The Mathematics of Juggling - WordPress.com · The Mathematics of Juggling Stephen Hardy March 25, 2016 Stephen Hardy: The Mathematics of Juggling 1

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.


Introduction

ReferenceMost of the material for this talk was gleaned from BurkardPolster’s great little book The Mathematics of Juggling:


What is Juggling?

What is Juggling?

Juggling takes many forms...


What is Juggling?

Examples

Jonathan Alexander http://adoreministries.com/, Kevin Rivoli http://www.syracuse.com/


http://adoreministries.com/

http://www.syracuse.com/

What is Juggling?

Diablo

Tony Frebourg - http://historicaljugglingprops.com/


http://historicaljugglingprops.com/

What is Juggling?

Flair Bartending

Ami Shroff - https://www.facebook.com/amibehramshroff


https://www.facebook.com/amibehramshroff

What is Juggling?

Contact Juggling

Michael Moschen - http://www.michaelmoschen.com/


http://www.michaelmoschen.com/

What is Juggling?

Bounce Juggling

Dan Menendez - http://pianojuggler.com/


http://pianojuggler.com/

What is Juggling?

Juggling is...

I Juggling is manipulating more objects than hands you areusing.

Bob Whitcomb, http://historicaljugglingprops.com/

I We are interested in toss juggling.


http://historicaljugglingprops.com/

What is Juggling?

Toss Juggling Rings

Anthony Gatto - http://www.anthonygatto.com/


http://www.anthonygatto.com/

What is Juggling?

Toss Juggling Clubs

Jason Garfield - http://jasongarfield.com/


http://jasongarfield.com/

What is Juggling?

Toss Juggling Knives

Edward Gosling - http://chivaree.co.uk/


What is Juggling?

Toss Juggling (Passing)

Cirque Dreams Jungle Fantasy - https://en.wikipedia.org/wiki/Juggling_ring


https://en.wikipedia.org/wiki/Juggling_ring

What is Juggling?

Toss JugglingWe will focus on a single person toss juggling with two hands.

Jason Garfield - http://jasongarfield.com/


http://jasongarfield.com/

Simplifications

Distractions

We are only interested in the various patterns of throws andcatches.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 theback...)


Simplifications

Simplifying Assumptions

I The balls are juggled at a constant rate, throws alternatingbetween 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 aball is caught, it is the one which is thrown on that beat. (Weignore the time between catch and throw).

These are simple, asynchronous juggling patterns.At the end of the talk we will mention some generalizations.


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.



Juggling PatternsThe 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 ofjuggling is called a shower. Since this pattern is asymmetricand requires throwing higher, is actually a bad way to startlearning.



Throws (Continued)

The throws 0, 1 and 2 are a bit unusual.

I You juggle zero balls by doing nothing (sometimes performersuse this opportunity to clap or pirouette or catch a propthrown 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 smallvertical throws, but more often this is an opportunity forflourishes, or biting an apple).



Throws (Continued)

I Humans can only throw so high accurately, even if ceilingsand wind aren’t an issue.

I When you start even 5 level throws are hard. I can manageabout a 7.

I The world records are around 13.

I Thus it makes sense we can cap our throws to a finite level.


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!


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.


Examples

More Three-Ball Patterns

I 504 – kind of weird.

I 441 – the asynchronous box.

I 4414413 – my favorite, fairly easy to do but impressivelooking.

I 50505 – the snake - good training for 5.

I 531 – very pretty when done properly.


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...


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...


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?


Questions

Observations

I The period of a pattern is the minimal length before itrepeats: 33333 = 3 has period 1, 441441 = 441 has period 3.

I Cyclic permutations give the same pattern up to chirality (51vs. 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:


A Theorem

Notation

Let’s tackle the question: given a valid juggling pattern, how manyballs are needed to juggle it?

I We let t(i) be the height of of the throw on the i th 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 tojuggle 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 validjuggling pattern.

I Conversely, given such a permutation we can recover t(i).


A Theorem

An Example


A Theorem

More Notation

I height(t) = maxi t(i) is the maximum throw height thatoccurs (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 noother possibilities since balls cannot vanish or appear).


A Theorem

Theorem (Average Theorem)

I The number of balls needed to juggle a valid pattern is theaverage height of the throws.

I If t is a valid juggling pattern with finite height, then

balls(t) = lim|I |→∞

∑i∈I t(i)

|I |

Where I = {a, a + 1, . . . , b} is an integer interval and|I | = b − a + 1 is the number of elements in that interval.


