Matthew Wright slides also by John Chase

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

2. Juggling & Mathematics???? • Juggling Sequences • You will understand juggle-speak, knowing precisely how the pattern 45141 should be juggled. • You will know some basic theorems involving juggling sequences, juggling functions, and site swaps. • Juggling States • You will learn to read juggling states like 110101. • We will use states to find juggling sequences, making state transition matrices. • Synchronous Juggling • We will see how to model synchronous patterns like (6x,4)(4,6x).

3. Simple Juggling Patterns • The juggler must juggle at a constant rhythm. • The pattern juggled must be periodic. It must repeat. It must repeat. • Only one throw may occur on each beat of the pattern.

4. The Basic b-ball Patterns • The 3-ball pattern: • Each ball is thrown every third throw. • Each throw last 3 beats. • Each throws is called a 3-throw. • Notice that a 3-throw lands in the opposite hand. • The 4-ball pattern: • Throws in this pattern are called 4-throws. • Notice that a 4-throw lands in the same hand.

5. The Basic b-ball Patterns Can you guess the rule? All odd throws land in the opposite hand. (These are called cascade throws.) All even throws land in the same hand. (These are called fountain throws.) (This is, of course, assuming that you juggle a simple juggling pattern with 2 hands R, L, R, L, R …. This rule would not holdif you juggled with only 1 hand, or 3, or 1 hand and 2 feet, etc.).

6. What is site swap notation? • Invented simultaneously by three groups of people, circa 1985: • Bengt Magnusson and Bruce Tiemann • Paul Klimak • Adam Chalcraft, Mike Day, and Colin Wright It looks like this: 91 441 744 45141 8 75 3 534 75751 b97531 These are juggling sequences!

7. After this slide, you will understand how to read juggling sequences. Now consider the following pattern: … 4, 4, 1, 4, 4, 1, … Each integer corresponds to the height of a throw. For the example above, the juggler would make a 4-throw, followed by a 4-throw, then a 1-throw, a 4-throw, a 4-throw, etc. This pattern is called 441 (we use the minimal representation, since it’s periodic).

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

9. Juggling Diagrams • We often depict juggling sequences by juggling diagrams. arcs represent throws 4 4 4 4 4 4 1 1 1 dots indicate beats 1 ∙∙∙ 2 3 4 5 6 7 8 9 A sequence of connected arcs is called an orbit, representing the throws made by one particular ball. Colors help keep track of orbits.

10. What does “site swap” mean? A “site swap” is simply an exchange in position of landing sites. Take the standard 4-ball pattern, for instance: 4 4 4 4 4 4 4 4 …and concentrate on the landing site of two of the throws. • 4 4 • • • • • • 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. • 5 3 • • • 4 4 • • • • •

11. Is every sequence a juggling sequence? No. Consider the sequence 544. 5 4 4 5 4 4 Clearly, making a 5-throw, followed by a 4-throw, results in a collision. I will explain later how to determine whether or not a given sequence is a juggling sequence.

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

13. Juggling Function A juggling function is a function: j : Z → Z≥0 This function tells us what throw to make on each beat. That is, on beat i, we juggle a j(i)-throw, for each integer i. The infinite sequence described by this function is jugglable if and only if the function j+: Z → Z j+(i) = i + j(i) is a permutation (that is, a rearrangement) of the integers.

14. Juggling Function Two important questions to ask about juggling functions: How high is the highest throw? height(j) = How many balls are required to juggle this function? balls( j) = number of orbits in the juggling diagram of j

15. The Average Theorem Question: How many balls are required to juggle a given sequence? Theorem: Let jbe a juggling function. If height(j) is finite, then exists, is finite, and is equal to balls(j), where the limit is over all integer intervals I = {a, a + 1, a + 2, ..., b} of integers, and |I| = b – a + 1 is the number of integers in I. Proof: The left and right expressions tend to balls(j) as |I| tends to infinity.

16. 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.

17. The Average Theorem • The number of balls necessary to juggle a sequence equals its average. • Example: 534 • ( 5 + 3 + 4 ) / 3 = 12 / 3 = 4 • So 534 is a 4-ball pattern. • Try these! • 441753175751 • 34 5 • The Average Theorem provides a test to see if a number sequence is not a siteswap. If the number of balls is not a whole number, then the sequence is unjugglable(consider 352, for instance).

