Once again, the behavior that results is always repetitive, and the repetition period can never be greater than the total number of possible positions for the dot. But as the picture shows, the actual repetition period jumps around considerably as the size of the system is changed. And as it turns out, the repetition period is again related to the factors of the number of possible positions for the dot—and tends to be maximal in those cases where this number is prime.
So what happens in systems like cellular automata?
The pictures on the facing page show some examples of cellular automata that have a limited number of cells. In each case the cells are in effect arranged around a circle, so that the right neighbor of the rightmost cell is the leftmost cell and vice versa.
And once again, the behavior of these systems is ultimately repetitive. But the period of repetition is often quite large.
The maximum possible repetition period for any system is always equal to the total number of possible states of the system.
For the systems involving a single dot that we discussed above, the possible states correspond just to possible positions for the dot, and the number of states is therefore equal to the size of the system.
But in a cellular automaton, every possible arrangement of black and white cells corresponds to a possible state of the system. With n cells there are thus 2n possible states. And this number increases very rapidly with the size n: for 5 cells there are already 32 states, for 10 cells 1024 states, for 20 cells 1,048,576 states, and for 30 cells 1,073,741,824 states.
The pictures on the next page show the actual repetition periods for various cellular automata. In general, a rapid increase with size is characteristic of class 3 behavior. Of the elementary rules, however, only rule 45 seems to yield periods that always stay close to the maximum of 2n. And in all cases, there are considerable fluctuations in the periods that occur as the size changes.
So how does all of this relate to class 2 behavior? In the examples we have just discussed, we have explicitly set up systems that have limited size. But even when a system in principle contains an infinite number of cells it is still possible that a particular pattern in that system will only grow to occupy a limited number of cells. And in any