By Brian Dubé

Claude Shannon

The following is an explanation of  Shannon’s Juggling Theorem, developed by legendary mathematical, engineer and juggler Claude Shannon. This posting is based on articles written by mathematicians Ron Graham and Joe Buhler.

In mathematical terms, a juggler juggles five variables. He is free to vary the number of balls (b) he is juggling, the number of hands (h) he juggles with, the flight time (f) of each ball between his hands, the length of time a hand is empty (e) between catches, and the length of time a ball dwells (d) in a hand between throws. We assume that for a given pattern that all the above variables are constant and that no two balls are ever in the same hand at the same time and that the pattern is periodic (each configuration
of balls occurs at fixed intervals). These assumptions imply that the pattern has a certain symmetry and stable rhythm. This is true for the cascade but not for the shower which requires more complex mathematical descriptions. Consider two time lines, each representing one period (p1 or p2), one from the ball’s perspective and one from the hand’s perspective:

Ball perspective

__d f__     __d f__
1st hand    2nd hand

Hand perspective

__d e__   __ d e__   __d e__
1st ball     2nd ball     3rd ball

From the ball’s perspective, the length of one period is equal to the
number of hands times the combined time that a ball dwells in one hand
and is in flight, for each hand it meets:

a) p1 = h(d+f)

From the hand’s perspective, the length of one period is equal to the
number of balls times the combined time that the ball dwells in a hand
and that the hand is empty, for each ball it meets:

b) p2 = b(d+e)

Since the two periods are equal (p1=p2) (just considered from two different perspectives) we have:

c) b(d+e)=h(d+f) or:

d) b/h=(d+f)/(d+e)

We call this Shannon’s Juggling Theorem.

We can see from this theorem that each variable in juggling is related to and affected by changes in the others. Many relationships can be explored with this theorem. For example, what are the limits of the juggler’s freedom to vary the speed of a juggling pattern? We can fix the number of balls, hands, and the throw height (and, therefore, the flight time). The juggler can slow the pattern down by holding the ball longer or can speed it up by releasing each ball faster. The limit in the first case where the ball is held as long as possible is: empty time=0. Using equation c) above and setting e=0, we get:

e) bd=h(d+f).

The limit in the second case where the ball is held as short as possible is:
dwell time=0. Setting d=0 we get:

f) be=hf.

The ratio of these two extremes (between fastest and slowest) is bd/be=d/e
Solving equation e) for d yields d=(hf)/(b-h).
Solving equation f) for e yields e=(hf)/b.
So the ratio of fastest to slowest speeds (or period lengths) is
d/e=[(hf)/(b-h)]/[(hf)/b] or:

g) b/(b-h)

According to this ratio, b/(b-h), the range of possible juggling speeds decreases with the number of balls (and increases with the number of hands). So for a juggler using two hands and three balls, the ratio between the fastest and slowest speeds is 3/(3-2) or three to one. For a jugglerusing two hands and SEVEN balls, the ratio is 7/(7-2) or only 1.4 to one. Thus, one can see how the addition of more balls dramatically constrains the juggler. Of course, the extremes of fast and slow juggling are only possible in theory since a juggler can’t make the dwell time or empty time actually equal to zero. Therefore, the ratio of achievable speeds is even smaller.

We would like to thank Ron Graham and Joe Buhler for use of their work…
and of course a special thanks to Claude Shannon.