Juggling by Design

The earliest known depiction of juggling is on the wall of an Egyptian tomb nearly 4,000 years old. The painting shows a woman keeping three balls aloft. It's only in the last decade or so, however, that juggling has become the subject of serious mathematical analysis.

The surprise may be that it took so long for mathematicians to get into the act. A large number of the roughly 3,000 members of the International Jugglers Association are involved with math or computers. Attracted by juggling's demand for a combination of dexterity, precision, invention, and experiment, they find it an immensely appealing pastime.

"Like music-making, it is a common ground between abstract form and physical dexterity; like mathematics, it is a form of pure play," mathematicians Joe Buhler, now deputy director of the Mathematical Sciences Research Institute (MSRI) in Berkeley, Calif., and Ron Graham of the University of California, San Diego remarked more than a decade ago in The Sciences.

Since then, mathematicians have developed a notation for and mathematical model of juggling that has allowed performers to gain a better understanding of their tricks and to develop new juggling routines to amaze their audiences.

A juggling pattern is usually periodic. The juggler repeats a sequence of movements at regular intervals, with the balls (or other objects) moving along precise trajectories to create a pleasing pattern.

One common pattern is known as the shower. A ball is thrown upward in a high arc by the right hand, caught by the left, then quickly passed in a low arc to the right. In effect, three or more balls chase each other along (more or less) a circular path.

The cascade pattern requires an odd number of balls. The left and right hands alternate throwing balls to each other, and the balls follow a looping path that resembles a figure 8 on its side (or the mathematical symbol for infinity). The world record for a sustained cascade is nine balls for 60 consecutive catches. On a good day, Buhler or Graham can handle seven.

In the fountain (or waterfall) pattern, a juggler uses an even number of balls and the balls never change hands. Early in this century, the famed juggler Enrico Rastelli managed 20 consecutive catches of a 10-ball fountain.

The initial step in the mathematical study of juggling was the development by several mathematicians around 1985 of a special sort of notation to convert juggling patterns into numbers.

The so-called site-swap notation represents the order in which balls are thrown and caught in each cycle of a juggle, assuming that the throws happen on beats that are equally spaced in time. In essence, only one ball is thrown at any instant and every ball is thrown repeatedly.

Let's look at a three-ball cascade. Ball 1 is thrown at time 0, again at time 3, then at time 6, and so on. Ball 2 follows the same pattern, thrown at times 1, 4, 7, and so on. Ball 3 is thrown at times 2, 5, 8, and so on.

The pattern can be characterized by using the intervals between throws. In a three-ball cascade, the time between throws of any ball is three beats, so its site swap is 3333. . ., or 3 for short.

Three-ball shower:
Ball 1: 0, 5, 6, 11, 12,. . .
Ball 2: 1, 2, 7, 8, 13,. . .
Ball 3: 3, 4, 9, 10,. . .

This pattern is designated 51, with 5 representing the duration of the high toss and 1 the time needed to pass the ball from one hand to the other along a low arc.

Four-ball fountain:
Ball 1: 0, 4, 8,. . .
Ball 2: 1, 5, 9,. . .
Ball 3: 2, 6, 10,. . .
Ball 4: 3, 7, 11,. . .
Its site-swap designation is 4.

The site-swap notation offers a snapshot of a juggling pattern. A "1" throw, for instance, goes from hand to hand in one beat; a "4" returns a ball to the same hand in four beats. A "0" represents a rest when no catch or toss is made.

Given a site-swap sequence, it's possible to figure out what a juggler has to do to perform that pattern.

Suppose the sequence is 531. Write down a row of integers, starting at 0, to represent consecutive beats. Beneath those integers, write the corresponding site-swap digits, repeating the sequence 5, 3, 1 as required.

Beat 0 1 2 3 4 5 6 7 8 9 10 1 . . .

Toss 5 3 1 5 3 1 5 3 1 5 3 1 . . .

Even integers (and 0) in the top row correspond to throws from the right hand, and odd integers to throws from the left. The throw height must increase as the interval between tosses of a ball gets longer.

In the 531 pattern, the first ball at time 0 is tossed high (5 beats, as seen in the second row below 0) by the right hand and caught by the left hand at time 5. The 1 in the second row beneath 5 means that the ball is then tossed low to the right hand, which catches it at time 6. The right hand then tosses it high again (5 beats), and the ball is caught by the left hand at time 11.

