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the bumps, and if there is randomness in these bumps it leads to corresponding randomness in the motion of the car.

A somewhat similar example is a ball rolled along a rough surface. A question such as where the ball comes to rest will depend on the pattern of bumps on the surface. But now another feature of the initial conditions is also important: the initial speed of the ball.

And somewhat surprisingly there is already in practice some apparent randomness in the behavior of such a system even when there are no significant bumps on the surface. Indeed, games of chance based on rolling dice, tossing coins and so on all rely on just such randomness.

As a simple example, consider a ball that has one hemisphere white and the other black. One can roll this ball like a die, and then look to see which color is on top when the ball comes to rest. And if one does this in practice, what one will typically find is that the outcome seems quite random. But where does this randomness come from?

The answer is that it comes from randomness in the initial speed with which the ball is rolled. The picture below shows the motion of a ball with a sequence of different initial speeds. And what one sees is that it takes only a small change in the initial speed to make the ball come to rest in a completely different orientation.

Captions on this page:

A plot of the position of a ball rolled with various initial speeds. Time goes down the page. The ball starts on the left, with an initial speed given by the initial slope of the curve. The ball slows down as a result of friction, and eventually stops. The ball is half white and half black, and the stripes in the picture indicate which color is on top when the ball is at a particular position. The divergence of the curves in the picture indicate the sensitivity of the motion to the exact initial speed of the ball. Small changes in this speed are seen to make the ball stop with a different color on top. It is such sensitivity to randomness in the initial conditions that makes processes such as rolling dice or tossing coins yield seemingly random output.

From Stephen Wolfram: A New Kind of Science [citation]