Siteswap

Siteswap is a mathematical notation used by jugglers to describe patterns of throws and catches. It encodes a juggling sequence as a repeating string of digits (and, in extended forms, letters) where each position corresponds to a beat in time. In the asynchronous form, a digit n indicates a throw of height n, that is, a ball thrown on that beat will land n beats later. A 0 means no throw on that beat. The pattern repeats after a fixed period.

To be valid, the pattern must avoid collisions: the landing times of all balls, computed as t + a_t modulo the pattern length, must be all distinct. The average of the digits in the pattern equals the number of balls in the pattern.

Synchronous siteswap handles patterns where both hands throw in the same beat; throws are written as pairs in parentheses, for example (a,b), and the pattern repeats. More generally, multiplex patterns allow more than one ball to be thrown on a single beat, written by placing multiple numbers within the same beat, often using parentheses.

Common examples include the three-ball cascade, written as 3, which repeats to form the classic pattern; and the five-ball cascade, written as 531, also repeating. Higher throws can be denoted with letters (for example, a representing 10).

Siteswap has become a standard tool in juggling literature and practice, enabling jugglers to analyze, compare, and choreograph patterns, and to explore feasible variations and transitions between patterns.