Juggling/Symmetric Passing Patterns

This system is a great way of generating all the possible passing patterns within given restrictions, as well as providing a useful intuitive understanding of passing patterns to people already familiar with [siteswap notation]. The idea behind the system was originally described in an article by French Juggler and mathematician Cristophe Préchac which was [posted on rec.juggling in 1999]. Nowadays it is taught and promoted primarily by Sean Gandini and more and more other jugglers. On his website, he offers a [detailed explanation of symmetric passing] as PDF.

Even though that PDF is well written and structured, a wikibook would be more dynamic, easier to access and could be integrated with videos and passing software like generators and animators. Please see the discussion page.

Préchac Transformation
The Préchac transformation is a clever way of transforming solo siteswaps into numerous fascinating passing patterns. In fact every siteswap can be transformed into many different passing patterns. The resulting patterns will bear a structural resemblance to the siteswap they emanated from. We call these patterns symmetric because each juggler does the same sequence of selfs and passes. In addition to being symmetric, the patterns are also staggered, by this we mean, that the jugglers do the same thing but at different times.

 Essentially the system takes a siteswap of $$n$$ objects and of period $$p$$ and transforms it into a passing pattern for 2 jugglers with $$2n-1$$ objects. It does this subtracting half of the period of any of the throw values in the siteswap, and transforming this into a pass.

Example
Let's look at an examples of this. Lets take the siteswap 3 1 the two ball shower. This is a period 2 pattern. We can subtract half the period from the 3. The result is a 2p 1. A relatively easy and intuitive 3 object 2 count juggling pattern that has the same feeling as the 2 ball shower, 3 1. Instead of a 3 you throw a 2p to your partner and receive a 2p from her exactly when the 3 from your 3 1 would land. Please find a detailed explanation of the notation.

Notice how the amount of objects is computed in the same way as in solo siteswaps: the average of 2p and 1 is 1.5, so every juggler has 1.5 objects, and together the 2 jugglers have 3 objects. Consider the timing: both juggler juggle the same pattern one half a period after the other, i. e. when one throws the 2p the other throws the 1 and vice versa.

Determine Passing Category
To determine how the passes will be thrown it's helpful to put the pattern into one of four categories: classic, equi, bi or instant-bi.

If the period is an even number there are two types of passes it might be – classic or equi. Transforming an odd number either up or down defines the pattern as 'classic'. Transforming an even number in the same way defines the pattern as 'equi'.

In the case of an odd period the two options are – bi or instant bi. Transforming an odd number up or an even number down defines the pattern as ‘bi’. Transforming an even number up or an odd number down defines the pattern as ‘instant-bi’