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. |