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Working backwards

By this point, I hope you're excited by the possibility that you could come up with new patterns just by correctly choosing numbers - even ones that would require a much more competent numbers juggler! The question is then:

When is a list of numbers actually a genuine siteswap?

There are two things to check:

  • at each time, we should be catching at most one ball (none if we're doing a 0 then)
  • we should actually catch one, to have one available to throw (unless we're doing a 0 then.)
  • (It is simpler to think of a 0-throw as one where there is actually a ball borrowed from the Astral Plane only long enough to throw and catch at the exact same time - then at each time, we're always throwing, and catching, exactly one ball. This is the main reason why it is mathematically nice to call these throws 0s.)

    It is possible to prove mathematically that these two conditions are equivalent - you only need to check one of them, and it doesn't matter which. (It seems that everybody checks that no two throws land at the same time, though!) It is also possible to prove that in this case, the average of the numbers must be the number of balls involved, which is convenient.

    MISCONCEPTION: word has gotten out that the average of the numbers is the number of balls, and many people believe this to be the property to be checked. (If the average isn't an integer, than obviously the list is not a real juggling pattern.) But this isn't good enough - for example, if you tried to juggle the "pattern" 543, you'd be throwing three balls that would land at the same time.

    When analyzing 4413 in the last section, I talked about following one ball, and figuring out its entire destiny. You can do this with any siteswap (and it's the first thing I do when trying to learn them, though other people learn them differently). This procedure breaks up the numbers into bunches called "orbits", where each ball stays within a particular orbit. So you can remove all the balls in an orbit by setting all those numbers to zero. (For example, from 4413 we can make the patterns 0413, 4013, 4400, and 4000.) So the averaging theorem implies the more precise fact that the number of balls in a given orbit is the sum of the numbers in that orbit, divided by the length of the whole pattern.

    For example, here are the orbits of the pattern 561:

    So in a sense, this pattern is a combination of the patterns 501 and 060.

    The total height in the 5,1 orbit is 6: since the pattern is length 3, this indicates that 6/3=2 balls are trapped going 1,5,1,5,1,5 forever. Similarly, the other two balls just go 6,6,6,6 forever, hopping up and down in the same hand. And since the delay is 3 between one 6 and the next, they are in opposite hands. So in this pattern, there are two balls hopping up and down exactly out of phase from one another, surrounding a 501 in the middle.

    In 72312, the orbits are (7,3) and (2,1,2). There are two balls in the (7,3) loop (going in opposite directions), and the baby in the (2,1,2) loop.

    In 5241, the orbits are (5,2,1), with two balls, and (4), with just one.

    In 5551, there is just one orbit (5,5,5,1). This pattern mixes up the balls completely.

    In 55514, each orbit holds just one ball. The orbits are (5), (5), (5), and (1,4).

    When I try to learn these, I fixate on a ball, or a couple of balls, that are doing something I understand, like one ball going 5,5,5,5 or two balls going 501.

    Rocky road siteswap: synchronous and multiplexed throws

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