The Mathematics of Juggling - The Mathematics of Juggling Yuki Takahashi University of California,...

The Mathematics of Juggling

Yuki Takahashi

University of California, Irvine

November 12, 2015

Co-sponcered by UCI Illuminations and Juggle Buddies

Yuki Takahashi (UC Irvine) November 12, 2015 1 / 51

Introduction: What is Juggling?

Figure : 2013, winter, 5-ball cascade

Juggling is the manipulation of objects (balls, clubs, rings, hats, cigarboxes, diabolos, devil sticks, yoyos, etc).


Club Juggling

Figure : Korynn Aguilar (Chair), April 2015, UC Irvine, One World Concert.


Passing

Figure : October 2015, University Hills, Fall Fiesta.


(Dragon) Ball Juggling

Figure : Vegeta, juggling 7 (dragon) balls


Papers about Siteswaps

A. Engsrom, L. Leskela, H. Varpanen, Geometric juggling withq-analogues, Discrete Math. 338 (2015), 1067–1074.

A. Ayyer, B, Arvind, S. Corteel, F. Nunzi, Multivariate jugglingprobabilities, Electron. J. Probab. 20 (2015).

A. Knutson, T. Lam, D. Speyer, Positroid varieties: Juggling andgeometry, Compos. Math. 149 (2013) 1710–1752.

C. Elsner, D. Klyve, E. Tou, A zeta function for juggling sequences, J.Comb. Number Theory 4 (2012) 53–65.

S. Butler, R. Graham, Enumerating (multiplex) juggling sequences,Ann. Comb. 13 (2010) 413–424.

F. Chung, R. Graham, Primitive juggling sequences, Amer. Math.Monthly 115 (2008) 185–194.


Juggling Patterns

Definition (Simple Juggling Patterns)

In this talk, we assume

the balls are juggled in a constant beat, that is, the throws occur atdiscrete equally spaced moments in time,

patterns are periodic, and

at most one ball gets caught and thrown at every beat.

Figure : Juggling diagram of the ?-ball cascade.


Example (1)

Figure : Juggling diagram of the 4-ball fountain.

This pattern is denoted by 4.


Example (2)

Figure : Juggling diagram of the 3-ball shower.

This pattern is denoted by ...?


Example (3)

Figure : 4-ball shower.

This pattern is denoted by 71.


Example (4)

This pattern is denoted by ...?

The number of ball juggled is ...?


Terminologies

We call a finite sequence of non-negative integers arising from ajuggling pattern a juggling sequence.(Example: 3, 4, 51, 441, 7, 51515151, are all juggling sequences)

The length of a finite sequence of integers is called its period.(Example: 441441 has period 6)

A juggling sequence is minimal if it has minimal period among all thejuggling sequences representing the same juggling pattern.(Example: 144 is minimal, but 441441 is not)


Example (4)

546 is NOT a juggling sequence.

Figure : What is wrong...?

In general, if s = n(n − 1) · · · , then s is not a juggling sequence.


The Average Theorem

Theorem (The Average Theorem)

The number of balls necessary to juggle a juggling sequence is itsaverage.

If the average is not an integer, then the sequence is not a jugglingsequence.

Example

The number of balls necessary to juggle 441 is 3.

562 is NOT a juggling sequence (note that 5+6+23 = 4.3333... is not

an integer).


Swapping

i j i j

Figure : a juggling sequence s is transformed into another juggling sequence si,j .

This operation is called swapping.


Example (1)

Let s = 441. Then s0,2 = ...?

Figure : s to s0,2.


Example (1)

Figure : 441 to 342.


Example (2)

Let s = 7531. Then s1,3 = ...?

Figure : s to s1,3.


Example (3)

Let s = 6313. Then s1,2 = 6223.

Figure : s to s1,2.


Swapping

Definition (Swapping)

Let s = {ak}p−1k=0 be a sequence of nonnegative integers. Let i and j beintegers such that 0 ≤ i < j ≤ p − 1 and j − i ≤ ai . Let si ,j be

si ,j(k) =

aj + j − i if k = i

ai − (j − i) if k = j

ak o.w.

i j i j


Summary of Swapping

For a juggling sequence s, we denote the number of balls necessary tojuggle it by ball(s). Then

The sequence s is a juggling sequence if and only if si ,j is a jugglingsequence.

The average of s is the same as the average of si ,j .

If s is the juggling sequence, then ball(s) = ball(si ,j).


Leeeeet’s Practice!!! =]

Example (Swapping)

Let s = 642. Then s0,1 = 552, and s0,2 = ...?(What does this result tell you about s? Is it a juggling sequence?)

