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, N1 other throws occur.
For instance, the 3ball cascade becomes
and the 3ball 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 showerbox, but it does include the cascade and (asynchronous) fountain. 
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 3cascade and 3shower) 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 N1 other throws are occur  so the throws are all N's.
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.


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! 

Working backwards from numbers to patterns