# Popcorns II: asymmetry, synchronicity and more people

Author: Sean Gandini

See the previous part, Popcorns I

For this article we shall assume that you have read last issues' article. Last time we looked at 2 person symmetric popcorns. This time round lets look at some different kinds of popcorns and how one expands the popcorn idea to more people. Again even though the article is very notation heavy and slightly theoretical it is very much rooted in real world juggling patterns. I highly recommend downloading Joe pass and the files for a lot of these patterns. Ok so here we go:

## Semi-Asymmetric popcorns

Up to now we have assumed that both jugglers lift into the same amount of objects. This is not always necessarily the case. We can build popcorns where both jugglers lift into different amounts. Lets imagine for example that you want juggler one to lift into 5 objects and juggler two to lift into 4 objects. This would be a 1 pass 8 object popcorn. We obviously won't do the charts for all of the possible asymmetric patterns but for reference below is part of the 8 object asymmetric 4/5 popcorn.

 554.5p4444|4444.5p333 4p4|44p 54p44|444p3 554p444|4444p33 5554p4444|44444p333 53.5p4|443.5p 553.5p44|4443.5p3 5553.5p444|44443.5p33 553p4|4443p

554p444|4444p33

Note that we now need to show both passers roles since the pattern is asymmetric. Note also that the pattern 554p444|4444p33 illustrated above should strickly speaking be written as 554p444|4p333444; that is the second part of the pattern starts on the pass. We find that for the purposes of this article and to illustrate the particularities of popcorns the notation we have chosen is clearer.

There is also a whole family of patterns which have the jugglers lifting into the same amount of objects but holding their patterns for different amounts of time. So there is a further expansion of each pattern in the original symmetric charts which involves elongating the numbers on one side whilst shortening the numbers on the other side. So for example below is the expansion of the classic popcorn into its 3 mutations.

 444p333 44p3333|4444p33 4p33333|44444p3

444p333

44p3333|4444p33

4p33333|44444p3

Every pattern can be mutated this way. In fact we can combine the two procedures above to generate a third family, popcorns of different lengths and different amount of objects. Lets look at the mutations of 554p444|4444p33 a pattern which we met before:

 555554p|4p33333 55554p4|44p3333 5554p44|444p333 554p444|4444p33 54p4444|44444p3

## Progressive popcorns

It is also possible to progressively lift into a given pattern. For example if we take the 8 object 2 pass popcorn: 554p4p33, the jugglers take turns at lifting from 3 to 5 objects. We can stagger the lifting by separating the passes. So lets look at the pattern 554p4p33 which we met in the 8 object 2 pass chart last issue. So by inserting some 4s the pattern becomes:

554p44p334,
the bold 4s are the inserted 4s.

then 554p444p3344 and so on and so forth. So essentially any popcorn pattern with more than 1 pass can be progressive!

 554p44p334 554p444p3344 554p4444p33444 554p44444p334444

Needless to say we can use the staggering procedure that we met above to make the patterns asymmetrical.

## More People

Ok so what happens if there are more jugglers involved. Once again we shall only look at patterns where all jugglers do the same thing at different times. Spatially the jugglers can stand wherever they want, for practical reasons however the easiest way to juggle these patterns in a triangle or a line formation.

So lets look at 3 jugglers with 10 objects. Here each juggler will take turns at lifting from 3 objects to 4 objects.
Below is the chart expanded in the same way as the 2 person chart.

Here the chart increases Horizontally by adding a 4 on the left side of the pass and 2 x 3 on the other side.
It increases Vertically by adding 0.3 to the pass.

