Juggling by Number

By Ivars Peterson

Muse, February 2000, p. 26.

Playing catch is easy. It's not hard to follow a single ball thrown back and forth between two people. But add another ball or two (or more), and take away one of the catchers. You end up with something that looks quite magical. The juggled balls seem to take on a life of their own.

Many of the roughly 3,000 members of the International Jugglers Association are mathematicians or computer scientists. Joe Buhler, a mathematician at the Mathematical Sciences Research Institute in Berkeley, California, started juggling years ago when he was in college. He did it just because it was fun. On a good day, he can keep as many as seven balls going in a pattern called a cascade.

The cascade pattern requires an odd number of balls. Each ball follows a looping path that resembles a figure 8 on its side. Once the balls are in motion, the juggler never holds more than one ball at a time. The world record for a sustained cascade is nine balls for 60 catches in a row.

With a little coaching and some practice, just about anyone can learn how to juggle, Buhler says. Once learned, juggling—like bicycle riding—is almost impossible to forget.

Mathematicians have invented a way to write down juggling patterns as sets of numbers. They look at the order in which balls are tossed into the air and then caught. Each toss or catch happens on a particular beat, as if the juggler were keeping time to music with a definite rhythm.

In the three-ball cascade, for example, ball 1 is thrown at beat 0, again on beat 3, then on beat 6, and so on. Ball 2 follows the same pattern. It's thrown at beat 1, then thrown again on beats 4, 7, and so on. Ball 3 is thrown at times 2, 5, 8, and so on. Because each ball is tossed on every third beat, mathematicians give the cascade pattern the label 3.

By combining numbers according to a few rules, they can invent new patterns for jugglers to try. One of the new (but tricky) juggling patterns is 441. It requires three balls, and each ball travels high twice, then low once, and then the sequence repeats.

The math captures only the order in which balls are tossed. It ignores crowd-pleasing features such as the style of throws and catches and their location—behind your back or between your legs, for example. The types of objects also make a difference. Buhler likes juggling with clubs, which look like oversized bowling pins. You might also see fruit, rings, plates, axes, flaming torches, or chainsaws sailing through the air!

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