The Mathematics Of Juggling

1
The Mathematics Of Juggling
2
Introduction

• Is juggling mathematical?
• Figure of 8 clearly proof it is!
• Not very rigorous however, as real mathematicians
  we want something better. We want

3
A model!

• Learning a three ball cascade is fairly easy,
  anyone can learn it if shown how.
• We want to be able to write it down, then extend
  it in such a way to be able to invent some
  tricks.

4
The obvious (uncrossed) position, left hand about
to throw.
Ul
Right hand over left, left hand about to throw.
Rl
Left hand over right, right hand about to throw.
Lr
5
Mills Mess State Transition Diagram (or MMSTD)

• There are clearly 6 states for the hands to be
  in Lr, Ll, Ul, Ur, Rr, Rl
• Write these as nodes on a graph.

6
The Three Ball and Reverse Cascade
7
Under the Hand
8
Mills Mess
9
Rubensteins Revenge
10
Limitations

• Behind the back
• Kick-ups

11
Clubs

• Can actually be modelled in the same way, if you
  make a simplifying assumption.
• Although more difficult how do we deal with
  orientation?

12
Simplify!

• Assume that height doesnt matter, and instead
  take interest in the number of spins a club does
  before being caught.