The physical constraints that affect mastery and limit the number of objects juggled arise from gravity - more specifically, Newtonian mechanics (h=1/2gt2). Each ball must be thrown sufficiently high to allow the juggler time to deal with the other balls. The need for either speed or height increases rapidly with the number of objects juggled.
These temporal constraints on juggling are summarized by Shannon's theorem. He defines relations that must exist among the times that the hands are empty or full and the time each ball spends in the air.
Shannon presented his theorems in a paper he wrote in the 1980s entitled Scientific Aspects of Juggling. Here he provides the first mathematical basis of juggling.
The uniform juggle
The variables Shannon uses to form his theorems are:
D - the dwell time (time a ball spends in a hand between when it's caught and when it's thrown)
F - the flight time (time a ball spends in the air between when it's thrown and when it's caught)
V - the vacant time (time a hand is empty between throwing one object and catching the next)
H - number of hands involved
B - number of balls juggled
He also only concentrates on uniform juggles:
A uniform juggle is one without multiplexing (no two or more balls may be caught in one hand at the same time) and with all dwell times the same, all flight times the same and all vacant times the same. Most basic patterns (three-ball cascade, four-ball fountain etc.) are uniform juggles. Since in practice a juggle can only be uniform for a period of time, Shannon's theorems require only that uniformity last for the period that it would take one ball to visit all the hands, that is H(F+D).
In a uniform juggle: (F+D)/(V+D)=B/H
That is, the number of balls and hands is proportional to the total time for each ball circuit and each hand circuit. This theorem is schematically represented for the three-ball cascade (fig. 4).
Fig. 4 - Shannon's theorem
If B and H are relatively prime (have no common divisor) then there is essentially a unique uniform juggle. The balls are numbered 0 to B-1 and the hands 0 to H-1 in such a way that each ball goes through the hands in cyclical sequence and each hand catches the balls in cyclical sequence.
If B and H are not relatively prime and n is their greatest common divisor then B=np and H=nq, where p and q are relatively prime. In this case, there are as many types of juggles as ways of partitioning n into a sum of integers.
Example: If n=5 a partition of 2+2+1 would correspond to three different juggles. There would be no possible interchange of balls among these three juggles. Each "2" of this partition would correspond to 2p balls circulating among 2q hands. The "1" in the partition is a cyclical juggle of the type in Theorem 2, with p balls circulating around q hands with no choice.
The number of partitions of n into sums increases rapidly with n as the table shows (fig.5).
Proof of theorems
The theorem is proved by following one complete cycle of the juggle from the point of view of the hand and of the ball and then equating the two.
From the point of view of the ball:
At time 0 a uniform juggle starts with the toss of a ball. The time required for one catch, i.e. the time needed for one ball to move from one hand to the other, is D+F. For a period of H catches, the total time is H(F+D). Since there are H hands, this means that the catches per second per hand are B/(H(D+F)).
From the point of view of the hand:
The time between catches of adjacent objects into a hand must be V+D, so the number of catches per second per hand is 1/(V+D).
Equating the two expressions for catches per second per hand and solving for B gives B=H(D+F)/(D+V), which shows how many objects can be juggled for any given combination of flight times, dwell times, and vacant times.
Shannon carried out a series of experiments to measure the various dwell times, vacant times and flight times involved in actual juggling. He called the system he used the Jugglometer. At first he used three electromagnetic stopclocks that were activated by a relay circuit. The relays were connected to a copper mesh, which was fitted over the fingers of the juggler's hands. The balls were also covered with conducting foil. When a ball was caught it closed the connection between the fingers causing a clock to start. Break contacts allowed the measurement of vacant time and contacts on two hands enabled measurement of flight times. Later the system was replaced with a computerized version where the fingers were connected to a computer and a computer program displayed the various times. The results showed that V is usually less than D, V ranging between 50% and 70% of D. The times depend on the tool used.
Uses of theorem
According to Shannon's theorem, increasing the number of balls leaves less room to vary the speed of the juggle. If one were to juggle many balls to a certain height, the theorem indicates that even the smallest variation in tossing speed would destroy the pattern.
The theorem also explains why juggling gets so hard so fast. It's easy to move from three to four-ball cascades but moving up to five is a lot harder. If all the terms in the equation are constant except B and F, F increases linearly with B. F also increases with the root of the height, which is proportional to energy. So energy and height requirements increase as the square of B. As the height increases, D also increases, since more dwell time is needed for the balls to accelerate, which makes it harder to catch incoming balls. Novice jugglers prefer larger dwell times so as to achieve accurate tosses. More proficient jugglers tend toward smaller values because of a greater flexibility to shift the pattern.
Shannon's formula can also be used to determine the greatest number of objects a person could possibly juggle. The "flight-dwell" ratio, F/D, is the amount of time a ball spends in the air relative to the time it spends in a hand. Adding more objects to a pattern requires throwing higher (increasing F), throwing faster (decreasing D), or both. Either way, increasing B requires increasing F/D. Since V is greater than 0, the formula of Theorem 1 can be rewritten in terms of the flight-dwell ratio as (F/D)>(B/H)-1 ((F/D)+1=B/H). This describes the minimum flight-dwell ratio required for juggling B objects. By measuring a juggler's maximum flight-dwell ratio, this formula determines the greatest number of objects a person could possibly juggle.
In general, by using the mathematical relationships set by Shannon, researchers were able to study how jugglers coordinate their limbs to move rhythmically and at the same frequency within certain constraints.