A Theorem

The Proof

Let I be an integer interval with |I | > height(t). Then each balllands at least once in the interval I . Let O be an infinite orbit.


A Theorem

Proof Continued

From the diagram, we can see that

|I | − height(t) ≤∑

i∈I∩Ot(i) ≤ |I |+ height(t)

Summing over all the orbits (singleton orbits do not contributebecause then t(i) = 0), we get

balls(t)[ |I | − height(t) ] ≤∑i∈I

t(i) ≤ balls(t)[ |I |+ height(t) ]

Dividing by |I | and taking the limit as |I | → ∞ we get the desired

result: balls(t) = lim|I |→∞

∑i∈I t(i)

|I |�


A Theorem

Applications

I Corollary: If the average of a string of natural numbers is notan integer, then that string is not a valid juggling pattern.

I Example: 5342 is not a valid sequence because(5 + 3 + 4 + 2)/4 = 15/4 is not an integer.

I This averaging-to-an-integer condition is not sufficient: 543 isnot a valid juggling pattern, but (5 + 4 + 3)/3 = 4

I However, 534 is a valid juggling pattern.

I We do have a partial converse due to Hall:Theorem: If a string of natural numbers has an integeraverage, then some permutation of it is a valid jugglingpattern.


A Theorem

Real Math

I Note that σ(i + p) = (i + p) + t(i + p)− balls(t) =i + t(i)− balls(t) + p = σ(i) + p

I

p∑i=1

(σ(i)− i

)=

p∑i=1

(t(i)− balls(t)

)=

p · balls(t)− p · balls(t) = 0

I Recall the affine Weyl group Ap−1 can be represented asbijective functions f : Z→ Z so that f (i + p) = f (i) + p, and∑p

i=1

(f (i)− i

)= 0.

I Thus the permutations coming from juggling patterns are asubgroup of the affine Weyl group Ap−1.

I In fact, Ehrenborg and Readdy used juggling to calculate thePoincare series of the affine Weyl groups!


More Results

More Questions

I How can we generate all of the valid juggling patterns? Howmany are there?

I Site Swaps: We switch the landing sites of two differentthrows to get a new valid pattern with the same number ofballs.


More Results

Site Swap Example


More Results

More Algorithms

I If our pattern is not the constant pattern, then there must bea throw which is higher than the average and other throwwhich is lower than the average. By swapping the landingsites of a higher throw with a lower throw, we get a patternwhich is closer to a constant pattern.

I Since the throw heights and period are finite, in finitely manysteps we must be able to flatten our valid sequence to theconstant sequence with the same number of balls.

I The Flattening Algorithm: By applying site swaps andcycling permutations, we can take any valid juggling patternwith b balls to the constant sequence b.


More Results

More Algorithms Continued

I Since site swaps and cyclic permutations are reversible, theseoperations actually generate all the valid juggling patternswith a given period and number of balls!

I Theorem: The constant pattern consisting of the throw brepeated p times can be transformed into any valid jugglingpattern of length p for b balls via site swaps and cyclicpermutations.


More Results

ExamplesI The 3-ball period 2 patterns.

I The 3-ball period 3 patterns.

I Mathematics brought new patterns into juggler’s repertoires!


More Results

More Results

I How many valid juggling patterns are there?

I Theorem: The number of all minimal b-ball juggling patterns(modulo cyclic permutations) of period p is

N(b, p) =1

p

∑d |p

µ(pd

) ((b + 1)d − bd

)Where µ is the Mobius function

µ(n) =

0, if n has repeated prime factors

1, if n = 1 or n has an even number of prime factors

−1, if n has an odd number of prime factors


Final Thoughts

Generalizations

I Synchronous patterns – throw and catch with both handssimultaneously.

I Multiple hands – allow for passing between people.

I Multiplexing – throw and catch multiple balls in the samehand at the same time.

I Braids – attach each ball to a string then tie other endstogether. Then patterns correspond to braids!


Final Thoughts

Final Thoughts

I Math is fun.

I Juggling is fun.

I Try playing withhttp://www.siteswap.net/JsJuggle.html

Thank You For Listening!


http://www.siteswap.net/JsJuggle.html

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The Mathematics of Juggling - WordPress.com · The Mathematics of Juggling Stephen Hardy March 25, 2016 Stephen Hardy: The Mathematics of Juggling 1