 5p33333 45p3333333 445p333333333 4445p33333333333 44445p3333333333333 444445p333333333333333 4.6p3333 44.6p333333 444.6p33333333 4444.6p3333333333 44444.6p333333333333 444444.6p33333333333333 4.3p333 44.3p33333 444.3p3333333 4444.3p333333333 44444.3p33333333333 444444.3p3333333333333 4p33 44p3333 444p333333 4444p33333333 44444p3333333333 444444p333333333333 3.6p3 43.6p333 443.6p33333 4443.6p3333333 44443.6p333333333 444443.6p33333333333 3.3p 43.3p33 443.3p3333 4443.3p333333 44443.3p33333333 444443.3p3333333333 43p3 443p333 4443p33333 44443p3333333 444443p333333333

Again note that the column on the left side of the chart has the 1-count, 2-count, 3-count, 4-count…patterns.

Lets now look at 11 object 3 person popcorns. There are 1 pass and 2 pass versions of this. Below is the 1 pass version.

 6.3p3333 446.3p33333 44446.3p333333 4444446.3p3333333 444444446.3p33333333 5.6p333 445.6p3333 44445.6p33333 4444445.6p333333 444444445.6p3333333 5p33 445p333 44445p3333 4444445p33333 444444445p333333 4.3p3 444.3p33 44444.3p333 4444444.3p3333 444444444.3p33333 3.6p 443.6p3 44443.6p33 4444443.6p333 444444443.6p3333 443p 44443p3 4444443p33 444444443p333

For fun lets look at a small selection from the 4 person charts:

Below is the 13 object 1 pass charts. The chart increases vertically by adding 0.25 to the pass and horizontally by adding one 4 on one side and 3 x 3 on the other.

 5p3333333 45p3333333333 445p333333333333 4445p33333333333333 4.75p333333 44.75p333333333 444.75p333333333333 4444.75p3333333333333 4.5p33333 44.5p33333333 444.5p33333333333 4444.5p333333333333 4.25p3333 44.25p3333333 444.25p3333333333 4444.25p33333333333 4p333 44p333333 444p333333333 4444p3333333333 3.75p33 43.75p33333 443.75p33333333 4443.75p333333333 3.5p3 43.5p3333 443.5p3333333 4443.5p33333333 3.25p 43.25p333 443.25p333333 4443.25p3333333

The 14 object chart for 4 jugglers, is the same as the 7 object chart for 2 juggler so it has not been included.
All the 2 person charts can be transformed into 4 person charts by doubling the jugglers and the amount of objects.

If you have got this far with me then you can imagine how to construct charts for more jugglers.

## Synchronous popcorns

Essentially one can draw the same kind of charts as the asynchronous popcorns. However things get slightly complicated. We mentioned in the last article that we defined popcorns as jugglers juggling a certain amount of objects in ground state and lifting/descending into a different amount of objects still in ground state.

If a lone juggler jugglers 4 objects asynchronously there is only one way of staying in ground state, that is throwing 4s. This is not the case for synchronous 4. One can throw (4,4) or (4x,4x), two different ways of staying ground state. For odd numbers of balls there are 4 different ways of staying ground state.

What this basically means is that every pattern has numerous equivalent versions.
However bearing this in mind the chart process still works. Below are examples of the various charts.

7 Objects:
 (4,5p)(4x,2x)(4x,2x) (4,4)(4,4p)(4x,2x)(4x,2x) (4,4)(4,4)(4,4p)(4x,2x)(4x,2x)(4x,2x) (4,4)(4,4)(4,4)(4,4p)(4x,2x)(4x,2x)(4x,2x)(4x,2x) (4,4p)(4x,2x) (4,4)(4,4p)(4x,2x)(4x,2x) (4,4)(4,4)(4,4p)(4x,2x)(4x,2x)(4x,2x) (4,4)(4,4)(4,4)(4,4p)(4x,2x)(4x,2x)(4x,2x)(4x,2x) (4,3p) (4,4)(4,3p)(4x,2x) (4,4)(4,4)(4,3p)(4x,2x)(4x,2x) (4,4)(4,4)(4,4)(4,3p)(4x,2x)(4x,2x)(4x,2x)

And for fun:

9 Objects:
 (7p,4)(4,4)(4,4) (4x,6x)(7p,4)(4,4)(4,4)(4,4) (4x,6x)(7p,4)(4,4)(4,4)(4,4) (6p,4)(4,4) (4,6x)(6p,4)(4,4)(4,4) (6x,4)(4,6x)(6p,4)(4,4)(4,4)(4,4) (5p,4x) (4,6x)(5p,4x)(4,4) (4,6x)(5p,4x)(4,4) (4,6x) (4p,4x) (4,6x) (4p,4)(4,4)

## More People synchronous

So needless to say we can make the synchronous charts for more jugglers.

