## The Phenomenon of Continuity

Many systems that we encounter in nature have behavior that seems in some way smooth or continuous. Yet cellular automata and most of the other programs that we have discussed involve only discrete elements. So how can such systems ever reproduce what we see in nature?

The crucial point is that even though the individual components in a system may be discrete, the average behavior that is obtained by looking at a large number of these components may still appear to be smooth and continuous. And indeed, there are many familiar systems in nature where exactly this happens.

Thus, for example, air and water seem like continuous fluids, even though we know that at a microscopic level they are both in fact made up of discrete molecules. And in a similar way, sand flows much like a continuous fluid, even though we can easily see that it is actually made up of discrete grains. So what is the basic mechanism that allows systems with discrete components to produce behavior that seems smooth and continuous?

Most often, the key ingredient is randomness.

If there is no randomness, then the overall forms that one sees tend to reflect the discreteness of the underlying components. Thus, for example, the faceted shape of a crystal reflects the regular microscopic arrangement of discrete atoms in the crystal.

But when randomness is present, such microscopic details often get averaged out, so that in the end no trace of discreteness is left, and the results appear to be smooth and continuous. The next page shows a classic example of this phenomenon, based on so-called random walks.

Each random walk is made by taking a discrete particle, and then at each step randomly moving the particle one position to the left or right. If one starts off with several particles, then at any particular time, each particle will be at a definite discrete position. But what happens if one looks not at the position of each individual particle, but rather at the overall distribution of all particles?

The answer, as illustrated on the next page, is that if there are enough particles, then the distribution one sees takes on a smooth and