September 1, 2006 Even old jugglers can learn new tricks from mathematics. Several computer algorithms are able to simulate the combinatorial patterns of juggling and generate new ones that even experienced jugglers had never thought of.
SAN DIEGO -- If you've ever been to a circus, you've no doubt witnessed a juggling show. It takes skill, concentration, and even a little courage! But do you have what it takes to juggle? It looks hard, but now computer science makes learning new tricks easier for both beginners and pros.
Mathematicians Ron Graham and Joe Buhler have been juggling for more than 30 years.
"It's a nice combination of abstract form and pattern and physical activity, but really it's a just a very pure form of play," Buhler, a professor and director of communications research at Reed College in Portland, Ore., tells DBIS.
"It's like bicycle riding," says Graham, a computer science and mathematics professor at the University of California, San Diego. "Once you learn it, you don't forget."
They learned by watching others. Now, computer programs apply mathematics to help jugglers form new patterns.
"You just didn't realize what some things were possible until you actually saw them simulated on the computer, and you say, 'Oh, yes, I see,'" Graham says.
The program assigns a number to each throw. A one is when the juggler passes the ball directly to his other side. A four goes straight up, and a five is a little higher. You can make the computer-generated pattern easy or hard, and select just about any object to watch the program juggle. You can also watch multiple people perform patterns and see the ball from all angles.
While the program helps, Buhler says there's really only one way to become as good as him and Graham: "Practice. Lots of practice!"
You can download juggling programs on the Internet. Many of the programs are free, but some cost around $20.
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.