The Academic Juggler

by Arthur Lewbel

I'm not alone. Other people are interested enough in academic juggling to write to me. Ted Stresen sent me seven pages dealing with the role of performers and theories of how best to teach juggling. Nolan Haims, Mike Dorris and Garey Jonson all wrote looking for additional pictures and articles about science and juggling. Adria  Sullivan sent a paper trying to count the number of possible juggling patterns.

For those interested in juggling automata (Summer 1988 "Academic Juggler"), Marc Donner at the IBM research center in Yorktown Heights, N.Y., is picking up where Claude Shannon left off by working on a new Juggling robot (in a future article I'll describe Claude's current academic juggling work). Referring to the same article, Ken Letko objected to my generalization that all good club jugglers slide a club to the knob, pointing out that different types of throws require different grips.

The Winter 1987-88 academic juggler described Shannon's juggling theorem, and  a number of people have asked me to explain it again. A simplification is: the time each ball spends in the air (throw time), divided by the time each ball spends in a hand (dwell time), must be greater than (N/ 2)-1, where N is the number of objects being juggled. Michael Kramer wrote to correctly point out that one can sometimes increase the number of balls juggled without increasing one's "throw-dwell" ratio by changing the frequency of throws. The point of the theorem is only that for any N, there is a certain minimum required throw­dwell ratio. Michael also has an interesting juggling notation, and graphs relating heights and rhythms of juggles.

A few people have asked about computer programs that illustrate juggling patterns. The fanciest one I've seen is David Greenberg's Juggling Simulator.

In the last "Academic Juggler," I showed that every possible club passing pattern has the property that, when repeated enough times without a drop, everybody gets back the clubs they started with. I said then that the proof can be recast in terms of mathematical group theory. In a prompt letter, Mike Keith took this point a step further by observing that each repetition of a passing pattern is a group operation, so upper bounds on the number of repetitions required are given by the maximum "order" of groups.

Let k be the maximum number of pattern repetitions required for everyone to get their own clubs back. By consulting stan­dard tables on group orders, Mike has shown that:

If the number of clubs is:

6     9      12            15             18

then k is less than or equal to:

6     20    60            105          210

What does k equal for common actual passing patterns? For simple patterns, the easiest way to find out is to get some jug­glers together and do the pattern (without drops or tricks!), counting repetitions until everyone has their own clubs back.