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The Mathematics of Juggling The Mathematics of Juggling Stephen Hardy March 25, 2016 Stephen Hardy: The Mathematics of Juggling 1 Introduction Goals of the Talk I Learn a little about juggling. I Learn a little mathematics. I Have some fun. Stephen Hardy: The Mathematics of Juggling 2 Introduction Reference Most of the material for this talk was gleaned from Burkard Polster's great little book The Mathematics of Juggling: Stephen Hardy: The Mathematics of Juggling 3 What is Juggling? What is Juggling? Juggling takes many forms... Stephen Hardy: The Mathematics of Juggling 4 What is Juggling? Examples Jonathan Alexander http://adoreministries.com/, Kevin Rivoli http://www.syracuse.com/ Stephen Hardy: The Mathematics of Juggling 5 What is Juggling? Diablo Tony Frebourg - http://historicaljugglingprops.com/ Stephen Hardy: The Mathematics of Juggling 6 What is Juggling? Flair Bartending Ami Shroff - https://www.facebook.com/amibehramshroff Stephen Hardy: The Mathematics of Juggling 7 What is Juggling? Contact Juggling Michael Moschen - http://www.michaelmoschen.com/ Stephen Hardy: The Mathematics of Juggling 8 What is Juggling? Bounce Juggling Dan Menendez - http://pianojuggler.com/ Stephen Hardy: The Mathematics of Juggling 9 What is Juggling? Juggling is... I Juggling is manipulating more objects than hands you are using. Bob Whitcomb, http://historicaljugglingprops.com/ I We are interested in toss juggling. Stephen Hardy: The Mathematics of Juggling 10 What is Juggling? Toss Juggling Rings Anthony Gatto - http://www.anthonygatto.com/ Stephen Hardy: The Mathematics of Juggling 11 What is Juggling? Toss Juggling Clubs Jason Garfield - http://jasongarfield.com/ Stephen Hardy: The Mathematics of Juggling 12 What is Juggling? Toss Juggling Knives Edward Gosling - http://chivaree.co.uk/ Stephen Hardy: The Mathematics of Juggling 13 What is Juggling? Toss Juggling (Passing) Cirque Dreams Jungle Fantasy - https://en.wikipedia.org/wiki/Juggling_ring Stephen Hardy: The Mathematics of Juggling 14 What is Juggling? Toss Juggling We will focus on a single person toss juggling with two hands. Jason Garfield - http://jasongarfield.com/ Stephen Hardy: The Mathematics of Juggling 15 Simplifications Distractions We are only interested in the various patterns of throws and catches. We will ignore: I Showmanship (costumes, unicycles, balancing...) I Props (kerchiefs, balls, rings, clubs, knives, bowling balls, torches, chainsaws...) I Variations (inside, outside, piston two-in-one-hand...) I Flourishes (pirouettes, throwing under the leg or behind the back...) Stephen Hardy: The Mathematics of Juggling 16 Simplifications Simplifying Assumptions I The balls are juggled at a constant rate, throws alternating between right and left hands. I Patterns are periodic (otherwise it is not a pattern). I At most one ball gets caught and thrown on every beat. If a ball is caught, it is the one which is thrown on that beat. (We ignore the time between catch and throw). These are simple, asynchronous juggling patterns. At the end of the talk we will mention some generalizations. Stephen Hardy: The Mathematics of Juggling 17 Representing Juggling Patterns Throws I We use numbers to denote throws. I A throw of height n lands n beats later. I If n is odd it changes hands. I If n is even it lands in the same hand it was thrown from. I A juggling pattern is denoted by a string of numbers. I The constant string n is the standard way to juggle n balls. Stephen Hardy: The Mathematics of Juggling 18 Representing Juggling Patterns Juggling Patterns The constant string n is the standard way to juggle n balls. I For odd n this is called a cascade. I For even n this is called a fountain { and you actually just juggle n=2 balls in each hand independently. I The circular pattern people think of when they think of juggling is called a shower. Since this pattern is asymmetric and requires throwing higher, is actually a bad way to start learning. Stephen Hardy: The Mathematics of Juggling 19 Representing Juggling Patterns Throws (Continued) The throws 0, 1 and 2 are a bit unusual. I You juggle zero balls by doing nothing (sometimes performers use this opportunity to clap or pirouette or catch a prop thrown by someone else). I You juggle one ball by fast horizontal throws back and forth. I You juggle two balls by just holding them (or very small vertical throws, but more often this is an opportunity for flourishes, or biting an apple). Stephen Hardy: The Mathematics of Juggling 20 Representing Juggling Patterns Throws (Continued) I Humans can only throw so high accurately, even if