n Balls, m Hands
An Introduction to Juggling Theory
slides powered by reveal.js & claude codeSiteswap Notation
- "Juggling" in this context:
- Hands (usually 2) throw alternately
- In each time step, 0 or 1 ball is thrown
- Juggling patterns are described as a sequence of numbers
- The number describes how many time steps later the ball lands again
Number ~ throw height
- With 2 hands:
- Odd number: throw crosses to the other hand
- Even number: throw lands in the same hand
- The pattern repeats infinitely
- Period: length of the repeating sequence
- 3 (period 1) is equivalent to 3333333...
- 531 (period 3) is equivalent to 531531531...
Which number sequences are jugglable?
- At each time step, exactly 1 ball must land
(except for a "0" throw, then no ball may land here) - One possible definition:
a siteswap $a_0a_1a_2\dots a_{n-1}$ is valid,
if the set $S=\{(a_i+i)\bmod n | 0\leq i\leq n-1\}$
has exactly $n$ distinct elements. - Example: 123 works ($S=\{{\color{green}1},{\color{red}0},{\color{blue}2}\}$), 321 doesn't ($S=\{0\}$)
- The average of the numbers $a_i$ equals the number of balls
- Necessary condition: integer average
Can these patterns be chained together?
That depends on the state!
Patterns like 3 are in the ground state
Patterns like 51 (Shower) are in an excited state
Example:
- we get from 3 to 51 with e.g. 4 or 52
- we get back with e.g. 2 or 41
More than 2 hands
Why not 86277?
- The number of hands is arbitrary
→ we just need to define a throwing order! - Popular variant: 4 hands, distributed across 2 people
- Both throw alternately, each alternating both hands
- Results in e.g.: a 6 is a self (changes hand, but not person), 7 is a pass (changes person)
- one person throws the 7 straight, the other across
There is of course much more ...
- What if we also throw simultaneously with both/multiple hands?
→ Notation with parentheses and x for crossing throws, e.g. everyone knows (6x,4)(4,6x), short 6x4 - What if we throw and catch multiple balls at once?
→ Notation with square brackets - What if the hands also move around somehow?
- What if the numbers are negative?!?
- Here we learn from Dirac's physics:
- Negative numbers represent anti-balls, or equivalently (Feynman/Stückelberg):
- Balls that fly backwards in time
- Ball/anti-ball pairs are created and annihilate during the juggling pattern