It turns out that they do not. And indeed the picture on the next page shows one of many examples in which starting from random initial conditions there continues to be very complicated behavior forever. And indeed the behavior that is produced appears in many respects completely random. But dotted around the picture one sees many definite white triangles and other small structures that indicate at least a certain degree of organization.
Captions on this page:
Examples of cellular automata that evolve from random initial conditions to produce a definite set of simple structures. For any particular rule, the form of these structures is always the same. But their positions depend on the details of the initial conditions given, and in many cases the final arrangement of structures can be thought of as a kind of filtered version of the initial conditions. Thus for example in the first rule shown here a structure consisting of a black cell occurs wherever there was an isolated black cell in the initial conditions. The rules shown are numbers 4, 108, 218 and 232.