Publications by Jochen Voss. A complete list of my pub­li­ca­tions. … read more

Juggling patterns. Patterns for Ball Juggling … read more

The Physics of Ball Juggling. A short description of some of the physical properties of juggling patterns, including Shannon's juggling theorems. … read more

# The Mathematical Theory of Juggling

By Jochen Voss, last updated 2012-02-18

The purpose of this text is to explain enough of the theory of juggling, that you can understand the juggling patterns which are explained on my juggling web pages. This page does not intend to give a complete explanation of site-swaps and indeed there are many interesting facts omitted. If you want to know more, Burkhard Polster's book The Mathematics of Juggling is a good reference. The references section below also contains some links to online resources.

## Site-swap Notation

### Numbers Describe Throws

The site-swap notation is an abstract method to describe different juggling patterns. The basic idea is to encode each throw of the pattern as a single number, the larger the number the higher the throw. For example

```3 3 3 3 3 3 4 4 1 3 3 3 3 3 3
```

describes a sequence of 15 throws, where most of them have the same height, but in the middle of the sequence there are two higher throws and a very low one.

### Assigning Hands

Two-handed jugglers usually will use their hand alternatingly, so the above pattern may be distributed among hands as

```left     3   3   3   4   3   3   3
right  3   3   3   4   1   3   3   3
```

or as

```left   3   3   3   4   1   3   3   3
right    3   3   3   4   3   3   3
```

And a three-handed juggler like Zaphod Beeblebrox might juggle the pattern as

```hand1  3     3     4     3     3
hand2    3     3     4     3     3
hand3      3     3     1     3     3
```

The point is that the site-swap description of a pattern does not describe which hand is used for which throw. It is independent of the number of hands used, but once you decide for a number of hands, it is easy to figure out which throw is done by which hand.

### The Height of Throws

The numbers in the site-swap encode the height of the throws. For example a `3` denotes a throw, which is caught and thrown again three beats later. Similarly a `4` is a throw, which is thrown again four beats later. This leads to the following pattern for our example:

Here we assume, that catching a ball and throwing it again happens at the same instant. In a real juggling pattern the catches happen of course some time before the throws, the hand needs some time to reach the correct position and the necessary speed to do a throw. These holding times are considered in the chapter about the physics of juggling, where we will calculate the actual height of the throws.

The meaning of the numbers is summarised in the table below. There are two special cases: a `0` denotes a beat, where the hand which would do a throw now is empty and has no ball to catch. Thus a `0` indicates a pause in the pattern. The second special case is a `2`. This indicates a throw, which is thrown again two beats later, i.e., assuming a two-handed juggler, the very next throw of this hand is done with exactly the same ball. Theoretically you could just throw up the ball a little bit, and catch it back before the next throw. But in reality it is very common to just hold the ball during these two beats. Thus a `2` also indicates a pause in the pattern, but during this pause the hand is holding a ball.

```0 = an empty hand
1 = a throw which is caught one beat later
2 = a throw which is "caught" two beats later (see above)
3 = a throw which is caught three beats later
...
n = a throw which is caught n beats later
...
```

The following picture illustrates a more interesting pattern, containing a `0` and a `2`.

### Cyclic Patterns

Most real-world juggling patterns, at least these with a name, are cyclic: the same sequence of numbers repeats again and again. A common abbreviation is to just write this sequence once. Some typical examples are the following patterns.

```3   = ...333333333333333333...   (the 3-ball cascade)
51  = ...515151515151515151...   (the 3-ball shower)
441 = ...441441441441441441...
```

The picture at the end of the previous section shows one way of switching from a 3-ball cascade `3` into a 3-ball shower `51` and back again.

Something which is not so obvious in the site-swap notation, is the required number of balls. For example how many balls would you need to juggle `97531`? For cyclic patterns it is easy to find this number: it is just the average value of the sequence! If you want to understand the reason for this, some mathematical knowledge comes in handy :-) You should try to prove this, it really is fun.

### Compound Patterns

The article Why men (and octopuses) cannot juggle a four ball cascade of Engels and Mauw defines the concept of a cascade as a pattern where each ball is thrown at the same height and the hands are used in turn. They show, that whenever the number of balls and the number of hands are not relatively prime (i.e. if the greatest common divisor is not 1) a cascade can be decomposed into a unique set of sub-patterns. For example the basic 4-ball pattern `4` with two hands can be decomposed as follows.

consists of the two prime patterns

and

If we consider these patterns `40` and `04` as one-hand patterns, these are indistinguishable from the one-hand pattern `2`.

The opposite operation, combining two sub-patterns into a larger one, comes into play when we consider multi-person patterns.

### Passing Patterns

Many of the more interesting juggling patterns involve more than one person. Here we will consider so called passing patterns. First we take a look at two-person patterns with 6 balls.

The greatest common divisor of 6 and 4 is 2, thus according to the previous section the cascade with six balls and four hands should consist of sub-patterns. Indeed these sub-patterns are just each of the two persons juggling three balls, one of them as `60` and the other as the pattern `06`. To get a true passing pattern, we need to consider something different.

The most popular passing pattern is the two-person 6-ball four-count, which can be illustrated as follows.

Obviously this cannot be described as a simple site-swap pattern as introduced above, because at each beat two different hands do a throw. But there are extended site-swap notations, which can handle these situations. A simple extension is used in my diagrams: a code like `3p` indicates a pass, i.e. a throw which is caught by your partner 3 beats after it is thrown.

Many different passing patterns are explained on my ball passing patterns page.