Page 27                                                     Summer 1989

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.


With two people passing in the shower pattern (every right hand throw being a  pass, also called "two-counts"), the entire pattern is just a single exchange of clubs, and it takes six repetitions for both jugglers to get their own clubs back, so k = 6. For passing "every others" (or "four counts"), k = 6 again, while for passing every thirds ("six counts"), k = 2.


George Strain wrote a computer program that simulates standard 3 club per person feeds, counting the repetitions k required for everyone to get their clubs back. The results for some feeds are:


People being fed to:

2   3   4   5   6   7   8   9   10

number of feeder's passes:

12   36   48   36   96   120   240 168 192

number of pattern repeats k:

6   9   6   3   6   6   10   6   6


These results are somewhat mysterious. Why does six repetitions work so often? Can anyone come up with a general formula that, for any number n, tells how many repetitions of an n-person feed is required for everyone to get their own clubs back? It would also be interesting to do similar counts for other passing patterns.


With all this talk of group theory and upper bounds, it is interesting to note that the work of mathematician and former IJA president Ron Graham is in the field of Ramsey theory (a branch of group theory), and deals with bounding values. In fact, Graham is listed in the "Guinness Book of World Records" for formulating the largest number ever used in a mathematics proof. In Ron's book about Ramsey theory, the chapter dealing with his number and simi­lar problems is entitled, "Eeeeeeeen­ormous Upper Bounds,"


The September 1987 issue of the Atlantic Monthly magazine has an article called, "The Man Who Loves Only Numbers," which is mainly about the mathematician Paul Erdos, but also talks a great deal about Ron Graham, and even briefly mentions the IJA.


"The Academic Juggler" is an occasional feature of Jugglers World, and is

devoted to all kinds of formal analyses of juggling. Anybody who has suggestions, comments, information or potential contri­butions for this feature is encouraged to write to: Arthur Lewbel; Lexington, MA.

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