Page 31                                             Winter 1987-88





Memorable Tricks And A Numbers Formula

 by Arthur Lewbel


Juggling 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.


This 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, Somerville , MA ).


Some 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.


I'll 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.


Juggling 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.


Consider 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.


These 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.


One 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.


By 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).


Let 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).


Equating 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 between times.


Define 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.

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