Matthew Wright slides also by John Chase

1. The Mathematics Juggling of Matthew Wright slides also by John Chase 1 4 5 1 4 1 4 5 1 4 4 5

2. Basic Juggling Patterns Axioms: • The juggler must juggle at a constant rhythm. • Only one throw may occur on each beat of the pattern. • Throws on odd beats must be made from the right hand; throws on even beats from the left hand. • The pattern juggled must be periodic. It must repeat. It must repeat. • All balls must be thrown to the same height. Example:basic 3-ball pattern arcs represent throws dots represent beats 1 2 4 5 6 7 8 9 ∙∙∙ 3

3. Basic 3-ball Pattern 1 2 4 5 6 7 8 9 ∙∙∙ 3 Notice: balls land in the opposite hand from which they were thrown Basic 4-ball Pattern 1 2 4 5 6 7 8 9 ∙∙∙ 3 Notice: balls land in the same hand from which they were thrown

4. The Basic -ball Patterns If is odd: • Each throw lands in the opposite hand from which it was thrown. • These are called cascade throws. If is even: • Each throw lands in the same hand from which it was thrown. • These are called fountain throws.

5. Let’s change things up a bit… Axioms: • The juggler must juggle at a constant rhythm. • Only one throw may occur on each beat of the pattern. • Throws on even beats must be made from the right hand; throws on odd beats from the left hand. • The pattern juggled must be periodic. It must repeat. It must repeat. • All balls must be thrown to the same height. What if we allow throws of different heights? Axioms 1-4 describe the simple juggling patterns.

6. Example Start with the basic 4-ball pattern: Concentrate on the landing sites of two throws. • Now swap them! • The first 4-throw will land a count later, making it a 5-throw. • The second 4-throw will land a count earlier, making it a 3-throw. • This is called a site swap.

7. Juggling Sequences Site swaps allow us to obtain many simple juggling patterns, starting from the basic juggling patterns. We describe each simple juggling pattern by a juggling sequence: a sequence of integers corresponding to the sequence of throws in the juggling pattern. The length of a juggling sequence is its period. A juggling sequence is minimalif it has minimal period among all juggling sequences representing the same pattern. Example:the juggling sequence 441 4 4 4 4 4 4 1 1 1 1 ∙∙∙ 2 3 4 5 6 7 8 9

8. Juggling Sequences 2-balls: 31, 312, 411, 330 3-balls: 441, 531, 51, 4413, 45141 4-balls: 5551, 53, 534, 633, 71 5-balls: 66661, 744, 75751 1 4 ∙∙∙ 5 1 4 1 4 5 1 4 4 5

9. Is every sequence a juggling sequence? No. Consider the sequence 54. collision! 5 4 Clearly, a 5-throw followed by a 4-throw results in a collision. In general, an -throw followed by an -throw results in a collision.

10. ? How do we know if a given sequence is jugglable? For instance, is 6831445 a jugglable sequence?

11. A juggling function is a function: This function tells us what throw to make on each beat. That is, on beat , we juggle a -throw, for each integer . The sequence described by this function is jugglable if and only if the function is a permutation of the integers. Two important properties of juggling functions: 1. Height of the highest throw: 2. Number of balls required to juggle the corresponding sequence: number of balls required to juggle

12. How many balls are required to juggle a given sequence? The Average Theorem: Let be a juggling function with finite height. Then exists, is finite, and is equal to , where the limit is over all integer intervals , and is the number of integers in . Proof:

13. interval , with minimum contribution of any particular ball to maximum contribution of any particular ball to Proof: The left and right expressions tend to as tends to infinity.

14. How many balls are required to juggle a given sequence? The Average Theorem: Corollary: The number of balls necessary to juggle a juggling sequence equals its average. Application: A finite juggling sequence must have an integer average. Examples: 534 441 7531 75751 352 5-ball pattern not jugglable! 4-ball pattern 3-ball pattern 4-ball pattern

15. How can we change one juggling sequence into another? We could perform a site swap. Consider the sequence of nonnegative integers: If , we can swap the landing positions of the balls thrown on beats and to obtain the sequence : Notice: • The sequence is a juggling sequence if and only if is. • The average of is the same as the average of . • If is a juggling sequence, then the number of balls used to juggle equals the number of balls used to juggle .

16. How can we change one juggling sequence into another? We could perform a cyclic shift. Again, let be a sequence of nonnegative integers: Now move the last element, , to the beginning of the sequence to obtain the sequence : Notice: • The sequence is a juggling sequence if and only if is. • The average of is the same as the average of . • If is a juggling sequence, then the number of balls used to juggle equals the number of balls used to juggle .

