this case orderly lanes are formed in each direction. Other researchers have found that when there are more than two flows, problems can arise when pedestrians travel in more than two directions. In these cases, it is much easier for several people to collapse, previous lanes to fade, and stable movement patterns to disappear.
For this reason, experts in this type of movement have long studied how to encourage the consistency of these unconscious pathways. One of the world’s best-studied cases is the stampede in the town of Mina near Mecca, Saudi Arabia, during the Hajj, the annual pilgrimage to Mecca by people of the Muslim religion. On September 24, 2015, a rush broke out when two groups of pilgrims traveling in opposite directions ran into each other on a narrow street. The avalanche killed at least 2,426 people. This case is often studied to understand how the spontaneous formation of lanes in a crowd can be done safely.
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