Origins of Discreteness

In the previous section we saw that even though a system may on a small scale consist of discrete components, it is still possible for the system overall to exhibit behavior that seems smooth and continuous. And as we have discussed before, the vast majority of traditional mathematical models have in fact been based on just such continuity.

But when one looks at actual systems in nature, it turns out that one often sees discrete behavior—so that, for example, the coat of a zebra has discrete black and white stripes, not continuous shades of gray. And in fact many systems that exhibit complex behavior show at least some level of overall discreteness.

So what does this mean for continuous models? In the previous section we found that discrete models could yield continuous behavior. And what we will find in this section is that the reverse is also true: continuous models can sometimes yield behavior that appears discrete.

Needless to say, if one wants to study phenomena that are based on discreteness, it usually makes more sense to start with a model that is fundamentally discrete. But in making contact with existing scientific models and results, it is useful to see how discrete behavior can emerge from continuous processes.

The boiling of water provides a classic example. If one takes some water and continuously increases its temperature, then for a while nothing much happens. But when the temperature reaches 100°C, a discrete transition occurs, and all the water evaporates into steam.

It turns out that there are many kinds of systems in which continuous changes can lead to such discrete transitions.

The pictures at the top of the next page show a simple example based on a one-dimensional cellular automaton. The idea is to make continuous changes in the initial density of black cells, and then to see what effect these have on the overall behavior of the system.

One might think that if the changes one makes are always continuous, then effects would be correspondingly continuous. But the pictures on the next page demonstrate that this is not so.