Photograph by Joe Schmelzer

Allen Knutson, a mathematician at the University of California at Berkeley, keeps five balls aloft in an intricate pattern. Knutson often juggles in class to demonstrate the basic premise of discrete mathematics: One input (or throw) will inevitably yield one output (or falling ball).

Allen Knutson’s office is a mess. There’s a unicycle with a flat tire in the corner, and a Bongo Board beside that, and a bunch of stuffed lizards crawling amid heaps of papers and books. The computer monitor is face down and unplugged on the desk. But Knutson, a 35-year-old tenured mathematics professor at the University of California at Berkeley, is completely focused on the challenge at hand. Long-haired, bearded, and wispy, with a quiet, otherworldly presence, he coolly recites a number sequence while juggling four balls in the air: “6-6-1-5-1-5-6-6-1-5-1-5-6-6-1-5-1-5 . . .”

Knutson is an authority on algebraic combinatorics, which involves, among other things, the counting of intersecting lines in multidimensional spaces. The number sequence he’s uttering would be familiar to anyone who knows siteswap, a mathematical language that describes juggling routines. Siteswap codifies motion by assigning each throw a number. A 3 is a throw that goes about chin high and stays aloft for roughly three beats of time; most novices toss ever-repeating 3s while learning to juggle three balls. A 6 is an over-the-head toss that stays in the air about twice as long as a 3, and so on. Odd-number throws are passed from one hand to another. Evens are both tossed and caught by the same hand. A 2 is a held ball, and a 0 denotes an empty hand.


An infinite number of sequences are possible, but the system is nonetheless orderly and—at least in its simplest form—governed by an ironclad law: No matter what the tempo, a juggler’s hand can make only one throw at a time. This means that the average of the numbers in a given throwing sequence must always be equal to the number of balls being thrown: 5-5-5-1 is unmistakably a four-ball pattern.

The beauty of siteswap is that it enables jugglers to write down and tinker with routines. A bland routine, in which three balls move back and forth between the right and left hands at exactly the same height, can be swapped for a sequence like a 4-4-1, in which two balls yo-yo straight up and down, staying aloft for four beats each, as a third ball is thrown from one hand to the other at waist level.

The system is so elegant, mathematically, and so beautifully spare in its notation that there is now a small cult of siteswapping numbers jugglers, a geeky, largely male group of computer programmers, academics, and engineers who eschew razzle-dazzle tricks—chain-saw juggling, for instance—in favor of such sublime challenges as keeping 10 balls aloft. Ron Graham, once the chief scientist for AT&T, is a siteswapper, as is a good portion of the math faculty at Reed College in Portland, Oregon. Siteswap even occasionally yields papers such as “Juggling and Applications to q-Analogues,” published in Discrete Mathematics a few years ago.

Allen Knutson became one with the cult in 1987, as a freshman at what may be juggling’s premier stronghold—Caltech in Pasadena. He was a Dungeons & Dragons devotee then and a video game aficionado who reveled in complex, rapid-fire keyboard maneuvers. (“You don’t have time for anything except direct computation,” he explains.) He found the same sort of Zen bliss in juggling. The calibrated hand work transported him “beyond emotion, to a point of mental clarity,” he says. He practiced an hour a day and soon became world class. In 1990 Knutson and a friend, Caltech physics major Dave Morton, juggled 12 balls together, establishing a world passing record that was not surpassed for five years (see box at the bottom of page 2).