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Paper Juggling


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I've been juggling seriously since the summer of 1979, when I saw a performance of the Pickle Family Circus in a park one gorgeous Saturday afternoon. I already knew how to juggle three balls -- shakily -- but that was the day Ireally discovered juggling. I had never seen clubs juggled up close and in person before (clubs are those bowling pin-like things that are thrownspinning end over end through the air), and in particular I had never seen jugglers throw things back and forth between each other (calledpassing ). The Pickle Family did lots of both.

I was stunned. I was bowled over. I was frozen in myseat, gaping and incredulous. I couldn't believe that what I was seeing was possible. Ihad to learn how to do that.

Fortuitously, the circus offered workshops in various circus arts, including juggling, so I immediately signed up. The following morning, I learned the basics of passing balls, forced my roommate to learn to juggle three balls so that I'd have someone to try it out with,and embarked on a long and fruitful juggling binge. Thefire that was lit that day burned white hot for over fiveyears, and will remain fitfully smoldering as long as I canstill lift my arms, close my fingers, and count to 3.

My favorite kind of juggling nowadays is getting together with other jugglers and passing clubs. We arrange ourselves in various formations about the floor, start juggling all together, and throw the juggling clubs back and forth in varied and complex -- but mostly predetermined -- ways.

Which brings me to the main topic of this column: how multiperson juggling patterns work, and one way to write them down on paper. I'm going dangerouslyfar out on a limb here, assuming that it will be interestingto you, even though it has precious little to do with programming computers, and even though you're probably not a juggler. This particular limb is propped up a little by the very high proportion of computer people, mathematicians, engineers, and other scientists among jugglers. (There have been long-winded and unresolved discussions about why this should be so, but whatever the reason, it's a fact.) It's also been my observation (at Apple at our weekly juggle, and at the Worldwide Developers Conference) that computer people, in their endearing analytical way, often stand around for a long time trying to figure out the patterns.

Once you understand the rules of how the objects interleave and the jugglers interconnect, you can search for new patterns on paper, whether or not you know how to juggle. It's like a puzzle, or like a mathematical game. It's even conceivable (though just barely) that a knowledge of juggling patterns could be useful to you. I saw a citation on the rec.juggling newsgroup a while back for a paper called "Juggling Networks," published in the proceedings of a conference on parallel and distributed computing. From the abstract:

. . . these constructions are based on a metaphor involving teams of jugglers whose throwing, catching, and passing patterns result in intricate permutations of the balls. This metaphor affords a convenient visualization of time-division-multiplex activities that should be of value in devising networks for a variety of switching tasks.

There have been several mathematical papers that deal with juggling in one way or another, and even so eminent a personage as Claude Shannon, the father of information theory, was an amateur juggler and was interested in the permutations and combinations in juggling patterns. He wrote a paper called "The Scientific Aspects of Juggling," and I heard that when he appeared at a juggling convention he drew thunderous applause from the assembled jugglers (another indication of how many jugglers are science types).

Club passing is by far my favorite kind of juggling. The jollies I get from it are all over the map; it's deeply satisfying for me on many, many levels. Part of it is social, of course. Like sex, it's just more fun with others. And a big part of it is the cooperation, being a part of this complicated group pattern that's built and maintained by everyone together. I suspect it's a lot like jamming with a band in that sense: we all agree on a framework -- 12-bar blues in E or a seven-club four-count with triples, as the case may be -- and then go for it, the members either struggling to keep up or embellishing wildly, according to their level of skill. Sometimes we'll hit a "groove," a day and a pattern and a distribution of people that just feels right, the beat solid, the hands sure of their grip.

Club passing can feel like being part of some giant, whirling, clockwork contraption, with everything ticking and clacking along. Talk about being a cog in the machine! The spinning clubs form this sort of living, writhing, flying tangle with its own weird existence, a kind of "energy net" connecting the jugglers involved. The old saw "what goes around comes around" has a particularly pointed truth in club passing: if one juggler throws a pass badly -- say without quite enough spin, or a little off target -- it causes the receiving juggler some, well, discomfort. That discomfort often manifests itself in another bad pass, causing the next receiver to struggle, and so on. It's often actually visible; you can see the disturbance making the rounds, until it either gets smoothed out by jugglers who manage to keep their cool, or amplifies itself so badly that the whole pattern comes crashing down around the jugglers' heads. (Interestingly, the disturbance often travels independently of the clubs themselves, in a different direction or at a different speed, like a wave passing through water.) And passing clubs fosters --requires , actually -- a sort of heightened awareness of the other people involved. Often a quick, nearly imperceptible motion on the part of one juggler, a tiny hesitation, or the beginning of a wrong throw, corrected almost before it happens, causes another juggler to react reflexively. Typically both burst out laughing, mostly because it's unbelievable that such a tiny signal is transmitted at all.

