1. 4154154141 4 5 Matthew Wright slides also by John Chase TheMathematics of Juggling

2. Basic Juggling Patterns Axioms: 1.The juggler must juggle at a constant rhythm. 2.Only one throw may occur on each beat of the pattern. 3.Throws on odd beats must be made from the right hand; throws on even beats from the left hand. 4.The pattern juggled must be periodic. It must repeat. It must repeat. 5.All balls must be thrown to the same height. 132457689∙∙∙ Example: basic 3-ball pattern dots represent beats arcs represent throws

3. 1 3 2457689∙∙∙ Basic 3-ball Pattern Basic 4-ball Pattern Notice: balls land in the opposite hand from which they were thrown 1 3 2457689 ∙∙∙ Notice: balls land in the same hand from which they were thrown

5. Let’s change things up a bit… Axioms: 1.The juggler must juggle at a constant rhythm. 2.Only one throw may occur on each beat of the pattern. 3.Throws on even beats must be made from the right hand; throws on odd beats from the left hand. 4.The pattern juggled must be periodic. It must repeat. It must repeat. 5.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 minimal if it has minimal period among all juggling sequences representing the same pattern. Example: the juggling sequence 441 132457689∙∙∙ 4 4 1 44 1 44 1

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 415415414∙∙∙ 1 4 5

9. Is every sequence a juggling sequence? No. Consider the sequence 54. 54 collision!

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

12. How many balls are required to juggle a given sequence?

14. 534441757517531 4-ball pattern 352 3-ball pattern 4-ball pattern 5-ball pattern not jugglable!

15. How can we change one juggling sequence into another?

17. The Flattening Algorithm

18. 444 The Flattening Algorithm Example: start with the sequence 642 642552525345534 Example: start with the sequence 514 swapshiftswapshiftswap jugglable! also jugglable! 514244424334443 swapshiftswapshift not jugglable also not jugglable

19. How do we know if a given sequence is jugglable?

21. How many ways are there to juggle? Infinitely many.

22. 2 3 14 01

23. 0 90 8 10 0 27 1 17 1 62 5 22 3 60 5 40 1 35 0 63 1 44 3 24 3 33

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. 4 14444141 ∙∙∙ Example: juggling sequence 441 414444141 juggling diagram constructed with juggling cards

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.

27. Counting Juggling Sequences

29. Questions?

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