such case, the pattern must repeat itself with a period of at most 2^{n} steps, where n is the size of the pattern.

In a class 2 system with random initial conditions, a similar thing happens: since different parts of the system do not communicate with each other, they all behave like separate patterns of limited size. And in fact in most class 2 cellular automata these patterns are effectively only a few cells across, so that their repetition periods are necessarily quite short.

## Captions on this page:

Repetition periods for various cellular automata as a function of size. The initial conditions used in each case consist of a single black cell, as in the pictures on the previous page. The dashed gray line indicates the maximum possible repetition period of 2^{n}. The maximum repetition period for rule 90 is 2^((n-1)/2) - 1. For rule 30, the peak repetition periods are of order 2^(0.63 n), while for rule 45, they are close to 2^{n} (for n = 29, for example, the period is 463,347,935, which is 86% of the maximum possible). For rule 110, the peaks seem to increase roughly like n^{3}.