Ivars Peterson's MathTrek
September 16, 2002
It was the first home game of the season for the University of Maryland football team. About 48,000 fans had crowded into Byrd Stadium to watch the Terrapins take on the University of Akron Zips. Inevitably, during a lull in what turned out to be a rather one-sided contest, the assembled spectators created their own entertainment. Several times, people seated near one end of the horseshoe-shaped stadium leaped to their feet with arms outstretched, then sat down again as a group, initiating a ragged wave of activity that propagated to the other end of the stadium.
From my seat high up in the stands near the middle of the horseshoe, I had an excellent view of each wave. At first, a few desultory efforts merely created small ripples that rapidly dissipated. Once the activity had the crowd's undivided attention, however, the phenomenon became remarkably distinct. Several waves traveled the full length of the stadium and were convincing enough that I expected a reflection from the far wall at the end of the horseshoe (though it didn't happen). It became clear to me that the best waves occur when spectators are paying far more attention to their neighbors than to the game.
This collective phenomenon is sometimes called the "Mexican wave," or La Ola, in honor of the soccer fans who introduced this form of spectator exercise to a worldwide audience at the 1986 World Cup in Mexico City. However, it was surely practiced earlier at college football stadiums in the United States and other venues. One source credits Mariner baseball fans in Seattle. In an amusing (but anachronistic) scene in the recent movie A Knight's Tale, a medieval throng performs the wave at a jousting match.
Now, researchers have created a mathematical model that illuminates what it takes to trigger such a wave, which they describe as concerted motion in an excitable medium. Illés Farkas and Tamás Vicsek of Eötvös University in Budapest and Dirk Helbing of the Dresden University of Technology report their findings in the Sept. 12 Nature.
The researchers analyzed video recordings of 14 waves in soccer stadiums holding more than 50,000 people. They discovered that a human-generated stadium wave typically traveled in a clockwise direction and moved at a speed of about 12 meters (or 20 seats) per second. Achieving a remarkably stable profile, such a wave generally had a width of 6 to 12 meters (corresponding to an average width of 15 seats) as it propagated through the crowd. It took no more than a few dozen people standing up simultaneously to get a wave going.
To simulate such behavior, Vicsek and his colleagues turned to mathematical models originally used to describe excitable media such as cardiac tissue, where a small core of so-called pacemaker cells regulates heart activity. "We found that well-established approaches to the theoretical interpretation of excitable media, which were originally created to describe processes such as forest fires or wave propagation in heart tissue, can be generalized to include human social behavior," the researchers remarked.
In their models, people are regarded as "excitable units." They can be activated when the number of people near them performing a certain action exceeds a threshold value. Once activated, each unit goes through the same cycle of active, passive, and excitable states. Details of the models, movies, and interactive simulations can be found at http://angel.elte.hu/wave/.
Triggering a wave requires "a critical mass of initiators," the researchers concluded. In large sports stadiums, for example, it may take 25 to 35 people to get a wave started. It's also much more likely to occur when there isn't much else of interest happening in the stadium.
"Our approach may have implications for crowd control," Vicsek and his colleagues suggested. Such models could help predict when and how quickly an excited crowd could get out of control, for example, especially when a small group of instigators gains the upper hand.
Copyright 2002 by Ivars PetersonReferences:
Farkas, I., D. Helbing, and T. Vicsek. 2002. Mexican waves in an excitable medium. Nature 419(Sept. 12):131-132. For additional information, see http://angel.elte.hu/wave/.
Comments are welcome. Please send messages to Ivars Peterson at email@example.com.
A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the MAA book Mathematical Treks: From Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.