FOR millennia, jugglers have relied on timing, precision and sheer chutzpah to see them through as they toss flaming torches, chainsaws and raw eggs before dazzled audiences. But these days there is a tool in their bag of tricks that you won’t see on stage – a little mathematics.
Over the past two decades, enthusiasts have developed a theory of juggling that has allowed them to invent many different new ways to juggle. And not content with that, they are now even exploring what happens when jugglers throw not in patterns, but randomly – and the results are mind-boggling.
Juggling maths took off in 1985, when several mathematically inclined jugglers came up with a notation for juggling patterns, called “site swap notation”. Until then, jugglers had shared new tricks by demonstration, but site swap notation made it possible to write down a trick, even one that had yet to be performed, and send it to a friend.
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“It’s like musical notation,” says Gregory Warrington, a juggler and a mathematics professor at the University of Massachusetts in Amherst. “If you hear someone play a tune, you can try to copy it just by listening, but if you both know how to read music, they can write it down for you explicitly and you can reconstruct it exactly.”
Site swap notation assumes that the balls are thrown and caught on rhythmic beats. It also assumes that balls are thrown as soon as they are caught and that the left and right hands throw alternately. Then all you do is simply list the number of beats that each ball stays in the air before it lands again.
Here’s how site swap notation would describe the most common juggling pattern – the three-ball cascade, in which the balls move back and forth in a figure-of-eight pattern. Each ball stays in the air for three beats. So the site swap notation for the cascade is just 3.
Another commonly performed pattern is the three-ball shower, in which balls travel in a circular path. In this trick, high five-beat throws alternate with quick one-beat passes from one hand to the other. So the site swap notation is 51. Mathematicians quickly proved that the number of balls being juggled is always equal to the average of the numbers in its site swap notation.
Before site swap notation, jugglers generally stuck to the cascade, the shower and another simple pattern called the fountain, which involves each hand juggling independently. The notation allowed jugglers to explore new territory and massively increase their repertoires. “Site swap has really expanded the kinds of tricks people do,” Warrington says.
And it turns out there is no limit to the number of juggling tricks. In 1994, Joe Buhler of Reed College in Portland, Oregon, proved that the number of different site swaps of length n using fewer than b balls is bn. So there are infinitely many different juggling tricks for any given number of balls.
Prime juggles
Some of the tricks developed using site swaps are very beautiful, says Ron Graham, a mathematician at the University of California, San Diego, who was formerly the president of the International Jugglers’ Association. Take 531, for instance, a three-ball trick in which one ball is tossed as if in a cascade, while the other two balls are, bafflingly, tossed as if in showers going in opposite directions. And 744 is a five-ball trick in which each third ball pops up above the others.
But before you start trying to juggle a series of numbers, be careful. Not every string translates into a juggling trick. Some don’t average out to a whole number, which means no number of balls will work, and others would make two balls land at the same time. For example, 345 works, but 354 doesn’t, since the second and third tosses both land on the sixth beat of the juggle.
So how do jugglers make sure that a site swap sequence they have concocted is truly jugglable? There’s an algorithm to do this and you can now download software packages that will check whether a pattern is a legitimate juggle, and if it is, will even produce an animation of it. These programs are ideal for jugglers looking to try out tricks before learning to perform them.
Some site swaps can be combined to make longer jugglable site swaps. For instance, 5551534 is performed by alternating between the juggling patterns 5551 and 534. But some sequences in site swaps can’t be broken down into smaller juggles. These indecomposable patterns are like prime numbers, which cannot be decomposed into factors, so Buhler and Graham call them prime juggles. They are now studying how many prime juggles there are of various lengths.
But does site swap notation really make for more entertaining performances? The truth is, most tricks developed using site swaps are lost on the uninitiated. As you marvel at balls whizzing around in front of you, you may not realise you are privy to an unusually long site swap. So most professional jugglers will try to impress you in other ways, by tossing under their legs, eating an apple while juggling, or juggling with a partner. “Site swaps are complicated and elegant, but to an audience they just look like more juggling,” says Roderick Kimball, one of the four Flying Karamazov Brothers.
