siteswapping

Siteswapping, or siteswap notation, is a mathematical framework used to describe juggling patterns. It encodes the timing and order of throws so that patterns can be analyzed, shared, and constructed.

In a siteswap, a pattern is represented by a finite or repeating sequence of numbers. Each number indicates the number of beats a thrown object will remain in the air before being caught and rethrown. A 0 represents a hold with no throw on that beat. The average of the digits in a repeating sequence equals the number of balls being juggled. For example, a repeating 3 corresponds to the three-ball cascade, while a repeating 5 corresponds to the five-ball cascade.

More complex patterns are described by extending the basic digits. Multiplex throws, where more than one ball is released on a single beat, are written with grouping symbols such as parentheses. The notation can represent synchronous (both hands throw on the same beat) and asynchronous patterns, and can be used to test the feasibility of a pattern by checking that no two throws land on the same beat.

Siteswap is widely used by jugglers to plan routines, communicate ideas, and verify patterns within competitions or training. It is also studied mathematically for properties such as reachability and symmetry. While the notation is powerful, not every physical throw corresponds to a simple digit; some patterns require more elaborate extensions.