The Mathematics of Juggling

One useful method that many jugglers rely on to summarize patterns is site-swap notation, an idea invented independently around 1985 by Paul Klimek of the University of California at Santa Cruz, Bruce Tiemann of the California Institute of Technology and Michael Day of the University of Cambridge. Site swaps are a compact notation representing the order in which props are thrown and caught in each cycle of the juggle, assuming throws happen on beats that are equally spaced in time.

To see how it works, consider the basic three-ball cascade. The first ball is tossed at time periods 0, 3, 6..., the second at times 1, 4, 7..., and the third ball at times 2, 5, 8.... Site-swap notation uses the time between tosses to characterize the pattern. In the cascade, the time between throws of any ball is three beats, so its site swap is 33333..., or just 3 for short. The notation for the three-ball shower (first ball 0, 5, 6, 11, 12..., second ball 1, 2, 7, 8, 13..., third ball 3, 4, 9, 10, 15....) consists of two digits, 51, where the 5 refers to the duration of the high toss and the 1 to the time needed to pass the ball from one hand to the other on the lower part of the arc. Other three-ball site swaps are 441, 45141, 531 and 504 (a 0 represents a rest, where no catch or toss is made).

The easiest way to unpack a site swap to discover how the balls are actually tossed is to draw a diagram of semicircles on a numbered time line. The even-numbered points on the line correspond to throws from the right hand, the odd-numbered points to throws from the left.

As an example, consider the pattern 531. Write the numbers 5, 3 and 1 a few times in a row, each digit under the next point in the number line starting at point 0 [see top illustration at right]. The number below point 0 is 5, so starting there, draw a semicircle five units in diameter to point 5, representing a throw that is high enough to spend five time units (beats) in the air. The number below point 5 is a 1, so draw a semicircle of diameter 1 from point 5 to 6. Point 6 has a 5 under it, so the next semicircle is from point 6 to 11. You have now traced out the path in time of the first ball, which happens to be the same as the first ball in the three-ball shower pattern 51 described above. Repeat the process starting at times 1 and 2, respectively, to trace out the path of the remaining two balls. The result is that the first and third balls both move in shower patterns but in opposite directions, and the second ball weaves between the two showers in a cascade rhythm. Leaving out this middle ball results in the neat and simple two-ball site swap 501.

Not all sequences of numbers can be translated into legitimate juggling patterns. For example, the sequence 21 leads to both balls landing simultaneously in the same hand (although more complicated variants of site-swap notation permit more than one ball to be caught or thrown at the same time, a feat jugglers call multiplexing).

Site-swap notation has led to the invention of some patterns that are gaining popularity because they look good in performances, such as 441, or because they are helpful in mastering other routines, such as the four-ball pattern 5551 as a prelude to learning to cascade five balls. Several computer programs exist that can animate arbitrary site swaps and identify legitimate ones. Such software enables jugglers to see what a pattern looks like before attempting it or allows them simply to gaze at humanly impossible tricks.

The strings of numbers that result in legitimate patterns have unexpected mathematical properties. For instance, the number of balls needed for a particular pattern equals the numerical average of the numbers in the site-swap sequence. Thus, the pattern 45141 requires (4+5+1+4+1)/5, or three balls. The number of legitimate site swaps that are n digits long using b (or fewer) balls is exactly b raised to the n power. Despite its simplicity, the formula was surprisingly difficult to prove.

Site-swap theory does not come close to describing completely all possible juggling feats, because it is concerned only with the order in which balls are thrown and caught. It ignores all other aspects of juggling, such as the location and style of throws and catches. Many of the most popular juggling tricks to learn, such as throwing balls from under the leg or behind the back, are done as part of a regular cascade and so have the same site-swap notation.