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The science of juggling

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Barry Sperling usually has his hands full. An avid juggler for 16 years, he tosses balls into the air for fun and relaxation, hoping to catch and toss them again."Juggling allows me to be in a different world and forget my troubles," he says. "It's something you can do by yourself. You don't have to have an audience."Mr. Sperling, who lives in Alexandria, is a member of the Fairfax Jugglers. The club typically meets Thursday nights at Key Intermediate School in Springfield.

Though many jugglers never consider the scientific principles governing their activities, mathematics and physics are at the heart of the hobby.

The body has a comfort with the rhythm of juggling, Mr. Sperling says. Objects follow a mathematical curve when they are thrown into the air, and jugglers don't need to look at the entire curve or path of the object. When the juggler looks at the top of the curve, the brain can figure out where the ball or club will come down.

"Good jugglers don't look at their hands at all," Mr. Sperling says. "They never see the catches. They just feel them. If you get really good at it, you can go completely on the feel of it and juggle blindfolded. The hands will make all the adjustments that are necessary."

The mathematical theory of juggling is referred to as siteswap, says Andrew Conway, a former board member of the International Jugglers Association, based in Carrollton, Texas. He lives in Freeport, Grand Bahama.

In the late 1980s, jugglers Bruce Tiemann and Bengt Magnusson created a juggling notation based on the changing places of balls when juggling, he says.

Siteswap theory allows strings of numbers to represent a juggling pattern, Mr. Conway says. The number given to a throw is the number of beats later that the ball lands in either hand.

A common three-ball pattern has a siteswap of three (3, 3, 3), he says. When each ball is thrown, it lands three beats later, he says.

In one four-ball pattern, such as 3, 3, 6, the right hand throws a ball that lands three beats later. The left hand throws a ball that lands three beats later, and the right hand throws a ball that lands six beats later. The fourth ball, which previously was thrown by the left hand, would be in the air. Then the pattern repeats, starting with the left hand.

"You can be throwing it under the leg, behind the back, bouncing it off the floor," Mr. Conway says. "It doesn't matter. It's not in the scope of the notation. It's just when it leaves your hand and when it gets back again."

The average of the numbers in the string equals the number of the balls in the pattern, he says.

To test whether a siteswap string is valid, take all the numbers in the string and add zero to the first number, one to the second number and two to the third number. Then divide each number by the amount of the numbers in the string and take the remainders. All of those remainders must be different, he says.

When 0, 1 and 2 are added to the string 3, 3, 6, the result is 3, 4, 8. When each number is divided by 3 -- which is the amount of numbers in the string -- the remainders are 0, 1 and 2.

Despite the analysis of siteswap theory, it doesn't ensure beauty in the presentation, Mr. Conway says.

"The better jugglers have taken siteswap tricks and managed to use them in routines where they are presented artistically," Mr. Conway says. "There are all sorts of things you can do. Rolling a ball off your head. Sticking a ball in your mouth. Rolling the ball off your knee."

The mechanics and motion of juggling are meant to be beautiful, says Donna Koczaja of Laurel. She is a member of the Baltimore Jugglers Association, which meets at Bedford Elementary School in Pikesville, Md. She is also a member of the United States Department of Juggling. The club meets at the Chevy Chase Community Center in Northwest. She appreciates the logic and order to juggling.

"There are laws that govern the juggling patterns," Ms. Koczaja says. "You could calculate the trajectories, if you wanted to really get crazy. It's all basically classical physics. It's coupled with chaos with added balls."

Each ball that is added greatly increases the difficulty of the routine, says Arthur Lewbel, economics professor at Boston College. He is a member of the International Jugglers Association.

"It's like taking a considerable amount off your time in the 100-yard dash," Mr. Lewbel says.

In addition to the number of balls adding complexity, the higher the ball is thrown, the less control the juggler has over where it returns, which only gives a little time before it comes down.

Claude Shannon, considered the father of information theory, created a juggling theorem in the 1970s that describes the timing of juggling, Mr. Lewbel says.

The equation says b/h=(d+f)/(d+e), when b equals the number of balls, h equals the number of hands, d equals the amount of time the ball spends in the hand, f equals the amount of time the ball spends in the air, e equals the amount of time the hand is empty.

Mr. Shannon also created the first juggling robot, Mr. Lewbel says. The machine bounced balls off a drum.

Later juggling robots helped scientists study how computers perceive and respond to information, Mr. Lewbel says.

"In a process of doing many tasks, we learn information to take shortcuts to get the robots to succeed at physical tasks that you want the robots to do," Mr. Shannon says.

The higher and faster the balls are thrown, the more balls can be juggled, says Erik Swanberg of Arlington. He is a member of the Fairfax Jugglers. He has a bachelor's degree in nuclear engineering and a master's degree in health physics and works as a nuclear scientist.

Albert Lucas, a Guinness Book of World Records record holder for juggling, once estimated that a human being couldn't juggle more than 14 objects, he says.

"Gravity makes the whole thing possible," Mr. Swanberg says. "Jugglers sometimes consider gravity to be the enemy, but juggling won't work at all without gravity. It's pretty much physics incarnate."

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