Reported September 2006

BACKGROUND: Mathematical models of juggling give performers a better understanding of the science behind their tricks and help them develop new juggling routines. Several computer programs are available that identify workable juggling patterns and animate them. Jugglers can see what a particular pattern looks like before trying it out in real life. They can even check out juggling feats that are humanly impossible. Mathematical theory has suggested new juggling patterns, some of which are beginning to gain in popularity. These models are now on the Web, available for easy download for anyone who wants to learn how to juggle like a pro.

PATTERN RECOGNITION: For a single juggler, there are three basic patterns. The "cascade" is the most common, in which an odd number of balls are tossed from one hand to the other. Then there is the "fountain," in which balls are thrown and caught with the same hand, traditionally used for an even number of objects. Finally, there is the "shower," in which all the objects are tossed in a circle. A juggler may also choose to "multiplex:" throwing more than one object from a single hand simultaneously.

MODELING BEHAVIOR: Any mathematical model for juggling must incorporate both ball motion and hand motion. The motion behind juggling can be modeled as standard projectile motion, involving multiple objects with interweaving paths. The patterns of those paths are periodic cycles: they repeat, rather than change continuously. And the number of possible patterns is relatively small. The most common objects used by jugglers are balls, clubs and rings, and each has very different physical characteristics that determine how they can best be modeled on a computer. Balls can be modeled with standard particle system dynamics, while the more oddly shaped clubs and rings work better with a rigid body system. But no two throws or catches will ever be exactly the same because there are so many variables associated with throwing, including angle of release, release velocity, and height of throws. Skilled jugglers are able to tightly control such variables to throw the objects as consistently as possible.

The American Mathematical Society and the Mathematical Association of America contributed to the information contained in the TV portion of this report.

If you would like more information, please contact:

Ron Graham, Ph.D.
Mathematician
University of California San Diego
graham@ucsd.edu

American Mathematical Society
Providence, RI 02904-2294
(800)321-4AMS (4267)
http://www.ams.org

Mathematical Association of America
Washington, DC 20036-1358
(800) 741-9415
http://www.maa.org

FACTOID...

The earliest historical record of juggling dates back to ancient Egypt (between 1994 ad 1781 B.C.), in which female dancers and acrobats throwing balls adorn the tomb of an unknown prince.

ON THE WEB...

Movie

Virtual Juggling

Juggling Information Service