Remarks for Jugglers Only

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The above description is geared towards the standard model: two hands throwing alternately, in constant rhythm. In fact there could be any number of hands and it is not necessary to assume that the rhythm is constant.
The practical meaning of the throw heights 0, 1, and 2 in the standard model requires a little thought. A throw height of 0 corresponds to an empty hand. A throw height of 1 corresponds to a rapid shower pass from one hand to another that is thrown again immediately. A throw height of 2 would ordinarily indicate a very low throw from a hand to itself that is thrown again by that hand immediately. This is actually rather unnatural in practice; the conventional interpretation ([11], [12]) is that a throw height of 2 is a held ball.
The paradigm for categorizing juggling patterns here is very interesting in practice, although many of the patterns require considerable proficiency. Several jugglers who have spent time in working on site swaps describe the same gain in flexibility and conceptual power that mathematicians seem to report from the use of well-chosen abstractions. The simplest non-obvious site-swap seems to be 441; it is similar to, but not the same as, the common 3-ball pattern of throwing balls up on the side while passing a ball back and forth underneath in a shower pass from hand to hand. (The latter pattern is not commonly performed with an even rhythm; if it is, it is 810.) The 3-ball 45141 pattern is also amusing, and the 4-ball 5551 pattern looks very much like the 5-ball cascade. The range of feasible and interesting tricks seems to be unlimited; we mention the following sample: 234, 504, 345, 5551, 40141, 561, 633, 55514, 7562, 7531, 566151, 561, 663, 771, 744, 753, 426, 459, 9559, 831.
A number of programs are available that simulate site swaps on a computer screen, sometimes with quite impressive graphics. These programs take a finite sequence of non-negative integers as input and dynamically represent the pattern. The Internet news group rec.juggling is a source of information on site swaps and various juggling animation software.
In order to find out which finite sequences represent juggling patterns we start by noting that a period-n pattern induces a permutation on the first n integers.

Contents Next: Counting Periodic Juggling Up: Periodic Juggling Previous: Periodic Juggling