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, cigar boxes, diabolos, devil sticks, yoyos, etc). Yuki Takahashi (UC Irvine) November 12, 2015 2 / 51 Club Juggling Figure : Korynn Aguilar (Chair), April 2015, UC Irvine, One World Concert. Yuki Takahashi (UC Irvine) November 12, 2015 3 / 51 Passing Figure : October 2015, University Hills, Fall Fiesta. Yuki Takahashi (UC Irvine) November 12, 2015 4 / 51 (Dragon) Ball Juggling Figure : Vegeta, juggling 7 (dragon) balls Yuki Takahashi (UC Irvine) November 12, 2015 5 / 51 Papers about Siteswaps A. Engsr¨om,L. Leskel¨a,H. Varpanen, Geometric juggling with q-analogues, Discrete Math. 338 (2015), 1067{1074. A. Ayyer, B, Arvind, S. Corteel, F. Nunzi, Multivariate juggling probabilities, Electron. J. Probab. 20 (2015). A. Knutson, T. Lam, D. Speyer, Positroid varieties: Juggling and geometry, 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. Yuki Takahashi (UC Irvine) November 12, 2015 6 / 51 Juggling Patterns Definition (Simple Juggling Patterns) In this talk, we assume the balls are juggled in a constant beat, that is, the throws occur at discrete 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. Yuki Takahashi (UC Irvine) November 12, 2015 7 / 51 Example (1) Figure : Juggling diagram of the 4-ball fountain. This pattern is denoted by 4. Yuki Takahashi (UC Irvine) November 12, 2015 8 / 51 Example (2) Figure : Juggling diagram of the 3-ball shower. This pattern is denoted by ...? Yuki Takahashi (UC Irvine) November 12, 2015 9 / 51 Example (3) Figure : 4-ball shower. This pattern is denoted by 71. Yuki Takahashi (UC Irvine) November 12, 2015 10 / 51 Example (4) This pattern is denoted by ...? The number of ball juggled is ...? Yuki Takahashi (UC Irvine) November 12, 2015 11 / 51 Terminologies We call a finite sequence of non-negative integers arising from a juggling 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 the juggling sequences representing the same juggling pattern. (Example: 144 is minimal, but 441441 is not) Yuki Takahashi (UC Irvine) November 12, 2015 12 / 51 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. Yuki Takahashi (UC Irvine) November 12, 2015 13 / 51 The Average Theorem Theorem (The Average Theorem) The number of balls necessary to juggle a juggling sequence is its average. If the average is not an integer, then the sequence is not a juggling sequence. Example The number of balls necessary to juggle 441 is 3. 5+6+2 562 is NOT a juggling sequence (note that 3 = 4:3333::: is not an integer). Yuki Takahashi (UC Irvine) November 12, 2015 14 / 51 Swapping i j i j Figure : a juggling sequence s is transformed into another juggling sequence si;j . This operation is called swapping. Yuki Takahashi (UC Irvine) November 12, 2015 15 / 51 Example (1) Let s = 441. Then s0;2 = :::? Figure : s to s0;2. Yuki Takahashi (UC Irvine) November 12, 2015 16 / 51 Example (1) Figure : 441 to 342. Yuki Takahashi (UC Irvine) November 12, 2015 17 / 51 Example (2) Let s = 7531. Then s1;3 = :::? Figure : s to s1;3. Yuki Takahashi (UC Irvine) November 12, 2015 18 / 51 Example (3) Let s = 6313. Then s1;2 = 6223. Figure : s to s1;2. Yuki Takahashi (UC Irvine) November 12, 2015 19 / 51 Swapping Definition (Swapping) p−1 Let s = fak gk=0 be a sequence of nonnegative integers. Let i and j be integers such that 0 ≤ i < j ≤ p − 1 and j − i ≤ ai . Let si;j be 8 a + j − i if k = i <> j si;j (k) = ai − (j − i) if k = j > :ak o.w. i j i j Yuki Takahashi (UC Irvine) November 12, 2015 20 / 51 Summary of Swapping For a juggling sequence s, we denote the number of balls necessary to juggle it by ball(s). Then The sequence s is a juggling sequence if and only if si;j is a juggling sequence. 