Since it would be silly to hold onto a ball forever, we assume that each ball is thrown repeatedly. We also assume that only one ball is thrown at any given time. With these conventions, a juggling pattern with b balls is described, for our purposes, by b doubly-infinite disjoint sequences of integers.
The three ball cascade is perhaps the most basic juggling trick. Balls are thrown alternately from each hand and travel in a figure eight pattern. The balls are thrown at timesThis pattern has a natural generalization for any odd number of balls, but can't be done in a natural way with an even number of balls --- even if simultaneous throws were allowed, in a symmetrical cascade with an even number of balls there would be a collision at the center of the figure eight.
Another basic pattern, sometimes called the fountain or waterfall, is most commonly done with an even number of balls and consists of two disjoint circles of balls.
The four ball waterfall gives rise to the four sequences of throw times, for a = 0,1,2,3.
The last truly basic juggling pattern is called the shower. In a shower the balls travel in a circular pattern, with one hand throwing a high throw and the other throwing a low horizontal throw. The shower can be done with any number of balls; most people find that the three ball shower is significantly harder than the three ball cascade. The three ball shower corresponds to the sequences
We should mention that although non-jugglers are often sure that they have seen virtuoso performers juggle 17 or 20 balls, the historical record for a sustained ball cascade seems to be nine. Enrico Rastelli, sometimes considered the greatest juggler of all time, was able to make twenty catches in a 10-ball waterfall pattern. Rings are somewhat easier to juggle in large numbers, and various people have been able to juggle 11 and 12 rings.
Now we return to our idealized form of juggling. Given lists of throw times of b balls define a function byThis function is a permutation of the integers. Moreover, it satisfies for all . This permutation partitions the integers into orbits which (ignoring the orbits of size one) are just the lists of throw times.
The function corresponds to the 3-ball cascade, which could be graphically represented as in Figure 4.
Similarly, the function represents the ordinary 4-ball waterfall. The three ball shower corresponds to a function that has a slightly more complicated description. The juggler is usually most interested in the duration between throws which corresponds, roughly, to the height to which balls must be thrown.
Definition: A juggling pattern is a permutation such that for all . The height function of a juggling pattern is .
The three ball cascade has a height function that is constant. The three ball shower has a periodic height function whose values are . The juggling pattern in Figure 5 corresponds to the functionwhich is easily verified to be a permutation. The height function takes on the values 4,4,1 cyclically. This trick is therefore called the ``441'' among those who use the standard site swap notation. It is not terribly difficult to learn but is not a familiar pattern to most jugglers.
Now let f be a juggling pattern. This permutation of partitions the integers into orbits; since , the orbits are either infinite or else singletons.
Definition: The number of balls of a juggling pattern f, denoted , is the number of infinite orbits determined by the permutation f.
Our first result says that if the throw height is bounded, which is surely true for even the most energetic of jugglers, then the number of balls is finite and can be calculated as the average value of the throw heights over large intervals.
If f is a bijection and is a non-negative and bounded then the limit exists and is equal to , where the limit is over all integer intervals
[Proof (Proof contains Figure 6)]