December 10, 2013

Pinball Chaos

Sending a steel ball speeding across a tilted board studded with bumpers can be an addictive pastime—a tantalizing blend of skill to keep the ball in play and unpredictability in the ball's erratic path, rebound by rebound.

The pinball machine can serve as a model of deterministic chaos—a system that embodies a sensitive dependence on initial conditions. Balls with slightly different starting points end up following very different paths when they ricochet through the array of bumpers. Moreover, any uncertainty in a ball's initial position makes it difficult to predict where the ball will be even after just a few bounces.

Mathematician Henri Poincaré introduced this notion of "sensitive dependence on initial conditions" in the early part of the 20th century, when he tangled with the intricacies of predicting planetary motion.

In his 1908 essay "Science and Method," Poincaré wrote, ". . . it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon."

Decades later, mathematician and meteorologist Edward N. Lorenz discovered the same effect embodied in equations used to model weather systems. At a meeting in 1972 he presented a paper with the provocative title "Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?"

"I avoided answering the question," he writes in his 1993 autobiography, The Essence of Chaos, "but noted that if a single flap could lead to a tornado that would not otherwise have formed, it could equally well prevent a tornado that would otherwise have formed."

Nonetheless, the term "butterfly effect" soon entered the lexicon to describe the unpredictable, even potentially drastic, consequences of a small change.

Lorenz also tells a story about pinball machines—the earlier kind without flippers or flashing lights, with nothing but arrays of pins to disturb the motion of the ball.

One spring in the 1930s during his undergraduate years at Dartmouth College, Lorenz recounts, a few pinball machines appeared in local drugstores and eateries.

"Soon many students were occasionally winning, but more often losing, considerable numbers of nickels," he writes. "Before long the town authorities decided that the machines violated the gambling laws and would have to be removed, but they were eventually persuaded, temporarily at least, that the machines were contests of skill rather than games of chance, and were therefore perfectly legal."

If that were true, however, students should have been able to perfect their skills and become regular winners, Lorenz notes. That didn't happen because of the deterministic chaos inherent in the pinball machine.

The Exploratorium in San Francisco gives you two chances to study pinball chaos. The Tinkering Studio includes a do-it-yourself model, where you can place various objects in different positions on a sloping table to investigate their effect on the trajectories of spring-projected balls.

Controlling just the initial speed of the ball, it is remarkably difficult to get a ball to follow the same path and come to rest at a given location at the bottom, even with just a few circular bumpers on the table.

Not too far away, you can also try your skill on arcade-style pinball, but with an Exploratorium twist. You don't need any coins to play, and the sides of the machine are transparent so that you can see what goes on inside.


Lorenz, E.N. 1993. The Essence of Chaos. University of Washington Press.

Peterson, I. 1993. Newton’s Clock: Chaos in the Solar System. W.H. Freeman.

______. 1990. Islands of Truth: A Mathematical Mystery Cruise. W.H. Freeman.

Photos by I. Peterson