Magical Patterns
May 2, 2016
Roger Antonsen

Juggling and Mathematics

(Published in Aftenposten May 3, 2016, in Norwegian.)

Why are so many mathematicians fascinated by juggling? Some well-known historical examples of researchers that have been drawn to juggling, and that have helped to develop mathematical theories for this activity, are the American mathematicians Claude Shannon (1916–2001) and Ronald Graham (1935–).

Shannon – who is best known as the father of information theory – analyzed the relationship between the time a ball is in the air and the time a ball is in a hand. He also constructed the world's first juggling robot. Graham has explored deep connections between combinatorics and juggling.

I am sure that it has something to do with patterns. Mathematicians love to discover patterns and find good representations of these patterns. In the video above you can learn a little about different juggling patterns and how these can be represented in the notation system siteswap .

There are many fascinating and enriching connections between juggling and mathematics, and many books and articles have been written on this. For example, Burkard Polster's book The Mathematics of Juggling gives a nice overview.


In the notation system siteswap, digits are used to represent throws, and sequences of digits are used to represent patterns. For instance, 333333..., or just 3, represents the simplest pattern you can juggle with three balls. The numbers indicate how many beats it takes for the same ball to get re-thrown. For example, 3 means that the ball is re-thrown after three beats.

Siteswap was invented in the 80's many different places in the world simultaneously and is today used by almost all jugglers, both as a language to talk precisely about what you do when you juggle – just likes notes are the language of music – and as a tool for constructing new juggling patterns.

The juggling group Gandini is well known for using siteswap to find new artistic expressions. They have also contributed to the development and expansion of the notation system.

Here are some simple examples of juggling patterns and what they are called. To make these, I used the program JugglingLab, which is one of many applications that can be used to find and visualize juggling patterns.

The juggling pattern 3

This is the simplest juggling pattern with three balls. All throws are similar and go to the opposite hand.

The Juggling Pattern 441

This is another way to juggle three balls. There are two types of throws: fours and ones. The fours are high throws in columns, which means that they land in the same hand as they were thrown from. The ones are low, quick throws that land in the opposite hand.

The Juggling Pattern 4413

Here we have the juggling pattern 4413, which is simply obtained by taking 441 together with 3. Here there are three types of throws: fours , ones and threes. Notice that the fours never change hands and that one ball continuously switches between being a one and a three.

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