## The Mathematics of JugglingLearn to juggle numbers! This book is the first comprehensive account of the mathematical techniques and results used in the modelling of juggling patterns. This includes all known and many new results about juggling sequences and matrices, the mathematical skeletons of juggling patterns. Many useful and entertaining tips and tricks spice up the mathematical menu presented in this book. There are detailed descriptions of jugglable and attractive juggling sequences, easy zero-gravity juggling, robot juggling, as well as fun juggling of words, anti-balls, and irrational numbers. The book also includes novel, or at least not very well known connections with topics such as bell ringing, knot theory, and the many body problem. In fact, the chapter on mathematical bell ringing has been expanded into the most comprehensive survey in the literature of the mathematics used by bell ringers. Accessible at all levels of mathematical sophistication, this is a book for mathematically wired jugglers, mathematical bell ringers, combinatorists, mathematics educators, and just about anybody interested in beautiful and unusual applications of mathematics. |

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### Contents

An Introduction | 1 |

Simple Juggling | 7 |

Multiplex Juggling | 65 |

Multihand Juggling 85 | 84 |

Practical Juggling | 117 |

Jingling or Ringing the Changes | 141 |

Juggling Loose Ends | 177 |

Stereograms of Hamiltonian Cycles | 201 |

209 | |

221 | |

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### Common terms and phrases

0-throws 2-hand juggling 3-ball cascade 4-bell extent arcs b-ball juggling sequences b-ball multiplex b-ball state graph beat points bells braid cascade juggled causal diagram Cayley graph change ringing Claude Shannon clubs consider construct contains corresponding juggling cosets cyclic shifts diagram in Figure edges elements entry Erin exactly example finite sequence fountain graph of height h hands Hamiltonian cycles height h integers inverse jugglable jugglers juggling animator juggling cards juggling diagram juggling function juggling matrices labeled Lewbel magic juggling sequence mathematical matrix maximal prime loop method multiplex juggling sequences multiplex state graph n-body problem necklace nonnegative integers Note number of balls orbits permutation test Plain Bob plain course prime juggling sequence principle quence result Reverse ringers ringing array robot Section sequences of height sequences of period Shannon’s simple juggling sequences spins starting swap symmetric group Table toss juggling transition sequence trick truncated octahedron uniform juggling pattern vertex vertices