Page 31 Winter 1987-88
THE ACADEMIC JUGGLER
Tricks And A Numbers Formula
by Arthur Lewbel
by Arthur Lewbel
is a simple pleasure, but it deserves serious thought. As an art,
juggling could be probed and dissected much as literature and music
often are. As a sport, its statistics can be compiled, marveled at,
and debated. In addition, juggling, like gymnastics, combines the
sciences of physics and biomechanics. Its geometric patterns can be
studied mathematically, and juggling's history and role in society
would make fascinating social studies.
column, which will be a semiregular feature in Jugglers World, is
devoted to all kinds of analyses of juggling. Anybody who has
comments or potential contributions to this feature is encouraged to
write me (Arthur Lewbel,
jugglers may view any attempt to analyze juggling with skepticism or
contempt. Don't worry. These studies can't hurt, or take anything
away from the simple joy of juggling. At best, analysis may improve
juggling as both sport and art. At worst, it'll be a waste of time,
but one that some of us find entertaining.
begin with a question of interest to all performers: "what
makes a juggling trick memorable?" Unlike other arts, there
exist no classes on juggling appreciation.
audiences consist almost entirely of people who know nothing about
juggling (though this may change if Jugglebug and the Klutz books
become successful enough at introducing juggling to the masses).
Therefore, to be memorable, a juggling trick must be easy to
recognize and describe by a casual observer.
some of the most popular juggling tricks around: Eating an apple
while juggling, juggling many objects at once, and juggling bowling
balls, chainsaws, or torches. With each of these tricks, someone who
knows nothing about juggling can tell at a glance what is being
done, and could describe it in a short sentence, as I have done.
two qualities, "easy to recognize," and "easy to
describe," make a trick memorable to the nonjuggler. Of course,
making an entire act memorable is much harder, since it requires
many other attributes, including ability, charisma, organization,
personality, and character.
of my favorite subjects is numbers juggling, so I'll finish this
first column with a numbers formula based on Claude Shannon's
juggling theorems. Let N be the number of objects
being juggled, T be the "throw time," that
is, the time (in seconds) that a single object spends in the air
between when it's thrown and when it's caught, and D
be the "dwell time," that is, the time a single object
spends in the hand between when it's caught and when it's thrown.
definition, D + T is the total amount of time between
when an object is thrown once and when it's thrown again. In that
time, all N objects must have been caught and thrown, to
get back to the first one again. Therefore, the amount of time you
have to deal with each catch is (D + T)/N, and inverting
this gives the number of catches per second as N/(D+T).
Since you have two hands, this means that the catches per second per
hand is N/(2(D+T).
B equal "between time," that is, the time a
hand spends empty between throwing one object and catching the next.
The time between catches of adjacent objects into a hand must be B
+ D, so the number of catches per second per hand is N/(B+D).
the two expressions for catches per second per hand and solving for N
gives N=2(D+T)/(D+B), which shows how many objects can
be juggled for any given combination of throw times, dwell times, and
the "throw-dwell" ratio as T/D. T/D is
the amount of time a ball spends in the air relative to the time it
spends in a hand. Any numbers juggler knows that adding more objects
to a pattern requires throwing higher (increasing T),
throwing faster (decreasing D), or both.
Either way, increasing N requires increasing T/D. Since B is greater than 0, the above formula can be rewritten in terms of the throw-dwell ratio as (T/D) is greater than (N/2)-1. This describes the minimum throw-dwell ratio required for juggling N objects. By measuring a juggler's maximum throw-dwell ratio, this formula determines the greatest number of objects a person could possibly juggle.