And then there's the patterns game: a significant portion of the time spent "juggling" is really spent standing around, fiddling with the clubs, and trying to come up with new formations, new ways to arrange ourselves and the clubs in space and time so that everything fits together. The landscape of possible patterns is vast and complex, but also highly structured in mysterious ways. As in other iterative systems (computers and economies spring to mind), the underlying rules are relatively simple but the results can be very complex and widely variable. It's a kind of combinatorics and is, I think, actually covered by the mathematics of group theory.

I wrote a computer program that implements one particular kind of juggling notation, introduced to me by a juggler named Martin Frost and known ascausal diagramming . This notation can be handy for doodling around trying to find new multiperson passing patterns. (Actually, Istarted writing the program. It's still rickety and unfinished, and will probably always remain so -- it was more an experiment in QuickDraw GX programming than anything else. Nevertheless, it's included on this issue's CD, for your edification and/or derision.) The program implements a kind of active graph paper, allowing you to draw only "legal" throws, and constraining your diagrams in appropriate ways (such as preventing you from drawing throws that go back in time, for a start).

Figure 1 shows the diagram for a juggler doing a basic three-object pattern (called acascade ), and will serve to show both how the notation works and how juggling works. First the diagram: Time marches off inexorably to the right, divided into nice, even steps (calledcounts ). A juggler is represented through time as a row of Ls and Rs, representing the juggler's left and right hands, alternately throwing things. A thrown object is represented by an arrow from the hand that throws it to the hand that catches it. The pattern wraps around at the dotted lines, and repeats endlessly -- or until someone drops something. (The program always shows two repeating cycles like this, with the repeated parts "faded.") Note that the arrows (throws) form an unbroken line traveling through time from left to right, and that each hand has exactly one "input" and one "output."

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Figure 1. A three-object juggle

Contrary to what you might think at first glance, the overall path the arrows make doesn't directly trace the path of an individual club. If it did, this would just be a diagram of throwing one club back and forth between two hands. (That's a necessary prerequisite to juggling, but is definitely not juggling.) Instead, each throwdisplaces a club that is always assumed to be held, waiting,in the receiving hand. Think of the juggler as holding a club in each hand, while the third is in the air. The incoming club displaces the club that's already there, forcing the juggler to throw it elsewhere. In a cascade, the displaced club is thrown back to the opposite hand, where it in turn displaces the club that's there, which goes back to the first hand, displacing the club that's there, and so on, ad infinitum. (Note that although I'm saying "club" here, all these principles apply equally well to balls or rings or rubber chickens.) So the chain of throws is really a conceptual one, not a material one; it's a chain of cause and effect through time.

Figure 2 shows two jugglers passing with each other (the repeated cycle was cropped for space reasons). Note that they juggle in time with each other, like musicians keeping a beat. (When juggling with clubs, you actuallyhear the beat, when the clubs slap into the jugglers' hands.) Both jugglers throw a club to each other at the same time, both from the right hand (though it could just as well be the left). Throwing a club to another juggler "breaks" the juggler's continuous line of throws, but the other juggler's club arrives in the nick of time, knitting the pattern back together. This is a requirement: any club thrown to another juggler must be replaced by an incoming one. Otherwise, juggling can't continue; the juggler just stops, a club in each hand, waiting. (Actually, there are common situations that force a juggler to "stall" like that for a count or two, but we'll limit ourselves to the nonstalling patterns here.)

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Figure 2. A four-count

Because of the close timing, both jugglers must agree on the pattern before starting. The pattern in Figure 2 is called afour-count because there's a pass every four counts. (Another name for this pattern is every other , referring to the fact that every other right-hand throw is a pass.) The four-count is a very common pattern, and for most club jugglers this is the default, "idling" pattern. Since there's so much time between passes, it's possible to do lots of fancy free-form tricks (affectionately known as "throwing trash") in the midst of the pattern. Of course, "so much time" isn't really much time at all: a club juggle is roughly 160 counts per minute, so there's just over a second between the passes in a four-count.

These diagrams show nothing about spatial relationships,by the way. The usual situation has the jugglers facing each other 6 or 8 feet apart, but the same patterns can be done standing side by side, back to back, or even with one juggler standing on the other's shoulders. These diagrams show only the "connectedness" of the pattern through time, and in fact you can draw patterns that work fine on paper but are difficult to actually do because of mid-air collisions.

Figure 3 shows another pattern that demonstrates some other important concepts. In this case, every right-hand throw is a pass (which makes this pattern atwo-count ). Although the jugglers are juggling to the same beat, note that they are out of sync; one juggler's right-hand throw is simultaneous with the other's left. Note also that each pass spends twice as long -- two counts -- in the air. In all the previous diagrams, the throws have beensingles , meaning that the club spins around once during transit. The passes in Figure 3 aredoubles; since they're in the air twice as long, they have time to spin around twice before being caught. (The left-hand "self" throws are still singles.)

