4 15 4 1 54 1 4 1 4 5

Matthew Wright

slides also by John Chase

The

MathematicsofJuggling

Basic Juggling PatternsAxioms: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.

1 32 4 5 76 8 9 ∙∙∙

Example: basic 3-ball pattern

dots represent

beats

arcs represent

throws

1 32 4 5 76 8 9 ∙∙∙

Basic 3-ball Pattern

Basic 4-ball Pattern

Notice: balls land in the opposite hand from which they were thrown

1 32 4 5 76 8 9 ∙∙∙Notice: balls land in the same hand from which they were thrown

The Basic -ball PatternsIf 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.

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.

ExampleStart 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.

Juggling SequencesSite 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

1 32 4 5 76 8 9 ∙∙∙

44

1

4 4

1

4 4

1

Juggling Sequences2-balls: 31, 312, 411, 330

3-balls: 441, 531, 51, 4413, 45141

4-balls: 5551, 53, 534, 633, 71

5-balls: 66661, 744, 75751

4 15 4 1 54 1 4 ∙∙∙1 4 5

Is every sequence a juggling sequence?

No.Consider the sequence 54.

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.

5 4

collision!

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

For instance, is 6831445 a jugglable sequence?

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

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:

Proof:

The left and right expressions tend to as tends to infinity.

interval , with

height ( 𝑗 )

minimum contribution of any particular ball to

maximum contribution of any particular ball to

height ( 𝑗 )

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 7575175314-ball

pattern

3523-ball

pattern4-ball

pattern5-ball

patternnot

jugglable!

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 .

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 .

The Flattening AlgorithmLet be a sequence of nonnegative integers:

The flattening algorithm transforms into a new sequence as follows:1. If is a constant sequence, stop and output this sequence.

Otherwise,2. 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,

3. perform a site swap of positions 0 and 1. Redefine to be the resulting sequence, and return to step 1.

444

The Flattening AlgorithmExample: start with the sequence 642

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 .

642 552 525 345 534

Example: start with the sequence 514

swap shift swap shift swap jugglable!

also jugglable!

514 244 424 334 443swap shift swap shift not jugglable

also not jugglable

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.

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 .

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 .

?How many ways are

there to juggle?

(Consider the basic -ball sequences for each integer .)Infinitely many.

How many -ball juggling sequences are there with period ?

How many -ball juggling sequences are there of period ?

: There is one unique sequence, namely, 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 333 and performing site swaps, we (eventually) obtain 13 sequences (unique up to cyclic shifts).

1

How many -ball juggling sequences are there of period ?

: Starting with 333 and performing site swaps, we (eventually) obtain 13 sequences (unique up to cyclic shifts).

090

810

027

117

162

522

360

540

1350

63

144

324

333

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.

4 14 4 4 41 4 1 ∙∙∙∙∙∙

Example: juggling sequence 441

4 14 4 4 41 4 1

juggling diagram

constructed with juggling

cards

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:

Counting Juggling SequencesWith many copies of these four cards, we can construct any (not-necessarily minimal) juggling sequences that is juggled with at most three balls.

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?

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 Möbius function:

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.

Counting Juggling Sequences

?Questions?

Reference:Burkard Polster. The Mathematics of Juggling. Springer, 2003.

Juggling Simulators:•www.quantumjuggling.com• jugglinglab.sourceforge.net