|Ivars Peterson's MathTrek|
March 3, 1997
On a level, rectangular table, a practiced player can return the balls to their original positions, repeat the shot, and obtain the identical result. The geometry of the table, however, can greatly influence the types of motion possible in a game. Going from a rectangle to a circle or an ellipse or to some sort of polygon introduces new elements into the game -- even chaos and unpredictability -- and offers a variety of intriguing mathematical puzzles.
One mathematician who thought of playing billiards on a circular table was Charles L. Dodgson, better known as Lewis Carroll. In 1890, he published a set of rules for a two-player game of circular billiards, and he apparently had a suitable table built for the game.
Carroll's rules specified that a circular table must have a cushion all around, no pockets, and a surface marked with three spots arranged in an equilateral triangle, where three differently colored balls are initially placed.
Hitting a cushion and then a ball scored one point for the player; hitting two balls scored two points; a ball, cushion, then ball, three points; cushion, ball, ball, four points; and cushion, ball, cushion, ball, five points. A player scoring more than one point would get another shot right away.
Carroll noted, "The circular table will be found to yield an interesting variety of Billiard-playing, as the rebounds from the cushion are totally different from those of the ordinary game."
In mathematical billiards, such motions are idealized to make it easier to look for and detect patterns. There's only one ball, which travels at a constant speed forever. The ball moves in a straight line until it hits the cushion, then bounces back, obeying the rule that the angle of reflection equals the angle of incidence. There are no pockets for the ball to roll into.
Here are examples of what a multiple-bounce trajectory looks like on a circular table. Depending on the ball's starting position and initial direction, it can follow paths that never penetrate an inner circular region of a certain diameter in the middle of the table.
Trajectory after five bounces on a circular table.
Trajectory after 100 bounces.
On an elliptical table, billiard-ball trajectories show additional, surprising features. Placed at one focus and shot in any direction, the ball hits the cushion, bounces, and passes over the other focus. If there is no friction to slow the ball down, it continues to pass over a focus with each subsequent bounce. After a few bounces, its course ends up closely following the major axis of the ellipse.
Path of a ball that started at one focus (100 rebounds).
Two different possibilities occur when the ball starts at a position away from either focus. If it is driven so that it doesn't pass between the two foci, it continues along a path that outlines a smaller ellipse -- an excluded region -- with the same foci as the ellipse of the table.
Trajectory of a ball driven so it doesn't pass between the two foci (100 bounces).
A ball driven between the two foci of an ellipse travels along a path that passes again between them after rebounding from the cushion. The motion repeats endlessly, and the trajectory never gets closer to the foci than a hyperbola with the same foci as the elliptical table.
Path of a ball that does pass between foci (100 bounces).
Setting up a computer simulation of idealized billiard-ball trajectories on circular and elliptical tables allows one to explore the effects of changing the starting conditions. In one little exercise, I varied the ball position step by tiny step as a way of locating a focus, just by observing from the resulting trajectories on which side of the focus I happened to start. Fascinating patterns emerged in that experiment and in some of my other efforts.
A stadium-shaped billiard table -- one in which a rectangle's two flat ends are replaced by half circles -- presents its own surprises. In stadium billiards, a slight difference in starting point can radically alter a ball's trajectory. The more often the ball rebounds from the curved walls, the less predictable its path becomes. If there were no friction at all, the ball's continuing motion would appear quite random.
Researchers have also explored idealized billiard trajectories on a family of tables with shapes reminiscent of not only ovals but also peanuts, violins, moons, flippers, and droplets. The sensitive dependence on initial conditions characteristic of chaos lurks in a significant proportion of these geometries.
There are times when billiards can be much more of a game of chance than skill, no matter how expert the player!
Copyright © 1996 by Ivars Peterson.
Gardner, Martin. 1996. The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Games, Puzzles, and Word Plays. New York: Copernicus.
______. 1995. The ellipse. In New Mathematical Diversions. Washington, D.C.: Mathematical Association of America.
Steinhaus, H. 1969. Mathematical Snapshots. New York: Oxford University Press.
Billiard-ball trajectories on polygonal, circular, elliptical, and stadium-shaped tables, presented by Victor J. Donnay and his students and colleagues at Bryn Mawr College, are featured at http://serendip.brynmawr.edu/chaos/doc.html.
Special thanks to Robert Dickau of Wolfram Research for his help in generating the graphics using Mathematica 3.0 ( http://www.wolfram.com).
Comments are welcome. Please send messages to Ivars Peterson at firstname.lastname@example.org.