18. Site Swaps, More Formally • Question: How can we change one juggling sequence into another? • Consider the sequence s of p nonnegative integers: • s: a0, a1, ..., ai–1, ai, ai+1, ..., ai+d–1, ai+d, ai+d+1, ..., ap–1 • If 0 < d ≤ ai, we can swap the landing positions of the balls thrown on beats i and j to obtain the sequence s': • s': a0, a1, ..., ai–1, ai+d + d, ai+1, ..., ai+d–1, ai – d , ai+d+1, ..., ap–1 • We notice: • The sequence s is a juggling sequence if and only if s' is. • The average of s is the same as the average of s'. • If s is a juggling sequence, then the number of balls used to juggle sequals the number of balls used to juggle s'.

19. Cyclic Shifts • Again, let s be a sequence of p nonnegative integers: • s: a0, a1, a2, ..., ap–1 • Now move the last element, ap–1, to the beginning of the series to obtain the sequence s→: • s→: ap–1, a0, a1, a2, ..., ap–2 • Just as with site swaps, we notice: • The sequence s is a juggling sequence if and only if s→ is. • The average of s is the same as the average of s→. • If s is a juggling sequence, then the number of balls used to juggle s is the same as the number of balls used to juggle s→.

20. The Flattening Algorithm • The flattening algorithm takes an arbitrary sequence s of • p≥ 1 nonnegative integers and transforms it into a new sequence as follows: • If s is a constant sequence, stop and output this sequence. Otherwise, • use cyclic shifts to arrange the elements of s such that one of maximum height, say m, comes to rest at beat 0 and one not of maximum height, say n, comes to rest at beat 1. If mand ndiffer by only 1, stop and output this sequence. Otherwise, • perform a site swap of beats 0 and 1. Redefine s to be the resulting sequence, and return to step 1.

21. The Flattening Algorithm The Flattening Algorithm can be used to decide whether or not a sequence is jugglable. If the input is a juggling sequence with average b, this algorithm outputs the b sequence of period p. If the input is not a juggling sequence, the program stops at step 2 and outputs a sequence of the form m, m– 1, ...

22. Flattening Algorithm: Example Suppose we run the flattening algorithm on the sequence 642: However, suppose we run the algorithm on the sequence 514: collision!

23. Modular Arithmetic • In arithmetic modulo n, we reduce numbers to their remainder after division by n. • For example: Why? • 7 modulo 5 is equal to 2 • If you divide 7 by 5, the remainder is 2. 7 (mod 5) = 2 • 9 modulo 4 equals 1 • If you divide 9 by 4, the remainder is 1. 9 (mod 4) = 1 What is 15 (mod 6) ? 15 (mod 6) = 3 8 (mod 2) = 0 How about 8 (mod 2) ?

24. Modular Arithmetic • You frequently use modular arithmetic when you think about time. 12 • If it is 10:00 now, what time will it be in 4 hours? 9 3 10 + 4 (mod 12) = 2 so it will be 2:00 6 • Today is November 11. In 25 days, what will be the date? 11 + 25 (mod 30) = 6 so it will be December 6

25. The Permutation Test • Theorem: Let s = {ak}, for integers k from 0 to p – 1, be a sequence of nonnegative integers and let [p] = {0, 1, 2, …, p - 1}. Then, s is a juggling sequence if and only if the function • s:[p] → [p] • s (i) = i+ ai (mod p) • operated on every element of s, is a permutation of the set [p]. • Example: Show 534 is a valid juggling sequence. • Let s = {5, 3, 4}. The period is 3, so p = 3. Note [p] = {0, 1, 2}. • Then { s(5), s(3), s(4) } • = { (0 + 5) mod 3, (1 + 3) mod 3, (2 + 4) mod 3 } • = { 5 mod 3, 4 mod 3, 6 mod 3 } • = { 2, 1, 0 } • This is a permutation of [p], so 534 is a valid juggling sequence.

26. The Permutation Test Theorem: Let s = {ak} as k goes from 0 to p – 1 be a sequence of nonnegative integers and let [p] = {0, 1, 2, …, p - 1}. Then, s is a juggling sequence if and only if the function s:[p] → [p] s (i) = i + ai (mod p) operated on every element of s, is a permutation of the set [p]. Proof: The function s is a permutation if and only if the vector v = (s(0), s(1), s(2), ..., s(p – 1)) contains all of the integers from 0 to p – 1. Suppose we apply site swaps and cyclic permutations to the sequence s to obtain sequence s' with corresponding vector v'. Then v'contains all of the elements of [p] if and only if v does. Therefore, given a sequence s, apply the flattening algorithm to obtain s'. Then s is a juggling sequence if and only if s' is a constant sequence, if and only if v' contains all of the elements of [p].