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 ). Yuki Takahashi (UC Irvine) November 12, 2015 21 / 51 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: p−1 Let s = fak gk=0 be a sequence of nonnegative integers. Then 8 a + j − i if k = i <> j si;j (k) = ai − (j − i) if k = j > :ak o.w. Yuki Takahashi (UC Irvine) November 12, 2015 22 / 51 Getting Sleeply? It's Show Time! :) Yuki Takahashi (UC Irvine) November 12, 2015 23 / 51 Cyclic Shifts Definition (Cyclic Shifts) p−1 Let s = fsk gk=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. Yuki Takahashi (UC Irvine) November 12, 2015 24 / 51 Example Let s = 441. Then ^s = 414. Figure : 441 to 414. Yuki Takahashi (UC Irvine) November 12, 2015 25 / 51 Summery of Cyclic Shifts Apparantly, we have the following: The sequence s is a juggling sequence if and only if ^s is a juggling sequence. The average of s is the same as the average of ^s. If s is a juggling sequence, then ball(s) = ball(^s). Yuki Takahashi (UC Irvine) November 12, 2015 26 / 51 Summery!!! Swapping and cyclic shifts both preserve 1 "jugglibility", 2 the average, and 3 the number of balls (if it is a juggling sequence). Yuki Takahashi (UC Irvine) November 12, 2015 27 / 51 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 of maximum height, say e, comes to rest at beat 0 and one not of maximum height, say f , comes to rest at beat 1. If e and f differ by 1, stop and output this new sequence. Otherwise, 3. perform a swapping of beats 0 and 1, and return to step 1. Yuki Takahashi (UC Irvine) November 12, 2015 28 / 51 Examples! =) Denote the map s 7! s0;1 by Sw, and cyclic shifts s 7! ^s by Cy. Example (flattening algorithm) Cy Cy Cy 2 1) 264 −! 642 −!Sw 552 −! 525 −!Sw 345 −! 534 −!Sw 444: (This implies ...what?) Cy 2 Cy 2 2) 514 −!Sw 244 −! 424 −!Sw 334 −! 433: (Similarly, this implies...?) This proves the Average Theorem! (why??) Yuki Takahashi (UC Irvine) November 12, 2015 29 / 51 Test Vector Definition (Test Vector) p−1 Let s = fak gk=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 the test vector is (2; 1; 0; 3). Let s = 543. Then (0 + 5; 1 + 4; 2 + 3) = (5; 5; 5), so the test vector is (2; 2; 2). Yuki Takahashi (UC Irvine) November 12, 2015 30 / 51 Permutation Test Theorem (The Permutation Test) p−1 Let s = fak gk=0 be a nonnegative sequence, and let φs be the test vector of 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 juggling sequence. Let s = 444. Then the test vector is (1; 2; 0), so 444 is a juggling sequence. Let s = 433. Then the test vector is (1; 1; 2), so 433 is NOT a juggling sequence. Yuki Takahashi (UC Irvine) November 12, 2015 31 / 51 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). Yuki Takahashi (UC Irvine) November 12, 2015 32 / 51 Proof of the Permutation test WTS: s is a juggling sequence if and only if the test vector of s is a permutation. Example (Flattening Algorithm and Test Vector) Cy 1) 642 (0; 2; 1) −!Sw 552 (2; 0; 1) −! 525 (2; 0; 1) Cy 2 −!Sw 345 (0; 2; 1) −! 534 (2; 1; 0) −!Sw 444 (1; 2; 0): (This implies ...what?) Cy 2 2) 514 (2; 2; 0) −!Sw 244 (2; 2; 0) −! 424 (1; 0; 0) Cy 2 −!Sw 334 (0; 1; 0) −! 433 (1; 1; 2): (Similarly, this implies...?) This proves the Permutation Test! :) (why??) Yuki Takahashi (UC Irvine) November 12, 2015 33 / 51 Corollary Corollary (Vertical Shifts) p−1 Let s = fak gk=0 be a sequence of nonnegative integers.