Structures in Class 4 Systems

The next page shows three typical examples of class 4 cellular automata. In each case the initial conditions that are used are completely random. But after just a few steps, the systems organize themselves to the point where definite structures become visible.

Most of these structures eventually die out, sometimes in rather complicated ways. But a crucial feature of any class 4 systems is that there must always be certain structures that can persist forever in it.

So how can one find out what these structures are for a particular cellular automaton? One approach is just to try each possible initial condition in turn, looking to see whether it leads to a new persistent structure. And taking the code 20 cellular automaton from the top of the next page, the page that follows shows what happens in this system with each of the first couple of hundred possible initial conditions.

In most cases everything just dies out. But when we reach initial condition number 151 we finally see a structure that persists.

This particular structure is fairly simple: it just remains fixed in position and repeats every two steps. But not all persistent structures are that simple. And indeed at initial condition 187 we see a considerably more complicated structure, that instead of staying still moves systematically to the right, repeating its basic form only every 9 steps.

The existence of structures that move is a fundamental feature of class 4 systems. For as we discussed on page 252, it is these kinds of structures that make it possible for information to be communicated from one part of a class 4 system to another—and that ultimately allow the complex behavior characteristic of class 4 to occur.

But having now seen the structure obtained with initial condition 187, we might assume that all subsequent structures that arise in the code 20 cellular automaton must be at least as complicated. It turns out, however, that initial condition 189 suddenly yields a much simpler structure—that just stays unchanged in one position at every step.

But going on to initial condition 195, we again find a more complicated structure—this time one that repeats only every 22 steps.

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