Space-time diagrams
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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).
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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:
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51
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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!)
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(4,2x)(2x,4)
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Here's 3 balls out of a 4-fountain.
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4440
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Here's a goofier one, much less well-known.
It has been christened "A complete waste of a 5-ball juggler".
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450
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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"!)
Vanilla siteswap
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