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# Juggling

As mathematicians are in the habit of doing, we start by throwing away irrelevant detail. In a juggling pattern we will ignore how many people or hands are involved, ignore which objects are being used, and ignore the specific paths of the thrown objects. We will assume that there are a fixed number of objects (occasionally referred to as ``balls'' for convenience) and will pay attention only to the times at which they are thrown, and will assume that the throw times are periodic. Although much of the interest of actual juggling comes from peculiar throws (behind the back, off the head, etc.), peculiar objects (clubs, calculus texts, chain saws, etc.), and peculiar rhythms, we will find that the above idealization is sufficiently interesting. Suppose that you are juggling b balls in a constant rhythm. Since the throws occur at discrete equally-spaced moments of time, and since in our idealized world you have been juggling forever and will continue to do so, we identify the times t of throws with integers .

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 times

This 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.

Figure 1: A cascade[QuickTime movie]

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.

Figure 2: A fountain (waterfall) [QuickTime movie]

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

Figure 3:A shower [QuickTime movie]

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 by

This 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.

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 function

which 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.

Figure 5: 441 [QuickTime movie]

Remarks:

• We refer to as the height function even though it more properly is a rough measure of the elapsed time of the throw. From basic physics the height is proportional to the square of the elapsed time. The elapsed time is actually less than since the ball must be held before being thrown; for a more physical discussion of actual elapsed times and throw heights see [11].

• Although there is nothing in our idealized setup that requires two hands, or even ``hands'' at all, we note that in the usual two-handed juggling patterns, that a throw with odd throw height goes from one hand to the other, and a throw with even throw height goes from one hand to itself.

• If , so that , then no throw takes place at time t. In actual practice this usually corresponds to an empty hand.

• Nothing in our model really requires that the rhythm of the juggling pattern be constant. We only need a periodic pattern of throw times. We retain the constant rhythm terminology in order to be consistent with jugglers' standard model of site swaps.

• The catch times are irrelevant in our model. Thus a throw at time t of height is next thrown at time , but in practice it is caught well before that time in order to allow time to prepare for the next throw. A common time to catch such a throw is approximately at time but great variation is possible. A theorem due to Claude Shannon ([13], [7]) gives a relationship between flight times, hold times, and empty times in a symmetrical pattern.

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.

### Theorem 1

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)]

Remarks:

• The limit is clearly a uniform limit in the sense that for all positive there is an m such that if I is an interval of integers with more than m elements then the average of df over I is within of .

• As an example illustrating the theorem we note if f is the 441 pattern described earlier, then the height function is periodic of period 3. The long term average of over any interval approaches the average over the period, i.e., , which confirms what we already knew: the 441 pattern is a 3-ball trick.

• The hypothesis of bounded throw heights is necessary. Indeed, if and, for nonzero t, is the highest power of 2 that divides t then the pattern has unbounded throw height and infinite , as in Figure 7. More vividly: you can juggle infinitely many balls if you can throw arbitrarily high.

Figure 7: Infinitely many balls

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