Juggling by numbers: How notation revealed new tricks

The mathematical formula of juggling

Juggling is usually associated with brightly coloured balls and clowning around, but it has more connections than you might think with the world of numbers.

Colin Wright is a mathematician who in the 1980s helped develop a notation system for juggling while at Cambridge University.

He was frustrated that there was no way to write down juggling moves.

"There was a juggling move called Mills Mess and when I tried to write it down I couldn't. When I finally did get a description after a few days of working, it took two and a half sides of A4 and I thought, for a juggling move that lasts one second there must be an easier way," says Wright.

The system he helped devise became known as Siteswap. Sequences of numbers were used to denote particular juggling moves.

These sequences encoded the number of beats of each throw, which is related to their height and the hand to which the throw is made.

Juggling clown Sequences of numbers are used to denote particular juggling moves also known as "Siteswap"

For example, throwing a three means the ball spends two beats in the air and one beat in the hand before it is thrown again, throwing a four means the ball spends three beats in the air then one beat in the hand before it gets thrown again, and so on.

The higher the ball is thrown, the bigger the number, so throwing a four means you are throwing the ball higher than a two.

Even numbers are used to represent a ball being thrown straight up and caught with the same hand and odd numbers for those balls caught in the opposite hand.

The numbers are then written into sequences. Juggling three balls for example would be written as 333 - each ball changes hands and spends three beats between throws.

Start Quote

Siteswap has allowed jugglers to share tricks with each other without having to meet in person”

End Quote James Grime Juggling enthusiast

The Siteswap system has had considerable success across the world particularly among professional jugglers according to Ieuan Evans, organiser of British Young Juggler of the Year.

"Some tricks - especially tricks with more than five balls or involving different heights - are hard on the eye. If you say a number sequence, like 515151 for example, it's easier to understand," says Evans.

The Siteswap system has helped communication between jugglers.

"Siteswap has allowed jugglers to share tricks with each other without having to meet in person or film themselves," says James Grime, juggling enthusiast and maths professor at Cambridge University.

And it has also led to new juggling tricks being discovered.

When known juggling tricks are written down in notation form, an overarching pattern emerges.

Gaps between written tricks reveal that other tricks exist which have not yet been discovered.

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To date, thousands of tricks have been discovered thanks to the Siteswap system.

"Most are uninteresting in the same way that some sequences of notes on a piano are unpleasant and won't be used in songs.

"However, possibly hundreds are technically interesting and possibly dozens are genuinely useful, fun and entertaining in their own right," says Wright who travels the world giving talks about juggling.

But while Siteswap is popular among professional jugglers, beginners might find it more difficult to get their heads around.

"If you're a beginner it may be quite confusing and you probably wouldn't use it," says Ieuan Evans.

Colin Wright agrees that Siteswap is not a system for everyone but does not think beginners should be put off trying it.

"It's not for everyone," he says. "But it provides a different way of thinking about patterns and for some people that's really useful."

More or Less is broadcast on BBC Radio 4 and the BBC World Service. You can listen again online by downloading the More or Less podcast

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