In some cases, the behavior is fairly simple, and the patterns obtained have simple repetitive or nested structures. But in many cases, even with simple initial conditions, the patterns produced are highly complex, and seem in many respects random.

The reversibility of the underlying rules has some obvious consequences, such as the presence of triangles pointing sideways but not down. But despite their reversibility, the rules still manage to produce the kinds of complex behavior that we have seen in cellular automata and many other systems throughout this book.

So what about localized structures?

The picture on the facing page demonstrates that these can also occur in reversible systems. There are some constraints on the details of the kinds of collisions that are possible, but reversible rules typically tend to work very much like ordinary ones.

So in the end it seems that even though only a very small fraction of possible systems have the property of being reversible, such systems can still exhibit behavior just as complex as one sees anywhere else.

Irreversibility and the Second Law of Thermodynamics

All the evidence we have from particle physics and elsewhere suggests that at a fundamental level the laws of physics are precisely reversible. Yet our everyday experience is full of examples of seemingly irreversible phenomena. Most often, what happens is that a system which starts in a fairly regular or organized state becomes progressively more and more random and disorganized. And it turns out that this phenomenon can already be seen in many simple programs.

The picture at the top of the next page shows an example based on a reversible cellular automaton of the type discussed in the previous section. The black cells in this system act a little like particles which bounce around inside a box and interact with each other when they collide.

At the beginning the particles are placed in a simple arrangement at the center of the box. But over the course of time the picture shows that the arrangement of particles becomes progressively more random.

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