In 1993 Claude Shannon, founder of information theory, wrote a popular analysis of juggling, and he even accompanied the article with a working model of a juggling robot. Building such a robot—in fact, juggling at all—is remarkable, because it seems to require faster reaction times than most of us can muster. Speed jugglers can achieve nearly 500 catches in a minute, a rate that allows just 120 ms per catch. Yet typical human reaction times are 250 ms, and even experts in high-speed sports such as tennis take 200 ms to adjust their responses.

So how do jugglers with reaction times no better than 200 ms catch balls every 120 ms? In part, multitasking may allow multiple balls to be processed simultaneously, though how that is done with 11 balls—the Guinness world record—is far from clear. And in part, balls are not thrown to random locations, so each ball need not be tracked and caught independently. Indeed, up to five balls can be juggled while the juggler is blindfolded. Jugglers rely on making accurate throws and predictions of where the balls will travel. The accuracy required is a measure of how unstable—and thus how difficult—a particular juggling pattern is.

Showers and cascades Section: Choose Top of page ABSTRACT Showers and cascades << Keep your eyes on the fli... Supplemental Material Additional resources

1 Figureshows two five-ball juggling patterns above plots that define their sensitivities to deviations in throw speed and angle. On the left is the most common pattern, the cascade, in which each hand catches and throws balls to equal heights across the body’s centerline. Hand motions in the cascade are left–right antisymmetric—that is, 180° out of phase. It is an amusing exercise to prove what all jugglers know—that only an odd number of balls can be juggled in the cascade. It’s impossible to juggle an even number of balls in a cascade without breaking the antisymmetry—for example, by throwing with both hands simultaneously or by throwing balls to different heights.

1b The shower pattern (shown on the right), by contrast, lacks symmetry and can be performed with either even or odd numbers of balls. Although the pattern is dynamically simpler, it’s also less stable—and thus more difficult—than the cascade. To appreciate why, consider the sensitivity plots (figure) for both cascade and shower.

2a v 0 and angle θ 0 to the throws and record how rates of catch success depend on throw deviations. Those rates are determined by counting the number of failures from 15 trials. Each trial comprises 50 throws for each combination of v 0 and θ 0 in the plots. Any throw in a trial that misses a catching hand 10 cm in diameter counts as a failure. Both panels show the results of simulated parabolic trajectories under gravity. Figureillustrates the ideal positions: The trajectories define throws that leave the exact center of the throwing hand and land precisely in the center of the catching hand. The hands move in ellipses, centered 50 cm apart and 180° out of phase, with horizontal radius 10 cm and vertical radius 5 cm. In simulations we add Gaussian deviations in initial speedand angleto the throws and record how rates of catch success depend on throw deviations. Those rates are determined by counting the number of failures from 15 trials. Each trial comprises 50 throws for each combination ofandin the plots. Any throw in a trial that misses a catching hand 10 cm in diameter counts as a failure.

1 The take-home message of figureis that the success region is larger for the cascade than for the shower. Cascade patterns can evidently tolerate greater variability in throw speed than can the shower, with balls thrown to the same height. For that reason, balls in a shower are typically thrown higher than those in a cascade. The higher throws provide more time between catches, but at the expense of requiring tighter control over the angle, as we’ll see.

For throws to the same height, the two patterns are indistinguishable in sensitivity to throw angle. That feature makes sense in terms of the dynamics of the problem. The trajectories of both patterns are parabolic and nearly identical, so changing the throw angle produces the same displacement of a catch from its expected location in both.

In the shower pattern, each ball travels first through a parabola and then through a quick shuffle, whereas in the cascade, each ball must travel through two parabolas to return to its starting point. So hands must move nearly twice as rapidly in the shower as in the cascade, which makes catches in the shower much more sensitive to timing.

2b The different effects of throw angle and speed can be quantified by plotting sensitivities of throws to independent deviations in the two parameters, shown in figurefor a five-ball cascade. Notice that throws to greater heights permit less angle deviation but more speed deviation than throws to lower heights. That phenomenon occurs because errors in throw angle produce variations in catch location that worsen as throws become higher.

Initial speed, on the other hand, chiefly affects the timing of the catch. And because greater heights are produced at lower hand frequencies, higher throws provide more tolerance to time variation. So for high, nearly vertical throws, it’s speed, not angle, that largely determines the time between throw and catch. The effects—particularly for higher throws—are that throw angle determines the location of a catch, whereas throw speed determines its timing. This distinguishes juggling from darts and other throwing games in which targeting accuracy does not depend on the relative timing between two hands.

The distinction between angle and speed is complicated by so-called siteswaps, juggling patterns defined by the order in which successive balls are manipulated, often resulting in throws to varying heights. When balls are thrown many meters high, throw angles must be accurate to within 0.1 degree. That’s more than an order of magnitude tighter than is achievable by world-class athletes, and the time between successive catches is shorter than human reaction times. For an analysis of sensitivity plots for several siteswaps, see the online supplement.