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.
References:
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
