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# Space-time diagrams

 Let's take the standard 3-ball cascade, at left. There are many things to notice about this pattern, like every throw being the same height, but for our purposes, we'll only note that the hands alternate, every throw goes from one hand to the other, and most importantly, while each ball is in its arc exactly two other throws occur (the other two balls are each thrown once).

The picture here is what you'd get, watching from above somebody juggling three glowing balls while walking forward, on time-lapse film. Alternately you can think of the horizontal axis as time, and the vertical axis as space (the space between the hands), so this is a "space-time diagram" of the 3-ball cascade. A rather simple trick you could do in the middle of this is throw a single 2-in-one-hand throw from the right hand.

Notice that the throw from the right hand isn't the only new thing in this diagram - the left hand holds once. (This looks more complicated than it has to be; why not just write L once up there? But this makes things more complicated in another way, though, as will be explained the "Vanilla siteswap" section. )
 Here are some more diagrams, for your amusement. The three-ball shower done asynchronously:

So every ball goes up from the right hand for a long time (long enough for the other two balls two visit each hand underneath) to the right hand, then is shuttled instantly back to the right.

 Here's the shower-box, which has synchronous throws; correspondingly, the space-time diagram has the R throws directly below the L throws. Note that you can follow the ball in the middle and see that it never goes up on the sides. (Admittedly we didn't need these diagrams to notice that!)

 Here's 3 balls out of a 4-fountain.

 Here's a goofier one, much less well-known. It has been christened "A complete waste of a 5-ball juggler".

What do these diagrams have in common? The main thing is that at every vertex, there are the same number of lines coming in as going out. (Usually that number is 1; in the last two it's sometimes zero.) Drawing multiplex patterns we'd have several balls coming in and the same number going out. On the other hand, given a picture like the above, it's easy to see how to juggle it, at least in theory - throw the balls so that they land in the right order. (Of course, you can draw something that requires you to throw balls the height of the Empire State Building - this is the meaning of "in theory"!)