Let s = 532. Then s0,1 = ...?

Let s = 123456789. Then s3,5 = ...?

Reminder:

Let s = {ak}p−1k=0 be a sequence of nonnegative integers. Then

si ,j(k) =

aj + j − i if k = i

ai − (j − i) if k = j

ak o.w.


Getting Sleeply? It’s Show Time! :)


Cyclic Shifts

Definition (Cyclic Shifts)

Let s = {sk}p−1k=0 be a sequence of nonnegative integers. Let

s = a1a2a3 · · · ap−1a0.

We call the operation of transforming s into s the cyclic shift of s.

Example: If s = 12345, then s = 23451.


Example

Let s = 441. Then s = 414.

Figure : 441 to 414.


Summery of Cyclic Shifts

Apparantly, we have the following:

The sequence s is a juggling sequence if and only if s is a jugglingsequence.

The average of s is the same as the average of s.

If s is a juggling sequence, then ball(s) = ball(s).


Summery!!!

Swapping and cyclic shifts both preserve

1 ”jugglibility”,

2 the average, and

3 the number of balls (if it is a juggling sequence).


Flattening Algorithm

The flattening algorithm takes as input an arbitrary sequence s.

1. If s is a constant sequence, stop and output this sequence. Otherwise,

2. use cyclic shifts to arrange the elements of s such that one ofmaximum height, say e, comes to rest at beat 0 and one not ofmaximum height, say f , comes to rest at beat 1. If e and f differ by1, stop and output this new sequence. Otherwise,

3. perform a swapping of beats 0 and 1, and return to step 1.


Examples! =)

Denote the map s 7→ s0,1 by Sw , and cyclic shifts s 7→ s by Cy .

Example (flattening algorithm)

1) 264Cy−→ 642

Sw−→ 552Cy−→ 525

Sw−→ 345Cy2

−→ 534Sw−→ 444.

(This implies ...what?)

2) 514Sw−→ 244

Cy2

−→ 424Sw−→ 334

Cy2

−→ 433.

(Similarly, this implies...?)

This proves the Average Theorem! (why??)


Test Vector

Definition (Test Vector)

Let s = {ak}p−1k=0 be a nonnegative sequence. Then we call the vector

(0 + a0, 1 + a1, · · · , (p − 1) + ap−1) mod p

the test vector of s.

Example

Let s = 6424. Then (0 + 6, 1 + 4, 2 + 2, 3 + 4) = (6, 5, 4, 7), so thetest vector is (2, 1, 0, 3).

Let s = 543. Then (0 + 5, 1 + 4, 2 + 3) = (5, 5, 5), so the test vectoris (2, 2, 2).


Permutation Test

Theorem (The Permutation Test)

Let s = {ak}p−1k=0 be a nonnegative sequence, and let φs be the test vectorof s. Then, s is a juggling sequence if and only if φs is a permutation.

Example

Let s = 6424. Then the test vector is (2, 1, 0, 3), so 6424 is a jugglingsequence.

Let s = 444. Then the test vector is (1, 2, 0), so 444 is a jugglingsequence.

Let s = 433. Then the test vector is (1, 1, 2), so 433 is NOT ajuggling sequence.


Idea of the Proof

The proof is based on the flattening algorithm.

The key fact is that:

”permutationability” is preserved under swapping and cyclic shift!!!

Example

Let s = 7531. Then s1,2 = 7441. The test vector of 7531 is(3, 2, 1, 0), and the test vector of 7441 is (3, 1, 2, 0).

Let s = 6534. Then s2,3 = 6552. The test vector of 6534 is(2, 2, 1, 3), and the test vector of 6552 is (2, 1, 3, 1).


Proof of the Permutation test

WTS: s is a juggling sequence if and only if the test vector of s is apermutation.

Example (Flattening Algorithm and Test Vector)

1) 642 (0, 2, 1)Sw−→ 552 (2, 0, 1)

Cy−→ 525 (2, 0, 1)

Sw−→ 345 (0, 2, 1)Cy2

−→ 534 (2, 1, 0)Sw−→ 444 (1, 2, 0).

(This implies ...what?)

2) 514 (2, 2, 0)Sw−→ 244 (2, 2, 0)

Cy2

−→ 424 (1, 0, 0)

Sw−→ 334 (0, 1, 0)Cy2

−→ 433 (1, 1, 2).

(Similarly, this implies...?)

This proves the Permutation Test! :) (why??)