Below is the chart for 3 jugglers and 10 objects.

 (6p,4)(2x,4x)(2x,4x)(2x,4x)(2x,4x)(2x,4x) (4,4)(6p,4)(2x,4x)(2x,4x)(2x,4x)(2x,4x)(2x,4x)(2x,4x)(2x,4x) 2 times (4,4)(5.3p,4) 9 times (2x,4x) (5.3p,4)(2x,4x)(2x,4x)(2x,4x)(2x,4x) (4,4)(5.3p,4)(2x,4x)(2x,4x)(2x,4x)(2x,4x)(2x,4x)(2x,4x) 2 times (4,4)(5.3p,4) 8 times (2x,4x) (4.6p,4)(2x,4x)(2x,4x)(2x,4x) (4,4)(4.6p,4)(2x,4x)(2x,4x)(2x,4x)(2x,4x)(2x,4x) (4,4)(4,4)(4.6p,4)(2x,4x)(2x,4x)(2x,4x)(2x,4x)(2x,4x)(2x,4x)(2x,4x) (4p,4)(2x,4x)(2x,4x) (4,4)(4p,4)(2x,4x)(2x,4x)(2x,4x)(2x,4x) (4,4)(4,4)(4p,4)(2x,4x)(2x,4x)(2x,4x)(2x,4x)(2x,4x)(2x,4x) (3.3p,4)(2x,4x) (4,4)(3.3p,4)(2x,4x)(2x,4x)(2x,4x) (4,4)(4,4)(3.3p,4)(2x,4x)(2x,4x)(2x,4x)(2x,4x)

## Generalising

One can of course combine the above ideas. So you can use the above as a cookbook and make your own recipes. For example you might enjoy making 3 person progressive synchronous asymmetric popcorns. One can combine synchronous and asynchronous patterns.

So too finish an 3 person asymmetric 11 object popcorn with one juggler lifting from 3 to 4, one juggler lifting form 4 to 5 and one juggler lifting synchronously from 3 to 4.

So that's the end of our exploration of popcorn patterns. I would like once again to stress that these patterns are a lot of fun to juggle. Getting confortable with the notation and the diagrams takes a while but the juggling rewards are huge. I lookforwards to any feedback you might have on these ideas and I hope that you get something out of them.

See the previous part, Popcorns I

As an addendum to the last article we would like to add that a 2 person pattern can be done in several different different ways, depending on the hand throwing order. So for example the classic 7 object 2 count 4p3 in 4 different ways:
- Both jugglers starting with their rights.
- Both Jugglers starting with their lefts.
- Juggler one starts right and Juggler 2 starts left.
- Juggler one starts left and Juggler 2 starts right.

This is the same for patterns of an even period. Patterns of an odd period have just 2 versions.

Now whereas the 2 person popcorn charts have either 2 or 4 different hand arrangements, the 3 person patterns have 4 or 8 different possibilities. So choosing the easiest or most convenient way of juggling a particular patterns will not always be easy. I suggest trying different possibilities using intuition to guide you.

### Bibliography

An extended version of this articles including all the charts and files for joe pass can be found at http://www.gandinijuggling.com/popcorns.htm
New Passing site on the internet www.passingdb.com has many films of patterns relevant to this article.
Wolfgang Westerboers fantastic passing animator http://www.koelnvention.de/software/index.html
For an understanding of states: Mark Thomas http://www.markthomasonline.co.uk/state.html