Maths and Juggling

A mathematical language for juggling

We have all seen, at least once in our life, a juggler tossing balls in the air. Why is that so impressive to our eyes?

Despite having just two hands, any respectable juggler can juggle three balls at the same time. Considering, for simplicity, that one can handle one ball per hand, how is that possible?

Animation: 3-ball cascade — Animation 1: the 3-ball cascade.

Let’s analyze the cascade above. Each ball is tossed from one hand to the other: the right hand throws to the left hand and vice versa. Just as a ball is about to be caught, the juggler launches another ball to free the hand for the catch. Juggling three or more balls is possible by iterating this principle.

Animation: alternative 3-ball pattern at different heights — Animation 2: still 3 balls, but at different heights (a different pattern).

As you can imagine, there is a wide variety of patterns and, if we were to assign a distinct name to each one, it would be a prohibitive memory exercise.

For this reason, Paul Klimek and Don Hatch independently invented a notation system in the early 1980s to describe and name juggling tricks—what we now call siteswap. When Klimek first introduced it, he referred to the notation as quantum juggling. The system was later extended by jugglers such as Bruce Tiemann, Jack Boyce, and Ben Beever (see this discussion).

Siteswap can describe (and name) juggling patterns with any number of balls and jugglers, covering both synchronous and asynchronous throws. Below is the same pattern executed asynchronously and synchronously.

Asynchronous version of a pattern — Asynchronous

Synchronous version of a pattern — Synchronous

For simplicity, we will focus on the so‑called vanilla siteswap: one juggler, two hands, alternating throws.

A limitation of the siteswap notation

The right‑hand pattern above (Mill’s Mess) is still a cascade, but with crossed and alternating hand positions. Even though the two animations look very different, they have the same siteswap. Siteswap encodes the timing, direction, and relative height of throws—not the body mechanics of how you execute the pattern.

A number for each toss

The idea is simple: assign a non‑negative integer to each throw, equal to the number of beats until the same ball is caught and thrown again. Odd numbers (1,3,5,…) cross to the other hand; even numbers (2,4,6,…) return to the same hand; and 0 indicates an empty beat.

0: a beat when the hand is empty.
1: a direct pass to the other hand (no time for any other catch/throw in between).
2: a tiny self toss; the throwing hand cannot do anything else in that beat.
3: a cross to the other hand, leaving one beat for both hands to handle other balls.
4: a self toss high enough that the throwing hand can handle another ball while the other hand can handle two.

The numbers therefore encode relative height with respect to pace. One can throw a 5 under the head quickly, or above three meters slowly; what matters is how many beats elapse while it’s in the air.

Patterns repeat cyclically. We notate only one period, e.g., 531 represents the repeating pattern 531531531…

Not every sequence of numbers is a pattern!

Consider the sequence 432. First throw a 4 with the right hand (a self), then a 3 with the left (a cross). While both balls are in flight, the right hand tries to make a 2 (a quick self). The right hand ends up with three balls arriving together: a collision. So 432 is not executable.

Characterization theorem of siteswaps.

A finite sequence of non‑negative integers $(s_1,s_2,\dots,s_n)$ is executable (collision‑free) if and only if the residues $$ (s_i + i) \bmod n $$ are pairwise distinct for $i=1,\dots,n$.

Applying it to 432 with $n=3$ gives the residues $2,2,2$ (not distinct), hence invalid. Swapping the last two digits to get 423 yields residues $2,1,0$ (all distinct), hence valid.

Theorem on the number of balls.

If $(s_1,\dots,s_n)$ is a valid siteswap, then the number of balls equals the average height: $$ \frac{s_1+s_2+\cdots+s_n}{n}. $$

For example, for $534$ the average is $(5+3+4)/3=4$ balls.

If you want to know more…

This page focused on vanilla siteswap. There are also notations for synchronous patterns, multiplex throws (one hand tossing multiple balls on the same beat), and passing between multiple jugglers. For a broader overview and examples, see the Siteswap page on Juggle Wiki.

Multiplex example [53][32]2 — Multiplex: [53][32]2

Synchronous (6x,4)(4,6x) — Synchronous: (6x,4)(4,6x)

Passing example — Passing: <4|4|5p1><5p2|5p1|3><4p3|3p3|3p2>

Republished from the author’s original post “Maths and Juggling” (2018) on Math is in the Air. Some wording and layout adapted for this site.