17. The Flattening Algorithm Let be a sequence of nonnegative integers: The flattening algorithm transforms into a new sequence as follows: • If is a constant sequence, stop and output this sequence. Otherwise, • use cyclic shifts to arrange the elements of such that a maximum integer in , say , is at position 0 and a non-maximum integer in , say , is at position 1. If , stop and output this sequence. Otherwise, • perform a site swap of positions 0 and 1. Redefine to be the resulting sequence, and return to step 1.

18. The Flattening Algorithm also jugglable! Example: start with the sequence 642 swap shift swap shift swap jugglable! 642 552 525 345 534 444 also not jugglable Example: start with the sequence 514 swap shift swap shift not jugglable 514 244 424 334 443 • Observe: • The Flattening Algorithm can be used to decide whether or not a sequence is jugglable. • If the input is a -ball juggling sequence with period , this algorithm outputs the basic -ball sequence of period . • If the input is not a juggling sequence, the program stops at step 2 and outputs a sequence of the form .

19. How do we know if a given sequence is jugglable? Theorem: Let , for , be a sequence of nonnegative integers and let . Then, is a juggling sequence if and only if the function defined is a permutation of the set . • Example: Show 534 is a valid juggling sequence. • Let . The period is 3, so . Note . • Then • This is a permutation of , so 534 is a valid juggling sequence.

20. Theorem: Let , for , be a sequence of nonnegative integers and let . Then, is a juggling sequence if and only if the function defined is a permutation of the set . Proof: The function is a permutation if and only if the vector contains all of the integers from to . Suppose we apply site swaps and cyclic permutations to the sequence to obtain sequence with corresponding vector . Then contains all of the elements of if and only if does. Therefore, given a sequence , apply the flattening algorithm to obtain . Then is a juggling sequence if and only if is a constant sequence, if and only if contains all of the elements of .

21. ? How many ways are there to juggle? Infinitely many. (Consider the basic -ball sequences for each integer .) How many -ball juggling sequences are there with period ?

22. How many -ball juggling sequences are there of period ? : There is one unique sequence, namely, 1. 1 : Starting with the sequence 22, we can perform site swaps to obtain two other sequences, 31 and 40(unique up to cyclic shifts). 2 2 3 1 4 0 : Starting with 333and performing site swaps, we (eventually) obtain 13 sequences (unique up to cyclic shifts).

23. How many -ball juggling sequences are there of period ? 3 0 6 5 1 0 4 2 6 1 1 5 4 4 7 1 1 2 2 5 3 0 8 3 7 2 0 1 4 2 0 3 6 : Starting with 333and performing site swaps, we (eventually) obtain 13 sequences (unique up to cyclic shifts). 3 0 3 3 0 9

24. Is there a better way to count juggling sequences? Suppose we have a large number of each of the following juggling cards: These cards can be used to construct all juggling sequences that are juggled with at most three balls.

25. Example: juggling sequence 441 juggling diagram ∙∙∙ 4 ∙∙∙ 4 1 4 4 1 4 4 1 constructed with juggling cards 4 4 1 4 4 1 4 4 1

26. Counting Juggling Sequences With many copies of these four cards, we can construct any (not-necessarily minimal) juggling sequences that is juggled with at most three balls. Similarly, with many copies of distinct cards, we can construct any (not-necessarily minimal) juggling sequence that is juggled with at most balls. Lemma: The number of all juggling sequences of period , juggled with at most balls, is:

27. Counting Juggling Sequences Lemma: The number of all juggling sequences of period , juggled with at most balls, is: It follows that: Lemma: The number of all -ball juggling sequences of period is: However, we have counted each cyclic permutation of every sequence, as well as non-minimal sequences. How can we count the minimal -ball juggling sequences of period , not counting cyclic permutations of the same sequence as distinct?

28. Counting Juggling Sequences Theorem: The number of all minimal -ball juggling sequences of period , with , is if cyclic permutations of juggling sequences are not counted as distinct. Here, denotes the Möbiusfunction: Proof: If divides , then each minimal juggling sequence of period gives rise to exactly sequences of period . Thus, The expression for follows by Möbius inversion.

29. ? Questions?

30. Reference: BurkardPolster. The Mathematics of Juggling. Springer, 2003. Juggling Simulators: • www.quantumjuggling.com • jugglinglab.sourceforge.net