The ratio between the maximum and minimum frequencies for a given height is the same as the ratio between the maximum and minimum periods that can be attained. The maximum possible period is obtained when the empty time e is 0, and the minimum period is obtained when the dwell time d is 0. By doing a little algebra one deduces that this ratio must be b/(b-h)

 

Of course, the extreme values are not actually possible (it is impossible to have the dwell time or empty time actually be 0) but this gives a good indication of the relative flexibility in a pattern. It also says that the "state space" of possible parameter values for a given juggling pattern with throws to a given height is one-dimensional, and if this is thought of as an interval on the real line then this gives the size of this interval. Note that if b=h, then the state space is of infinite length; one could juggle infinitely slowly by having each hand hold onto a ball for an infinite time.

 

One of us attempted to approximate the acceleration due to gravity for an introductory physics class by measuring the frequency of juggling patterns. An attempt was made to throw to a constant height of 1 meter and to fix the remaining parameter that defined the juggle by releasing a ball exactly when the preceding ball was at its apogee.

 

When this was done with the three ball cascade, the accuracy was disappointing; the maximum / minimum frequency ratio was b/(b-h)=3, which is large. This means that the juggling pattern was poorly defined for experimental purposes. When the experiment was repeated with a five ball cascade, the accuracy was much better. Indeed the ratio in this case (ie, the size of the state space) is b/(b-h)=5/3 so that the pattern was much more tightly constrained. The acceleration due to gravity was determined up to an error of about five percent, much to the satis­faction of the class.

 

. . . The process of learning to juggle is a paradigm for the acquisition of many motor skills. One starts with basic throwing and catching skills (possessed by nearly everyone) and refines them systematically. This also involves a refinement not only of these reflex skills but also spatial perception skills. One finds that it is necessary to construct various mental representations of the skill and that these mental models also evolve as one acquires a trick.

 

The process of refinement of mental constructs and of spatial perceptions can be quite startling to the neophyte juggler; it seems almost as if the balls begin to travel much more slowly. it would be very interesting to describe what is happening in the brain during this process, but of course our knowledge is entirely inadequate, both in terms of the structure and function of the brain and our knowledge of the way that intelligence is organized.

 

It has been observed that learning to juggle is like learning mathematics. One starts from obvious or known facts or skills and is led by a careful process to apparently new facts or skills. In mathematics, teachers often persuade students by their reasoning that the new facts were entirely necessary consequences of the known ones so that in some sense they are not new facts. In this abtruse sense, someone who accepts Euclid's axioms knows geometry and someone who can throw and catch and see knows how to juggle!