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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).
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Club Juggling
Figure : Korynn Aguilar (Chair), April 2015, UC Irvine, One World Concert.
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Passing
Figure : October 2015, University Hills, Fall Fiesta.
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(Dragon) Ball Juggling
Figure : Vegeta, juggling 7 (dragon) balls
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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.
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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.
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Example (1)
Figure : Juggling diagram of the 4-ball fountain.
This pattern is denoted by 4.
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Example (2)
Figure : Juggling diagram of the 3-ball shower.
This pattern is denoted by ...?
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Example (3)
Figure : 4-ball shower.
This pattern is denoted by 71.
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Example (4)
This pattern is denoted by ...?
The number of ball juggled is ...?
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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)
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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.
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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).
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Swapping
i j i j
Figure : a juggling sequence s is transformed into another juggling sequence si,j .
This operation is called swapping.
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Example (1)
Let s = 441. Then s0,2 = ...?
Figure : s to s0,2.
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Example (1)
Figure : 441 to 342.
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Example (2)
Let s = 7531. Then s1,3 = ...?
Figure : s to s1,3.
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Example (3)
Let s = 6313. Then s1,2 = 6223.
Figure : s to s1,2.
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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
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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).
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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.
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Getting Sleeply? It’s Show Time! :)
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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.
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Example
Let s = 441. Then s = 414.
Figure : 441 to 414.
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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).
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Summery!!!
Swapping and cyclic shifts both preserve
1 ”jugglibility”,
2 the average, and
3 the number of balls (if it is a juggling sequence).
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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.
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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??)
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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).
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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.
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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).
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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??)
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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.
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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.
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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.
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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.
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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.
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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) = ...?
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Points of Intersection
Example (1)
Figure : s = 4. Then cross(s) = 3.
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Points of Intersection
Example (2)
Figure : s = 441. cross(s) = ...?
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Points of Intersection
Example (2)
Figure : s = 441. Then cross(s) = 4.
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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.
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Affine Weyl Group 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.
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Affine Weyl Group Ap−1
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.
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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.
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Multiplex Juggling
Example
Figure : Juggling diagram of the multiplex juggling sequence [14]1.
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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.
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JuggleBuddies
Figure : 2015, October, Aldrich park, UCI Illuminations.
We are recruiting new members. Everyone is welcome! =]
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JuggleBuddies Information
Webpage: www.jugglebuddies.webs.com
Facebook: https://www.facebook.com/groups/858609887507070/
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Thank you! :)
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