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The Mathematics of . . . Juggling

An algebra whiz reveals the secrets of keeping a lot of balls in the air

By Bill Donahue
Dec 3, 2004 6:00 AMJul 19, 2023 3:52 PM


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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).

Knutson can still do seven balls by himself, but he is semiretired from juggling now. He keeps 20 or so glittery balls in his desk drawer, partly to show students how to digest complex sets of numbers. His approach is always low tech. While many jugglers rely on computer programs that generate possible throw sequences, such programs tend to propose a slew of aesthetically dull sequences. Knutson prefers to start with nothing but hunches, scratching out little rows of numbers with arrows that describe the balls’ orbits. When pondering the five-ball 3-4-5-6-7 pattern, for instance, he can determine that the first throw, a 3, will land three beats later, as the 6 is being thrown:


The 4 will stay aloft for four beats, landing as the second 3 is being thrown:


The 5 will float for five beats and land as the next 5 goes airborne:


In about five seconds, Knutson can see that the 3, 6, 7, and 4 spin in a single orbit, while the 5—always landing back on itself—does its own lonesome, solipsistic dance amid the fray.


Siteswappers assign a number to each throw in a juggling routine. The higher the number, the higher the throw. Odd-numbered throws are passed from hand to hand, and even-numbered throws stay in the same hand. “You could even do a throw of –1,” Allen Knutson muses. “It would go backward in time and become antimatter.”

In swapping sites, jugglers mutate orbits and follow a simple rule: You can increase the height of one throw in a sequence so long as you equally decrease the height of another throw that lands later—and also pay heed to how much later it occurs. If it’s one throw away, you can add one to the height. In other words, the 4-4-1 mentioned above could be further mutated into a 5-3-1, wherein one ball does an incessant 3 while the other two fly through a 5-1 orbit.

Sometimes, for kicks, siteswappers devise what Knutson calls jugglers-only tricks, improbable sequences whose difficulty is obvious only to the cognoscenti. Knutson’s proudest invention is a five-ball trick, 8-5-7-4-1, that forces the juggler to throw lofty 8s from each hand while continually changing gears to make a wide variety of lesser throws in rapid succession. “I came up with that one to annoy Bruce Tiemann,” Knutson gloats, alluding to a Caltech grad whom many consider the world’s top numbers juggler. “He said, ‘Oh, I can do that,’ and then, after a couple of hours, he gave up in frustration.”

By now, Knutson has exhausted most of the possible juggling sequences. “Either I have them down, or they’re simply beyond me,” he says. “They’re eight- or nine-ball patterns—things I can’t do.” So last fall, along with a Berkeley undergrad named Peter Dolan, he began to consider randomness in juggling.

Pure randomness occurs when a hypothetical juggler is equally likely to make a throw that stays aloft for one, two, three, four, or five beats. Knutson and Dolan want to know what happens when a random juggling pattern becomes increasingly predictable. Is there a mathematical “freezing point” between randomness and predictability, like the phase transition that separates a liquid from a solid?

As they work, Knutson and Dolan are evolving a parameter, q, that denotes where a juggling sequence lies on the continuum between randomness and predictability. If they find, for instance, that juggling routines freeze when q is 32, then computerized juggling programs should be able to use the parameter to weed out what Knutson calls “silly and boring” sequences and focus only on the most delightful. Knutson’s freezing point has a magical aura for siteswappers—like musicians, they yearn for a happy medium between rote predictability and random arm stabbing—and he and Dolan believe they’ve almost found it. “The point we’re

The world record for passing juggling balls was set by Americans Joey Cousin and Bruce Sarafian, who kept 13 balls aloft for 54 catches in 1995 and for 176 catches in 1997. Sarafian also holds the record for the most balls juggled by one person: 10. He once had a car with a Florida license plate that read “11-BALLS.”

seeking does exist,” Knutson says. Mathematicians just need to look a little further. A good theorem, he adds, is like a good juggling routine: “It holds together. It makes sense, and it also delivers pleasant surprises.”

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