The point then is that a human rolling the ball will typically not be able to control this speed with sufficient accuracy to determine whether black or white will end up on top. And indeed on successive trials there will usually be sufficiently large random variations in the initial speed that the outcomes will seem completely random.

Coin tossing, wheels of fortune, roulette wheels, and similar generators of randomness all work in essentially the same way. And in each case the basic mechanism that leads to the randomness we see is a sensitive dependence on randomness that is present in the typical initial conditions that are provided.

Without randomness in the initial conditions, however, there is no randomness in the output from these systems. And indeed it is quite feasible to build precise machines for tossing coins, rolling balls and so on that always produce a definite outcome with no randomness at all.

But the discovery which launched what has become known as chaos theory is that at least in principle there can be systems whose sensitivity to their initial conditions is so great that no machine with fixed tolerances can ever be expected to yield repeatable results.

A classic example is an idealized version of the kneading process which is used for instance to make noodles or taffy. The basic idea is to take a lump of dough-like material, and repeatedly to stretch this material to twice its original length, cut it in two, then stack the pieces on top of each other. The picture at the top of the facing page shows a few steps in this process. And the important point to notice is that every time the material is stretched, the distance between neighboring points is doubled.

The result of this is that any change in the initial position of a point will be amplified by a factor of two at each step. And while a particular machine may be able to control the initial position of a point to a certain accuracy, such repeated amplification will eventually lead to sensitivity to still smaller changes.

But what does this actually mean for the motion of a point in the material? The bottom pictures on the facing page show what happens to two sets of points that start very close together. The most obvious effect is that these points diverge rapidly on successive steps. But after a while, they reach the edge of the material and cannot diverge any