The Mathematics of . . . Applause

For all its sound and fury, a crowd that claps in unsion is not so different from a chorus of chirping crickets

By Josie Glausiusz|Saturday, July 01, 2000

Trumpeting cherubs, gold-leafed and puff-cheeked, peer down from the walls of Budapest's opulent music academy. As pianist Dezsö Ránki pounds and weaves at the keys, his breathing is at times so stertorous one could swear that someone in the audience is snoring. But there are no somnolent spectators here. When the final chords of Beethoven's Piano Concerto in C Major crash out, the crowd bursts into applause - at first tumultuously, but then, suddenly, in perfect unison. There is no signal, no leader; the synchrony is spontaneous. The pianist bows, his pageboy hairdo flopping about his solemn face. He disappears backstage, then returns to applause that grows stronger. Yet as the clapping gathers strength, its synchrony dissolves. Ránki retreats and reappears repeatedly, and so does the rhythm of the applause - one moment chaotic, the next a perfect beat. Then without warning the entireaudience stops, as one, on a single clap.

Tamás Vicsek, a physicist at Budapest's Eötvös University, twists around from his second-row seat and explains. "This synchronized clapping is called 'iron' applause in Hungarian," he says. "There was a time when an iron curtain would descend after a performance between the stage and the audience, which would clap rhythmically to induce the conductor or actors to appear in front of the curtain, through a little door at its center."

The iron curtain is gone - in more ways than one - from this country, but the rhythmic applause remains. In fact, it's hardly unique to the theaters of Budapest: When hockey player Wayne Gretzky retired from the Rangers, for instance, the crowd at Madison Square Garden burst into rhythmic applause, and the same response met Cecilia Bartoli when she sang an aria at the Teatro Olimpico in Vicenza, Italy. The reason, Vicsek and his colleagues in the United States and Romania believe, has as much to do with mathematics as it does with aesthetics and psychology.

According to Steven Strogatz, a mathematician at Cornell University who has studied synchronization for 20 years, the same set of mathematical principles governs the phenomenon wherever it occurs - be it among applauding people, flashing fireflies, or roomfuls of grandfather clocks. Strogatz has been fascinated by synchronization since he experimented with pendulums in his first science class. Now he seems to see it wherever he looks. Asian fireflies flash together each night in the mangrove trees along a riverbank. Crickets chirp in unison, and cicadas emerge from the ground at the same moment every 17 years. The moon rotates around its own axis at exactly the same rate as it orbits around Earth, which is why one side of the moon is never seen. Pacemaker cells in the heart oscillate in harmony. There is even the curious case of menstrual synchrony among women who live together.

To understand the mechanics of synchrony, Strogatz suggests, imagine several athletes running around a circular track. "Suppose these runners are friends, and they would prefer to run together so that they can talk," he says. "If their speeds are not too different - that is, if the slowest one can keep up with the fastest one, then you can get a group of runners all going in sync. But first, they have to be sensitive to each other. They have to be willing to adjust their speeds from what they would prefer. The fastest ones have to slow down, and the slowest ones have to speed up, to find some compromise. And that same principle - that slow oscillators have to speed up and fast ones have to slow down, and that this happens because of mutual interactions - is a pretty universal principle for synchrony."

The interactions, Strogatz adds, can be obvious: The runners see each other, the clappers hear the other claps. They can also be subtle. Two grandfather clocks can synchronize their pendulum swings - an effect first observed in 1665 by their inventor, the Dutch physicist Christiaan Huygens - through imperceptible vibrations traveling through the wall against which they both lean. But the story can also be more complicated than that. Picture, for example, some crickets living alone in soundproof chambers, as they did in Strogatz's lab. They can listen to their neighbors' sounds only when controlled levels of chirping are piped in. "If we make the sounds loud enough - if we let enough of the chirping sound through - at some point, there will suddenly be enough mutual influence that they can synchronize. Below that point, they can't. This is what physicists call a phase transition," Strogatz says. "There's a critical amount of interaction when synchrony will burst out. It doesn't just build up gradually."

A similar phase transition lies behind an audience's sudden switch to synchronized clapping. "Say the people are clapping in a disorganized way, but they all know that they're trying to synchronize," Strogatz explains. "However, they don't hear a beat. But then - and this is rather mysterious - suppose a beat just happens to emerge a little bit, maybe because a few clappers get lucky. That beat will then be audible above the din of the disorganized rest of the audience. And since everyone knows that they're trying to clap in unison, the cooperative clappers will try to join in with that beat. The pulse becomes stronger, and then it takes off."

Eighteen months ago, Tamás Vicsek, together with physicists Albert-László Barabási of the University of Notre Dame in Indiana and Zoltan Néda of Romania's Babes,-Bolyai University in Cluj-Napoca, set out to probe the dynamics of clapping more closely. Néda began the project by suspending microphones from the ceilings of concert halls in Romania and recording the applause. He and the other researchers then analyzed the recordings and found a fairly consistent pattern: several rounds - up to six or seven - of synchronized clapping, interspersed with incoherent cacophony. Moreover, the periods between claps doubled during synchronization.

To study the process further, Néda and his graduate student Erzsébet Ravasz asked 73 high-school students to stand alone in a room and clap quickly, as they would after an outstanding performance. Then he asked them to clap as if they were synchronizing with others. He found that their clapping rates varied quite widely when they were asked to clap quickly - between three and five claps per second - but when asked to simulate synchrony, most people clapped at around the same, relatively slow rate: about two claps per second.

As for the waves of chaotic clapping that interrupt the synchrony periodically, Barabási and Vicsek think they're a matter of crowd psychology. When synchrony sets in, the overall noise of the applause decreases; when it disappears, the noise level rises. Synchronization, they conclude, triggers a cozy feeling of togetherness among audience members, whereas faster clapping feels more enthusiastic. Conflicting desires swing the clappers between the two modes.

There was a time, Barabási adds, when comradeship reigned supreme in countries like Hungary and Romania and exuberance had no place, when swings between synchrony and chaos couldn't be heard. At the giant rallies common to Barabási's childhood in communist Romania - a childhood marked by the tyrannical regime of Nicolae Ceaus,escu - audiences applauding the "great leader" would clap monotonously and dutifully in synchrony in response to party speeches. But there was no enthusiasm to spur their clapping into disorder. Then one day in late December 1989, the synchronized applause ended - abruptly.

"When Ceaus,escu was overthrown, he organized a huge rally of 250,000 people in Bucharest to show support for him," Barabási remembers. "People were supposed to clap synchronously, and at the beginning they did. But then the clapping stopped. Some people threw their banners down, and then the shooting started and the revolution began." Four days later, on Christmas day, Ceaus,escu was shot dead.

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