Fans of juggling, of course, do appreciate the elegance of a good site swap. Say “441” at a juggling conference and you’ll spark instant comprehension. “It’s like when comedians refer to joke number 17 and they all burst out laughing,” Buhler says.
Besides, site swap practice does make for better shows, by helping the performers hone their skills. Although Kimball rarely performs site swaps, they are his favourite way to practise. Most jugglers habitually focus their eyes on the top of the pattern, but that doesn’t work for site swaps because they often include tosses to many different heights, making them harder to master. Several jugglers have told Graham that practising site swaps has increased both their flexibility and their conceptual power. “It’s a different level of juggling,” Graham says.
Even if you’re never able to master a 441 or a 744, learning to juggle can spur a cognitive leap. In January, researchers reported in Nature that volunteers who learned the basic three-ball cascade increased their brain mass by 3 to 4 per cent in two regions associated with perceiving motion.
So what’s the next step for jugglers? Until recently, mathematicians focused almost exclusively on periodic juggling, in which the same pattern repeats again and again. Now, however, they are turning their attention to random juggling, in which a juggler chooses at random how many beats each ball will be in the air for, taking care only to prevent two balls from landing at the same time.
In a study to appear in the February issue of the American Mathematical Monthly, Warrington has worked out for the first time what random juggling would look like, if you were ever lucky enough to meet someone able to do it. Imagine taking a photo on each beat of a random juggle. Given a maximum number of beats in a throw, and a fixed number of balls, there are a finite number of possible configurations for the balls in the photo. On each beat, the existing configuration becomes one of several possible new ones.
Warrington organised these configurations into a network rather like the World Wide Web. The configurations were the nodes of the network, and hyperlinks existed from each configuration to its possible successors. Warrington then explored the web of configurations in the same way Google explores web pages. Just as Google ranks web pages by noticing how often the software’s web crawler returns to the same page during a random walk around the web, Warrington ranked configurations, to work out which would appear most often during a random juggle.
“Random juggling is virtually impossible to perform”
The most common configuration by far, Warrington says, is a “low-energy” pattern in which the three balls are about to land in the next three beats. The least common configuration is the high-energy configuration in which all three balls must travel for the greatest possible number of beats before they land. So, for example, a random juggle with three balls and a maximum throw of five beats will be in a low-energy configuration about 27 times as often as it will be in a high-energy one.
Warrington, together with Allen Knutson of the University of California, Berkeley, and Matthew Levine of Akamai Technologies in Cambridge, Massachusetts, wrote a computer program to simulate random juggling. But when Knutson first watched the program’s animations, he was disappointed: the random juggles seemed rather boring.
On closer inspection, however, Knutson noticed that there seemed to be two different ways for random juggles to be boring. The juggling program lets you choose roughly how often to allow extra-high throws. When there are few high throws, the animation settles into a tedious low cascade with only the occasional breakout. When there are many high throws, the juggle is out of control, with the balls being thrown so high that they take forever to come down. “It’s like computer-generated music, which can be bad either because it’s too repetitive and uninteresting or because it’s too chaotic and cacophonous,” Knutson says.
Just as there are specific temperatures at which water changes from solid to liquid, and from liquid to gas, Knutson hypothesises that there is a corresponding “phase transition” in random juggling. Maybe there is a frequency of high throws at which the pattern would be poised between one kind of boringness and another.
If mathematicians ever find this magic formula for a specially entertaining random juggle, they warn not to expect to see it performed any time soon. Random juggling may sound easier than periodic juggling, but in fact it is virtually impossible to perform, Graham says. “Too much is happening too fast” for the juggler to process, he says.
On the other hand, maybe those mathematicians are just a little too pessimistic. After all, some intrepid jugglers have already followed in the footsteps of the software by juggling long site swaps that looked impossible to learn. Perhaps it is only a matter of time before some clever entertainer pioneers a show of live, random juggling. Make sure you get a ticket – although you might want to sit at the back.
