Siteswap
Introduction
Siteswap is a mathematical notation used to describe and analyze juggling patterns. It provides a systematic way to represent the sequence of throws and catches in a juggling routine, allowing jugglers to communicate and explore new patterns with precision. This notation is particularly useful for understanding the underlying structure of juggling sequences and for discovering new patterns that might not be immediately obvious through experimentation alone.
Historical Background
The development of siteswap notation can be traced back to the late 20th century when mathematicians and jugglers began to explore the mathematical foundations of juggling. The notation was independently discovered by several individuals, including Bruce Tiemann, Bengt Magnusson, and Mike Day, in the 1980s. Their work laid the foundation for a formalized system that could be used to describe juggling patterns in a concise and unambiguous manner.
Basic Principles of Siteswap
Siteswap notation represents juggling patterns using a sequence of numbers. Each number corresponds to the number of beats (or time units) until the ball is thrown again. For example, the sequence "3" represents a three-ball cascade, where each ball is thrown every three beats. The notation assumes a constant rhythm, with each throw taking the same amount of time.
Types of Throws
In siteswap, different types of throws are represented by different numbers. The most common types of throws include:
- **Even Numbers**: These represent throws that cross to the opposite hand. For example, in a three-ball cascade, each ball is thrown to the opposite hand, represented by the number "3".
- **Odd Numbers**: These represent throws that return to the same hand. For example, a "5" throw in a five-ball cascade returns to the same hand after five beats.
- **Zero (0)**: Represents a pause or empty hand, indicating that no ball is thrown during that beat.
Validity and Ground State
For a siteswap sequence to be valid, it must satisfy the "ground state" condition, which ensures that the number of balls in the pattern remains constant. This condition can be checked by calculating the average of the numbers in the sequence. If the average is an integer, the pattern is valid for that number of balls.
Advanced Concepts in Siteswap
Multiplexing
Multiplexing is an advanced concept in siteswap that involves throwing multiple balls simultaneously from the same hand. This is represented by enclosing the numbers for the simultaneous throws in square brackets. For example, the sequence "[43]" indicates that two balls are thrown simultaneously, one with a 4-beat throw and the other with a 3-beat throw.
Synchronous Siteswap
Synchronous siteswap extends the basic notation to account for patterns where both hands throw simultaneously. In this notation, pairs of numbers are enclosed in parentheses to indicate simultaneous throws. For example, "(4,4)" represents a synchronous pattern where both hands throw a 4-beat throw at the same time.
Passing Patterns
Siteswap can also be used to describe passing patterns between multiple jugglers. In these patterns, the notation includes additional symbols to indicate which juggler makes each throw. This allows for complex collaborative routines to be represented and analyzed.
Applications of Siteswap
Siteswap notation has a wide range of applications in both recreational and professional juggling. It serves as a tool for jugglers to explore new patterns, develop routines, and communicate ideas with others. Additionally, siteswap has been used in educational settings to teach mathematical concepts such as permutations, sequences, and modular arithmetic.