As a juggler the author likes to finish his performances with a stunt that combines props and techniques from a variety of juggling disciplines. Imagine him idling on a giraffe unicycle, while balancing a spinning basketball on a mouth stick, and toss-juggling a sword, a toilet plunger, and a rubber chicken. As a mathematician he is also interested in the treasure trove of beautiful mathematics used to model the different activities in a juggler's repertoire. In this book he provides an intellectually stimulating collection of mostly self-contained mathematical essays that introduce the reader to many elegant results and techniques from a wide range of mathematical disciplines such as combinatorics, graph theory, knot theory, mechanics, differential equations, control theory, and robotics. "The Mathematics of Juggling" is the first comprehensive account summarizing and expanding the results in the literature on juggling tricks and skills, as well as the mathematics behind these tricks and skills. Anybody who is not put off by the word "mathematics" in the title of this book should have a good time reading it.
Author(s): Burkard Polster
Edition: 1
Publisher: Springer
Year: 2002
Language: English
Pages: 245
Cover......Page 1
The Mathematics of Juggling......Page 4
Copyright - ISBN: 0387955135......Page 5
Preface......Page 8
Contents......Page 16
1.1 What Is Juggling?......Page 20
1.2 A Very Short History of Juggling......Page 21
1.3 rec.juggling......Page 23
2.1 Simplifying Juggling Patterns......Page 26
2.2 Juggling Diagrams......Page 28
2.3 Basic Juggling Patterns......Page 30
2.4 Average Theorem......Page 33
2.5 Site Swaps and Flattening Algorithm......Page 36
2.6 Permutation Test......Page 41
2.6.1 A Method to Construct All Juggling Sequences......Page 43
2.6.2 Inverse of a Juggling Sequence......Page 44
2.6.3 Pick a Pattern Procedure......Page 47
2.6.4 Converse of the Average Theorem......Page 48
2.6.5 Scramblable Juggling Sequences......Page 53
2.6.6 Magic Juggling Sequences......Page 54
2.7 How Many Ways to Juggle?......Page 56
2.7.1 Juggling Cards......Page 57
2.7.2 Weights of Juggling Sequences......Page 61
2.8.1 State Graphs......Page 63
2.8.2 Ground-State and Excited-State Sequences......Page 66
2.8.3 Throws from States......Page 68
2.8.4 Prime Juggling Sequences and Loops......Page 69
2.8.5 Complements of State Graphs......Page 77
2.8.6 Transition Matrices......Page 81
3 Multiplex Juggling......Page 84
3.1 Average Theorem and Permutation Test......Page 85
3.2 Number of Multiplex Juggling Sequences......Page 87
3.3 Weights of Multiplex Juggling Sequences......Page 92
3.4 Multiplex State Graphs......Page 94
3.4.1 Prime Multiplex Juggling Sequences and Loops......Page 96
3.5 Operations Involving Juggling Sequences......Page 100
4.1 Juggling Matrices......Page 104
4.2 Average Theorem and Permutation Test......Page 107
4.3 Multihand State Graphs......Page 109
4.4 Operations Involving Juggling Matrices......Page 111
4.5 Special Classes of Juggling Matrices......Page 113
4.6 Uniform Juggling and Shannon’s Theorems......Page 115
4.7 Shannon’s Theorems for Juggling Sequences......Page 122
4.8 Cascades and Fountains......Page 126
4.9 Juggling Balls and Hands......Page 129
4.10 Juggling Labeled Balls......Page 131
4.11 Decomposing Simple Juggling Sequences......Page 132
5.1 Jugglable Juggling Sequences......Page 136
5.2 Juggling Made Easy......Page 142
5.2.1 Zero-Gravity Juggling......Page 143
5.2.2 Bounce Juggling......Page 145
5.2.3 Robot Juggling......Page 146
5.3 Real-World Juggling with Gravity and Spin......Page 148
5.3.1 Accuracy and Dwell Time......Page 149
5.3.2 Why Clubs and Balls Line Up......Page 151
5.4 What Is All this Numbers JugglingGood for?......Page 156
6.1.1 Basic Definitions......Page 160
6.1.2 History and Practice of Change Ringing......Page 163
6.2.1 Turning Bells into Balls......Page 165
6.2.2 Turning Extents into Site Swaps......Page 168
6.3 Mathematical Notation and Basic Operations......Page 169
6.3.1 Notation......Page 170
6.3.2 Ringing Sequences from Ringing Sequences......Page 171
6.4.1 Principles......Page 173
6.4.2 Methods......Page 174
6.4.3 Extents Based on Principles or Methods......Page 176
6.5.1 Cayley Graphs......Page 178
6.5.2 Four Bells......Page 179
6.5.3 Five Bells......Page 182
6.5.4 Many Bells......Page 185
6.5.5 Names......Page 186
6.6 Extents from Groups......Page 187
6.6.1 Left Cosets and Plain Bob......Page 188
6.6.2 Right Cosets and No-Call Principles......Page 192
6.7 Computers, Bobs, and Singles......Page 194
7.1 Does God Juggle?......Page 196
7.2 Juggling Braids......Page 200
7.3 Spinning Top of a Palm-Spun Pyramid......Page 205
7.4.1 Juggling Words......Page 208
7.4.2 Juggling Rational and Irrational Numbers......Page 210
7.4.3 Antiballs, Antithrows, and Causal Diagrams......Page 211
7.5.1 Riddle......Page 216
7.5.2 Lord Valentine’sCastle......Page 217
7.6 Further Reading......Page 218
Appendix: Stereograms of Hamiltonian Cycles......Page 220
References......Page 228
A......Page 240
G......Page 241
J......Page 242
P......Page 243
S......Page 244
W......Page 245