The second ball starts off at time 1, is tossed by the left hand in a moderately high arc (3 beats), and is caught by the right hand at time 4, by the left hand at time 7, and so on.

The third ball is thrown at time 2, travels in a low arc for 1 beat (going to 3), then in a high arc for 5 beats (going to 8).

In effect, the first and third balls each move in a shower pattern, but in opposite directions. The second ball weaves between the two showers in a relatively slow cascade rhythm.

Not all possible sequences lead to legitimate juggling patterns. The sequence 21, for example, has two balls landing simultaneously in the same hand. Other illegal sequences require a juggler to toss two balls at once.

Several computer programs are now available to identify legitimate juggling patterns and animate them (see http://www.cix.co.uk/~solipsys/new/SiteSwap.html). An avid juggler can see what a particular pattern looks like before trying it out and even check out juggling feats that are humanly impossible.

The site-swap sequences 234, 504, 345, 5551, 40141, 561, 633, 55514, 7562, 7531, 566151, 561, 663, 771, 744, 753, 426, 459, 9559, and 831 all represent legitimate patterns in the site-swap characterization of juggling. Indeed, the mathematical model indicates that infinitely many potential juggling patterns exist--though it might take a multi-armed, superdexterous robot to perform most of them.

It turns out that the strings of numbers corresponding to legitimate patterns have unexpected mathematical properties. Buhler and Graham, along with David Eisenbud, now MSRI director, and Colin Wright of the University of Liverpool, discovered those results several years ago in the course of developing a mathematical theory of juggling, based on the numerical sequences resulting from the site-swap notation introduced earlier by others.

The number of balls needed for a pattern, for example, equals the average of the digits in the site-swap sequence. Thus, the pattern 45141 would require (4 + 5 + 1 + 4 + 1)/5, or 3, balls.

You can also determine if a sequence is legitimate from the digits of its site-swap designation. For example, suppose the sequence is 566151, which consists of 6 digits. Add each of the six digits of the sequence to the values 0, 1, 2, 3, 4, and 5 in order and in turn to get 5 + 0, 6 + 1, 6 + 2, 1 + 3, 5 + 4, and 1 + 5, or 578496. If any resulting value is 6 or greater, subtract 6. The sequence 578496 then becomes 512430. If that sequence is a permutation of 012345 (all six digits in any order), it is, in principle, possible to juggle the given pattern. A similar analysis would show that the sequence 561651 is not a permissible juggling pattern.

Buhler also worked out a remarkably simple formula for counting the number of different juggling patterns. The number of legitimate site swaps of n digits using b or fewer balls is exactly b raised to the nth power.

Site-swap juggling theory actually captures only a subset of all possible juggling feats. It concerns only the order in which balls are tossed and caught and ignores such features as the location and style of throws and catches (behind your back, under your leg, and so on), which contribute greatly to juggling showmanship.

Nonetheless, mathematical theory has suggested novel juggling patterns, and some have started to gain popularity. Next time you see a juggling performance, watch out for 441!

References:

Beek, P.J., and A. Lewbel. 1995. The science of juggling. Scientific American 273(November):92. Available at http://www2.bc.edu/~lewbel/jugweb/science-1.html.

Buhler, J., and R. Graham. 1984. Fountains, showers, and cascades. The Sciences (January-February):44.

Buhler, J., D. Eisenbud, R. Graham, and C. Wright. 1994. Juggling drops and descents. American Mathematical Monthly 101(June-July):507. Also available at http://www.cecm.sfu.ca/organics/papers/buhler/.

Donahue, B. 1996. Jugglers now juggle numbers to compute new tricks for ancient art. New York Times (April 16).

Horgan, J. 1997. Ronald L. Graham: Juggling act. Scientific American 276(March):28.

Stewart, I. 1995. Juggling by numbers. New Scientist (March 18):34.

Colin Wright's SiteSwaps Web page at http://www.compulink.co.uk/~solipsys/pub/ssintro.htm features a "gentle" introduction to juggling and juggling patterns.

The Juggling Information Service Web site at http://www.juggling.org/papers/ provides articles and links related to mathematics and juggling.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.