Corollary

Corollary (Vertical Shifts)

Let s = {ak}p−1k=0 be a sequence of nonnegative integers. Let d be an

integer such that s ′ = {ak + d}p−1k=0 is a sequence of nonnegativeintegers. Then s is a juggling sequence if and only if s ′ is a jugglingsequence.

We call this operation the vertical shifts.

Example

Let s = 441. Then s ′ = 996 is also a juggling sequence.


Converse of the Average Theorem

Theorem

Given a finite sequence of nonnegative integers whose average is aninteger, there is a permutation of this sequence that is a juggling sequence.

Example

43210 is NOT a juggling sequence, but 01234, 02413, 03142 are alljuggling sequences.


Scramblable Juggling Sequences

Definition (Scramblable sequences)

Scramblable sequences are juggling sequences that stay juggling sequenceno matter how you rearrange their elements.

(Example: 3333, 1999, 147 are all scramblable sequences)

Theorem

A finite sequence of nonnegative integers is a scramblable jugglingsequence of period p if and only if it is of the form {akp + c}p−1k=0, where cand ak are nonnegative integers.


Magic Juggling Sequences

Definition (Magic juggling sequence)

A magic juggling sequence is a juggling sequence of some period p thatcontains every integer from 0 to p − 1 exactly once.

(Example: 0123456 is a magic juggling sequence)

Theorem

Let p and q be two positive integers such that p is odd, q > 1, and p isrelatively prime to both q and q − 1. Then

{(q − 1)k mod p}p−1k=0

is a magic juggling sequence.

Let q = 2. Then we see that 0123 · · · (p − 1) is a magic jugglingsequence for any odd number p.


How Many Ways to Juggle?

Theorem

The number of all minimal b-ball juggling sequences of period p is

1

p

∑d |p

µ(p

d

)((b + 1)d − bd

),

where µ is the Mobius function.

µ(n) =

1 if n has an even number of distinct prime factors,

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

0 if n has repeated prime factors.

Example: if b = 3, and p = 3, then there are 12 ways.


Points of Intersection

Given a juggling sequence s, let cross(s) be the number of points ofintersection of arcs in the juggling diagram.

Example (1)

Figure : s = 4. cross(s) = ...?



Example (1)

Figure : s = 4. Then cross(s) = 3.



Example (2)

Figure : s = 441. cross(s) = ...?



Example (2)

Figure : s = 441. Then cross(s) = 4.


Affine Weyl Group Ap−1

Definition

Let s = {ai}p−1i=0 be a juggling sequence. Define a map ϕs : Z→ Z by

ϕs : i 7→ ai mod p + i − b,

where b is the number of balls juggled.

Example

Let s = 552. Then b = 4, and we have(· · · −1 0 1 2 · · ·· · · ϕ(−1) ϕ(0) ϕ(1) ϕ(2) · · ·

)=

(· · · −1 0 1 2 · · ·· · · −3 1 2 0 · · ·

)

For any juggling sequence s, ϕs is in fact a permutation of Z.

Denote the set of all permutations arising in this way by Ap−1.



Let us define simple reflection tk : Z→ Z by

i 7→

i + 1 for i = k mod p,

i − 1 for p = k + 1 mod p,

i o.w.

Given σ ∈ Ap−1, let length(σ) be the smallest integer such that σ canbe written as the product of this number of simple reflections.



Theorem

Let s be a b-ball juggling sequence of period p. Then

cross(s) = (b − 1)p − length(ϕs).

Example

Let s = 441. Then b = 3, and

ϕs =

(· · · 0 1 2 · · ·· · · 1 2 0 · · ·

).

Then length(ϕs) = 2.

Therefore, cross(s) = (3− 1) · 3− 2 = 4.


Multiplex Juggling

Multiplex juggling is a natural generalizations of the simple jugglingpatterns.

Definition (Multiplex Juggling Patterns)

Multiplex juggling patterns satisfy

the balls are juggled in a constant beat, that is, the throws occur atdiscrete equally spaced moments in time,

patterns are periodic, and

all the balls that get caught on a beat also get tossed on the samebeat.


Multiplex Juggling

Example

Figure : Juggling diagram of the multiplex juggling sequence [14]1.


The Average Theorem for Multiplex Juggling

Theorem (The Average Theorem)

The number of balls necessary to juggle a multiplex juggling sequenceequals its ”average”.

Example

The number of balls necessary to juggle [14]4 is (1+4)+12 = 3.


JuggleBuddies

Figure : 2015, October, Aldrich park, UCI Illuminations.

We are recruiting new members. Everyone is welcome! =]


JuggleBuddies Information

Webpage: www.jugglebuddies.webs.com

Facebook: https://www.facebook.com/groups/858609887507070/


Thank you! :)


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