Vanilla siteswap

Drawing pictures is nice, but plainly the pictures above contain lots of redundant information. How can we write them more compactly?

The Siteswap Idea: for each throw, write down the number of throws that occur before that ball is thrown again. In other words, call a throw an N if while that ball is in the air, N-1 other throws occur.

For instance, the 3-ball cascade becomes

and the 3-ball shower becomes

Of course, you don't have to write all that, since these patterns repeat - just write one period.

We can simplify this a bit further with the Vanilla Siteswap restrictions:

The hands always alternate - in particular, only one hand at a time.
Only one throw is done at a time: no multiplexing.


This rules out lots of patterns, including the shower-box, but it does include the cascade and (asynchronous) fountain.

3 4

With this assumption in place, it's time to actually write down some patterns like you'll see them on rec.juggling. The patterns written above (the 3-cascade and 3-shower) get written simply as "3" and "51".

Since we assume you're going left, right, left, right, you can always tell whether a throw goes across or back to the same hand - it's whether the number corresponding to it is odd or even. Similarly, if a pattern has an odd number of numbers in it, when it repeats it'll be starting at the other hand, and so will be symmetrical.

In the standard cascade or fountain with N balls, while each ball is in the air one throws each of the other balls exactly once, so N-1 other throws are occur - so the throws are all N's.

4413 For instance, let's take "4413", and try and work backwards to juggling. Remember, this is a shorthand for ...4413441344134413... Let's follow the first ball thrown, the initial 4. It goes up (to the same hand, 4 being even) and comes down 4 throws later, which is again a 4, in fact the next copy of the same 4. Then it goes up as a 4 again, and again, forever. Similarly for the second 4. Now follow the 1, which crosses quickly enough that it's thrown the very next throw - it's a handoff. That next throw is the 3, which also crosses, landing on the next copy of the 1. So that ball is handed off, then tossed over, then handed off, then tossed over, and so on. And it's the only ball that crosses. So the pattern looks like an anemic shower-box - one of the hands isn't fast enough to do the handoff, so both hands rest as that hand makes a little toss every time.

0's and 2's

It might seem like the vanilla restrictions are really fierce - they rule out many reasonable patterns, like 3-out-of-a-4-fountain, which has two right-hand throws in a row. We will fix this by two clever kludges. The first is to say that when a hand holds a ball while the other hand does two throws in a row, it is in fact doing a tiny, tiny little throw. 333342
Such a throw is thus even, since it's going back to the same hand; it's a 2. Look again at the pattern ...33342333..., and you'll see where the funny 2-throw occurred.

We also have to assign a name to when a hand is empty while the other hand does two in a row, and this name will be 0. I can mainly say that this turns out to be a nice idea mathematically; however, in its defense I will also point out that since N's are the throws one uses in the standard pattern with N (cascade or fountain), 0's should be the "throws" when one's hands are empty!

Aspects not visible in the siteswap

There are many aspects of a pattern that are not visible in the siteswap. Perhaps the most notable is normal (inside) vs.
reverse (outside) throws. Another thing lost is the distinction between bouncing a ball and tossing it through the air.

3 3r 3b

Working backwards from numbers to patterns

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