ceilings and wind aren't an issue. I When you start even 5 level throws are hard. I can manage about a 7. I The world records are around 13. I Thus it makes sense we can cap our throws to a finite level. Stephen Hardy: The Mathematics of Juggling 21 Examples Two-Ball Patterns I 31 { the 2-ball shower. A bad habit. I 330 { training for 3 balls. I 40 { two in one hand - training for 4 balls. I 501 { the interesting and fairly challenging 2-ball pattern! Stephen Hardy: The Mathematics of Juggling 22 Examples Three-Ball Patterns I 3 { lots of variations, inside/outside/tennis etc. I 42 { two in one hand. A good opportunity for showmanship. I 423 { a good first trick, time to bite the apple. I 522 { how you juggle when you learn. I 55500 { the flash - add a clap or pirouette to impress. I 51 { the shower - be sure to practice both ways. Stephen Hardy: The Mathematics of Juggling 23 Examples More Three-Ball Patterns I 504 { kind of weird. I 441 { the asynchronous box. I 4414413 { my favorite, fairly easy to do but impressive looking. I 50505 { the snake - good training for 5. I 531 { very pretty when done properly. Stephen Hardy: The Mathematics of Juggling 24 Examples Four-Ball Patterns I 4 { lots of variations: inside/outside/pistons, etc. I 552 { kind of ugly, not great practice for 5. I 55550 { better practice for 5. I 53 { the four-ball half-shower. I 71 { the four-ball shower. I 5551 { fun. I 534, 7531, 7131, 633, 741... Stephen Hardy: The Mathematics of Juggling 25 Examples Five-Ball Patterns I 5 I 73 { the five-ball half-shower. I 91 { the five-ball shower. I 64 { three in one hand, two in the other. I 645, 771, 726, 66661, 663, 744, 77731... Stephen Hardy: The Mathematics of Juggling 26 Questions Questions I What strings of natural numbers are valid juggling patterns? I How can we generate all valid juggling patterns? I How many balls are needed to juggle a pattern? Stephen Hardy: The Mathematics of Juggling 27 Questions Observations I The period of a pattern is the minimal length before it repeats: 33333 = 3 has period 1, 441441 = 441 has period 3. I Cyclic permutations give the same pattern up to chirality (51 vs. 15 changes direction of the shower). I Not all strings of natural numbers give valid juggling patterns. We can never have n(n − 1) for example: Stephen Hardy: The Mathematics of Juggling 28 A Theorem Notation Let's tackle the question: given a valid juggling pattern, how many balls are needed to juggle it? th I We let t(i) be the height of of the throw on the i beat. That ball lands on the i + t(i)th beat. I We assume this is a periodic function with some period p: t(i + p) = t(i). I We want to calculate balls(t), the number of balls needed to juggle this pattern. I We define σ(i) = i + t(i) − balls(t). I This will be a permutation of Z exactly when t(i) is a valid juggling pattern. I Conversely, given such a permutation we can recover t(i). Stephen Hardy: The Mathematics of Juggling 29 A Theorem An Example Stephen Hardy: The Mathematics of Juggling 30 A Theorem More Notation I height(t) = maxi t(i) is the maximum throw height that occurs (again we assume this is finite). I balls(t), the number of balls needed to juggle the pattern t, corresponds to the number of infinite orbits in our diagram (singleton orbits correspond to zero throws, and there are no other possibilities since balls cannot vanish or appear). Stephen Hardy: The Mathematics of Juggling 31 A Theorem Theorem (Average Theorem) I The number of balls needed to juggle a valid pattern is the average height of the throws. I If t is a valid juggling pattern with finite height, then P t(i) balls(t) = lim i2I jI j!1 jI j Where I = fa; a + 1;:::; bg is an integer interval and jI j = b − a + 1 is the number of elements in that interval. Stephen Hardy: The Mathematics of Juggling 32 A Theorem The Proof Let I be an integer interval with jI j > height(t). Then each ball lands at least once in the interval I . Let O be an infinite orbit. Stephen Hardy: The Mathematics of Juggling 33 A Theorem Proof Continued From the diagram, we can see that X jI j − height(t) ≤ t(i) ≤ jI j + height(t) i2I \O Summing over all the orbits (singleton orbits do not contribute because then t(i) = 0), we get X balls(t)[ jI j − height(t)] ≤ t(i) ≤ balls(t)[ jI j + height(t)] i2I Dividing by jI j and taking the limit as jI j ! 1 we get the desired P i2I t(i) result: balls(t) = lim jI j!1 jI j Stephen Hardy: The Mathematics of Juggling 34 A Theorem Applications I Corollary: If the average of a string of natural numbers is not an integer, then that string is not a valid juggling pattern.
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