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Figure 3. A two-count with right-handed doubles

A warning about these multiple-spin throws: It's tempting, on paper, to make heavy use of long arrows (throws that spend lots of time in the air between jugglers). A little physics tells you, though, that the time in the air is proportional to the height of the throwsquared . So a double needs to be thrown four times the height of a single, and a triple must thrown nine times higher. A quadruple -- a "quad" -- must besixteen times the height of a single, and that's about as far as you can reasonably go with any sort of accuracy (or safety!). I generally stop at triples.

Now take a look at Figure 4 (again, cropped for space). This shows a three-person pattern called a feed . In this case one person (juggler 2) acts as the feeder and the others are feedees . The feeder is passing twice as often as the feedees; the feeder is doing a two-count, while the feedees are each doing a four-count, interleaved with each other in time. The feeder switches back and forth between the two feedees. This is another very common pattern, and can be added to indefinitely: Juggler 3 could pass with a new juggler, juggler 4, on the first count, at the same time jugglers 1 and 2 are exchanging clubs. That makes juggler 3 a feeder as well, feeding 2 and 4.

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Figure 4. A feed

I think by now you can see how the patterns fit together. It's like building a network, where everything has to eventually connect up and balance out. Go ahead, give it a try. A favorite pattern of mine is a three-count, with a pass every third count; both left and right hands pass. How about a feed where the feedees do three-counts? How many three-count feedees can one feeder possibly handle? Try a ten-club feed (the feeder does two-count doubles, as in Figure 3, and the feedees each do four- count doubles). Admire the attractive and tidy braids that result. Go wild. There are some interesting and nonobvious things about this notation that are probably worth pointing out. You can tell how many clubs there are in a pattern by taking a vertical slice through the diagram anywhere, counting the throws you intersect, and adding two clubs per juggler. (Note that Figure 3 is a seven-club pattern!) Also, if you start anywhere and follow the line of arrows around, wrapping back at the first dotted line, they always form closed paths, eventually arriving back where they began. Some patterns form one long continuous cycle; they're knit from a single strand, like a sweater. All the examples here are like that. Other patterns form distinct "orbits," where there are two or more strands making up the pattern; the three-count is an example. Each strand is an independent line of cause and effect, really an independent subpattern, that has no effect on the other parts of the pattern. You can actually decompose such patterns into their constituent parts, and juggle just one strand of the pattern at a time.

Also, the fate of any particular club isn't obvious at all in these diagrams. You can trace it, if you like -- a club leaves a hand two counts after it arrives -- but it's a bit of a pain (hmm, that might make a good addition to the program). Of course, tracing the paths of individual clubs isn't of primary interest to jugglers (though it's fun sometimes), in the same way that the path of an individual dollar is rarely of interest to economists and the trials and tribulations of an individual electron don't concern circuit designers. In contrast, I'd bet that the paths of the individual clubs are of great interest to the folks who wrote the network paper cited earlier. This notation would probably be a poor choice for them.

Finally, of course, theexperience of juggling is nowhere to be found in these diagrams. In contrast to their clean, orderly lines, passing clubs is a very physical thing, full of grimacing effort, plagued with fumbling and mistakes, and occasionally bone-whackingly painful. It's more like chopping wood than like doing math; it's more like pounding nails than like tying macramé, despite the nice braided look of the diagrams. But when things get cooking, when everyone is warmed up and throwing well, when the pattern grows and takes shape between our hands and fills the air with intricate, swirling, impossible motion, there's nothing else quite like it in the world.


  • "Scientific Aspects of Juggling," in Claude Elwood Shannon Collected Papers (IEEE Press, 1993).
  • "The Academic Juggler," in Juggler's World, Vol. 45, No. 4 (Winter 1993-94). A discussion of the origins of juggling notations.
  • The Juggling Information Service on the World Wide Web at juggle/. You'll find juggling software, FAQs, archives of net discussions, movies, and lots more.
  • Operating Instructions, A Journal of My Son's First Year by Anne Lamott (Ballantine Books, 1993).
  • June 29, 1999 by David Wiesner (Clarion Books, 1992).

DAVE JOHNSON first met his wife, Lisa, in a stage combat class, learning to swashbuckle in dramatic fashion. Dave took fencing in college for a couple of years, always secretly wishing there were more yelling, ducking, slashing, and diving, instead of the tightly controlled, linear, minimalist motions of good foil fencing. Then he discovered the world of stage combat, and he's never gone back. He and Lisa are currently enrolled in a new class: Elizabethan Swordplay, using rapier and dagger. En garde! *

Thanks to Lorraine Anderson, Jeff Barbose, Martin Frost, Bo3b Johnson, Lisa Jongewaard, and Ned van Alstyne for their enlightening review comments.*

Dave welcomes feedback on his musings. He can be reached at JOHNSON.DK on AppleLink, on the Internet, or 75300,715 on CompuServe.*


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