27. Constructing the 3-Ball Juggling Sequences of Period 3

28. 3 0 6 1 2 6 5 0 4 1 7 1 5 1 2 2 4 4 1 5 3 8 0 1 0 3 7 2 4 2 0 3 6 0 0 9 3 3 3 Site Swap Graph The 13 (up to cyclic shifts) 3-ball juggling sequences of period 3:

29. How Many Ways to Juggle? How many juggling sequences are there? Infinitely many (consider the b-ball sequences for each integer b ∈ N) More interesting results The number of all juggling sequences of period p and at most b balls is S≤(b,p) = (b + 1)p The number of all b-ball juggling sequences of period p is S(b,p) = S≤(b,p) – S≤(b – 1,p) = (b + 1)p – bp The number of all minimal b-ball juggling sequences of period p, with b ≥ 1, is M(b, p) = if cyclic permutations of juggling sequences are not counted as distinct. Here, μ denotes the Möbius function.

30. Juggling States Though closely related to juggling sequences, juggling states are sequences of numbers that mean something different. They are not the same as juggling sequences. Juggling states are landing schedules. Juggling states look like this: 001011 111 111001 10111 10011 11111 10101

31. Ball lands Ball lands No ball lands Ball lands No ball lands Ball lands After this slide, you will understand how to read juggling states. A juggling state is a landing schedule. It tells you what event will occur on each of the following beats. We use a 1 to denote that a ball will land. We use a 0 to denote that a ball will not land. Examples The standard 3-ball cascade continually enters the state 111. This means that a ball is expected to land, then another ball will land, then another. Here’s a more complicated state. Clearly, this is a 4-ball state: 1 1 0 1 0 1

32. State Transitions How does one enter or exit a certain state? When you make a 4-throw from the state 111, take off the first number listed in the state and concatenate it so that it is in the 4th position. 1 1 0 1 Consider the juggling sequence 441 (remember, this is a juggling sequence, not a juggling state!). This is thrown out of the 3-ball ground state, 111. 4 1 4 1110 1101 1011 1110 4 1 4 1101 1011 1110 Note: The state 111 is the same as 1110 or 11100 or 111000. Adding zeros to the end of a state does not affect the state. In doing so, we just “state” the obvious .

33. 6 1 5 110110 101101 111010 Ground and Excited States We said that 441 is thrown out of the 3-ball ground state, 111. We call 441 a ground state pattern because it enters from and exits to the ground state. Ground State: The basic b-ball state (i.e.. 111) Excited State: A state with “holes” in it (i.e. 1011) But not all patterns can be entered from the ground state. Such patterns we call excited state patterns. Consider the excited state pattern 561, which must be entered from the 11101 state. 6 1 5 111010 110110 101101 111010 Juggling states help you see what throws are available given a certain state. For instance, in the state 11101, we see that a 3-throw, a 5-throw, or any greater throw is possible but that a 1-throw, 2-throw or 4-throw would produce a collision.

34. State Transition Matrix We can use a state transition matrix to show us in what ways a juggling state can be entered and exited. A 3-ball transition matrix with maximum throw height 5: Pick a state to start from. Let’s pick the ground state, 111. Choose an available throw from the row. Follow the columnto the gray box. This will put you in the row associated with the new state. And so on… A valid ground state jugging sequence: 531

35. State Transition Matrix

36. Synchronous Juggling Synchronous patterns break the rule that only one throw may be made at a time. With synchronous patterns, throws are made simultaneously with the right and left hands. Example: The 4-ball synchronous fountain The juggling sequence notation is (4,4), denoting that a 4-throw is being made in both the right and the left on the same beat (left, right). We denote a crossing throw by attaching an “x” to the crossing throw. For instance, the standard 4-ball crossing synch pattern is (4x,4x). Each beat in a synch pattern lasts two beats. So there are no odd throws. Examples: (4,2x)(2x,4) (4,6x)(2x,4) (8,4)(4,8)

37. Computers can juggle too. jugglinglab.sourceforge.net

38. Review • Hopefully you learned something… • Juggling Sequences • You now know precisely how the pattern 45141 should be juggled • You should be familiar with some basic juggling sequence theorems. • Juggling States • You know that 110101 is a simple landing schedule. • We saw how juggling sequences could easily be created with state matrices. • Synchronous Juggling • You know about synch juggling patterns like (6x,4)(4,6x).

39. References The Mathematics of Juggling by BurkardPolster Juggling Lab http://jugglinglab.sourceforge.net/

40. Questions?