Over more years than he'd probably care to remember, Master William Lee has become a weekend fixture in New York City's Washington Square Park. Most people know him as "that juggling Chinaman," which might strike readers as appallingly racist were it not how Lee refers to himself. "And how many of you want to see the Chinaman burn?" he routinely asks gathered onlookers as he teases them with the promise of juggling flaming torches, among other feats. It's all in fun, of course: Lee is a consummate entertainer, combining his crack juggling skills with a few illusionist tricks and a running commentary of jokes, punctuated by occasional pleas for donations. And he's the latest in a long line of juggling jokesters that dates back to ancient Egypt.
That's right, the earliest historical record of juggling (estimated date is between 1994 and 1781 BC) can be found in hieroglyphic depictions of female dancers and acrobats throwing balls that adorn the tomb of an unknown prince (see figures in third row from top). Master Lee's Chinese ancestors weren't far behind: by 770-476 BC, Chinese historians were writing about juggling warriors, such as Lan Zi, who supposedly juggled seven swords; and Xiong Yiliao who juggled nine balls at the same time during a battle between two warring states. One wonders why such warriors were not simply killed outright, but purportedly their skilled pyrotechnics were so impressive they could end conflicts before they began. Hmmm.
By 400 BC, juggling had spread to Greece. In Rome, an officer in the Roman legion named Sidonius Apollinaris supposedly entertained his troops by performing juggling tricks. Even the Irish got into the act: the legendary hero Cuchulainn was able to juggle nine apples, while a few centuries later, historical records show that a royal court buffoon diverted King Conaire by juggling nine swords, nine silver shields and nine balls of gold. And in 1066 -- yes, that pivotal year -- the warrior-bard of William of Normandy (named Taillefer), juggled before enemy lines and reportedly made the first kill at the Battle of Hastings. So it was a very effective diversionary tactic.
Nonetheless, by the mid 1800s juggling was pretty much comprised of street and court performers and "filler" acts at theaters and music halls throughout Europe and North America. And why not? It's fun, and good, clean (mostly) entertainment. But is there anything truly useful to be learned from it? The experts say sure! Peter Beek and Arthur Lewbel wrote a November 1995 article in Scientific American on the subject (which you can find on Lewbel's science of juggling Website), asserting the following:
Juggling definitely has uses beyond hobby and enterntainment. It is complex enough to have interesting properties and simple enough to allow the modeling of these properties. Thus, it provides a context in which to examine other, more complex fields.... One is the study of human movement and the coordination of the limbs. Another is robotics and the construction of juggling machines. The third is mathematics: juggling patterns have surprising numerical properties.
According to Beek (a movement scientist at the Free University in Amsterdam) and Lewbel (a professor of economics at Boston College), the first scientific paper on the subject appeared in 1903 in the American Journal of Psychology. The author, Edgar James Swift, documented the rate at which some students learned to toss two balls in one hand. Within 40 years, the advent of computers meant that scientists could calculate the trajectories of thrown objects. (This is also about the same time International Jugglers Association was founded.) Mostly, the focus through the 1960s was building on Swift's work: using juggling as a task to compare general methods of learning sensorimotor skills.
Juggling finally started getting more respect in the 1970s, when MIT scientist Claude E. Shannon created the first juggling machines out of an Erector set, and also formulated his "juggling theorem" correlating the position of the balls and the action of the hands while juggling. By the 1980s, mathematicians had gotten into the act. And they're still coming up with newer and better computer models for this simple human activity.
At least there are well-defined parameters for a computer modeling simulation. I won't go into excessive detail, because these have already been elaborated at great length by people far more knowledgeable than we are: namely, Beek and Lewbel, but also former mechanical engineer-turned entertainer Jack Kalvan, who once worked on his own juggling robot while employed with IBM Research, and has since analyzed spatially optimal patterns for juggling using computer modeling.
We'll reiterate this much information: The three most common objects used by jugglers are balls, rings, and clubs. For a single juggler, there are three basic patterns. The "cascade" is the most common, in which an odd number of balls are tossed from one hand to the other. Then there is the "fountain," in which balls are thrown and caught with the same hand, usually used for an even number of objects. Finally, there is the "shower," in which all the objects are tossed in a circle. A juggler might also choose to "multiplex": throwing more than one object from a single hand simultaneously.
Sounds simple enough in theory, but as anyone who's tried to juggle could could tell you, it's harder than it looks. That's even more true when it comes to figuring out how to mathematically model the process. True, the fundamental motion behind juggling is essentially standard projectile motion, involving multiple projectiles with interweaving paths. And the patterns of those paths are periodic cycles: they repeat, rather than change continuously.
Furthermore, the number of possible patterns is relatively small. But no two throws or catches will ever be exactly the same because there are so many variables associated with throwing: angle of release, release velocity, height of the throws, etc. Skilled jugglers are able to tightly control such variables to throw the objects as consistently as possible. (Add a second person to the mix, and it becomes even more complicated, even though the patterns between multiple jugglers are generally based on the single person patterns.)
Any mathematical model for juggling must incorporate both ball motion and hand motion. Hand motion modeling is pretty difficult, since the dynamics of the human arm are incredibly complex. It helps, though, that the hand motions in juggling closely correspond, by necessity, to the positions and velocities of the objects during throws and catches. Objects other than balls, like rings or clubs, have markedly different physical characteristics and must be treated differently in any computer model. Balls can be modeled with standard particle system dynamics, while the less-uniformly shaped clubs and rings work better when modeled as a rigid body system.
