The complex math behind juggling

Date:9 October 2017 Tags:, , ,

It might seem like fun and games, but juggling has some serious numbers behind it.

By Avery Thompson

Juggling is math. Follow me on this one.

The connection between the two came out of juggler’s attempts to develop a kind of notation for their tricks. That notation, called siteswap, can be used to describe nearly any juggling pattern, and it turns out that standard mathematical tricks can be used to develop new patterns nobody has ever seen before.

In the video above mathematician and juggler Colin Wright describes the math behind juggling on Numberphile.

The basic unit of siteswap is the throw. Each time a ball is thrown, it’s represented in siteswap as a number. For instance, in the most basic juggling pattern, 3-ball cascade, each throw is simply represented as the number 3. In the most basic pattern with four balls, 4-ball fountain, each throw is represented as the number 4. Essentially, each throw is given a number based on how many beats in the pattern it spends in the air before landing.

We can mix and match these numbers to create new patterns. For example, we can mix up a four ball pattern by swapping the places of the balls. If we want to exchange, say, the first ball with the second ball, we’ll need to throw the first ball a little higher and the second ball a little lower, so that they land where the other usually does. In siteswap notation, that means we throw a 5 and a 3, creating a new pattern, called 4-ball half-shower.

One easy way to tell if a specific siteswap pattern is valid is that all of the numbers of throws has to average out to the total number of balls in the pattern. So for four-ball juggling, the average of all the throws has to be 4. For a 4-ball fountain this is obvious, because every throw is a 4. For half shower the throws are 5 and 3, which indeed averages to 4.

We can go further. Instead of 5 and 3, we can do 6 and 2, which is actually pretty boring, or we could do 7 and 1, which is a 4-ball full shower. And there are other things we can try. Instead of 5 and 3, we could throw two throws as a 5, which means the third throw has to be a 2. The math works out, and indeed this is a real pattern.

We can use these mathematical tricks to discover new patterns. Wright used these tricks to invent 5551, which is a four ball pattern, and 441, which is a three ball pattern. No one had ever seen these tricks before, yet the math behind siteswap predicted them. Siteswap can get lots more complicated than this, and the math behind it gets more complicated too. For instance: What happens if you try throwing two balls at once?

Source: Numberphile
From: PM USA
Image credit: Juan Pablo Rodriguez on Unsplash

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