According to Ron Grahan, a mathematician at the University of California, San Diego, mathematical models of juggling might give performers a better understanding of the science behind their tricks, and possibly help them to develop new juggling routines. (He should know: he's a past president of the International Jugglers Association, as well as the American Mathematical Society and the Mathematical Association of America.) Using one of the many computer programs currently available that identify "legitimate" patterns and animate them -- many now available on the Web for easy download, such as Kalvan's Optimal Juggler -- jugglers can see what a particular pattern looks like before trying it out in real life. They can even check out juggling feats that are (to the best of our knowledge) humanly impossible, just for fun. In fact, mathematical theory has already suggested a few novel juggling patterns beginning to gain in popularity. (To check out a bit of virtual juggling, go here. Or watch this movie here.)
One would think that anyone with a background in math or physics would have a bit of an "edge" over the rest of us mere mortals when attempting to learn juggle. But such might not be the case; it is, ultimately, a learned motor skill involving excellent hand/eye coordination. While there is certainly a physics aspect to the act of juggling, having a physics background doesn't necessarily make one a better juggler, according to David Ehrenstein, a physicist and editor of Physical Review Focus. He learned to juggle as a child, long before he began studying physics, taught by a couple of fellow performers in a theater production ("Channukah '78!") at the Jewish Community Center in Rockville, Maryland. "I doubt it would have helped," he says now. "The learning is so subconscious, like riding a bike."
Not surprisingly, his skills improved the more he practiced -- and he practiced a lot in those early years, juggling every day for more than two months to master the basics: juggling three lacrosse balls, which he says are easier to juggle than tennis balls. By 14, he'd learned to juggle five balls, a much more difficult feat than performing tricks with three objects (even if, in Master Lee's case, one or more of those objects is on fire). "The difficulty goes up exponentially with each additional object," explains Ehrenstein, a fact most audiences don't appreciate. "People assume the dependence is more linear, so there's not a good audience-appreciation ratio" to the amount of effort required to learn that particular skill.
Numerous studies on how people master the art of juggling support Ehrenstein's assertion about the difficulty of juggling more than three objects. They show that learning the simple shower pattern can take a few hours or days, using three balls, but learning times increase to weeks and months for four balls, extending to years for five balls. If you're curious, the world record for the greatest number of objects juggled is 12 rings, 11 balls, or eight clubs. An object's shape, once again, is an important variable. But so is basic Newtonian mechanics. Beek and Lewbel's article points out that the need for either speed or height increases rapidly with the number of objects being juggled.
Okay, so maybe a math or physics background doesn't necessarily give you an edge in learning to juggle, but Ehrenstein concedes that there does seem to be quite a few examples of physics, math or computer nerds who also love to juggle. We can only speculate as to the reasons for this, but Ehrenstein thinks it might partly be a social thing: "Who else would have time -- at least during high school -- to practice juggling alone rather than going out with friends?" Jen-Luc Piquant is skeptical about this hypothesis, given the burgeoning number of high-tech gamers who rarely see the light of day. But it's more convincing if you consider Ehrenstein's second point: that there's a natural allure to juggling because it's "a mechanical, logical process that has the kind of certainty we all like in math and physics, as opposed to, say, the humanities."
As for the ongoing proliferation of mathematical models, juggling robots, and anaylses of the underlying physics, he notes, "Once you have a bunch of physicists doing something, it can't be long until they start to use physics principles to analyze it. (This is true. Geeks will scientifically analyze just about anything, even which Applebee's restaurant is the most expensive. Jen-Luc stumbled upon this classic exchange at Overheard in New York: First Geek: "Literally, it is the most expensive Applebee's in the universe." Second Geek: "Ah, not so. In a constantly expanding universe the probability approaches 100% that somewhere out there exists a more expensive Applebee's." First Geek: "... Let's just eat at KFC." Jen-Luc, who is quite the food snob, is appalled at the very thought of Applebee's and KFC, and suggests our two Geeks check out the cheaper places in Chinatown or Little India instead.)
For Ehrenstein, at least, all this combined with his early love of theater and performing: for him it was a chance to be noticed. And while adult responsibilities have cut down his juggling time drastically, he still indulges from time to time, performing for small, informal audiences: things like World Year of Physics nights at the New Deal Cafe in Maryland. Once he attended a nuclear physics conference to give a talk on science communication. At the end, he brought out some bean bags to perform a few simple tricks, describing them as protons and neutrons and cracking jokes about fission and fusion. "They just ate it up," he recalls. "I'm always amazed at how easy it is to get audiences to laugh at even incredibly stupid jokes if you're juggling at the same time."
Which might explain the enduring popularity of Master Lee. True, he's a bona fide comedian, and a damn good one at that, but we doubt very much he presents his "A" list material for free on Saturday afternoons in Washington Square Park. Not that his audience minds: they love his act, love being teased and invited to participate in feats like slicing through a cucumber balanced on the volunteer's bared belly with a very large sword... while blindfolded. No mathematical model can prepare someone for that. We wonder what happened to his very first volunteer. And whether he has liability insurance.