The answer is that for the vast majority of rules—including rules (c) through (g) in the picture on the facing page —using different schemes yields quite different behavior—and a quite different causal network. … Indeed, the pictures in the second image show how it can fail, for example, for rule (e) from the facing page . … One still has a The behavior of rules (a) and (b) from the facing page when replacements are performed at random.

Thus, for example, as shown below, rule 94 can effectively be described as enumerating even numbers. Similarly, rule 62 can be thought of as enumerating numbers that are multiples of 3, while rule 190 enumerates numbers that are multiples of 4. … In analogy to the previous page , the positions of white cells at the bottom of the rule 94 picture correspond to even numbers, on the left in rule 62 to multiples of 3, in rule 190 to multiples of 4, and in the center column of rule 129 to powers of 2.

But the result says nothing about whether such rules are somehow typical, or are instead very rare and special. And in practice, almost without exception, the actual rules that have been established to be universal have tended to be quite complex. … Indeed, from our discussion in the previous chapter , we already know that among the 256 very simplest possible cellular automaton rules at least rule 110 and three others like it are universal.

The nested structure seen in this pattern can then be viewed as a consequence of the fact that rule 184 is able to emulate itself. And the picture below shows that rule 184—unlike any of the additive rules—still produces recognizably nested patterns even when the initial conditions that are used are random. … Rule 184 evolving from a random initial condition.

[No text on this page] The effects of various levels of external randomness on the behavior of continuous cellular automata with generalizations of rules 90 and 30. … For the generalization of rule 90, the values of the left and right cells are added together, and the value of the cell on the next step is then found by applying the continuous generalization of the modulo 2 function shown at the right. For the generalization of rule 30, a similar scheme based on an algebraic representation of the rule is used.

Indeed, this kind of rapid increase in network complexity is a general characteristic of most class 3 and 4 rules. But it turns out that there are a few rules which at first appear to be exceptions. … The first two rules that are shown exhibit very simple class 2 behavior.

few of the 256 possible elementary rules. But for underlying rules that have more complex behavior—like rules 22, 30, or 110—it turns out that in the end it is always possible to emulate all 256 elementary rules. … And this suggests that such cellular automata will in the end turn out to be universal—with the result that out of the 256 elementary rules one expects that perhaps as many as 27 will in fact be universal.

The particular rule shown involves next-nearest as well as nearest neighbors and has rule number 4067213884. As in rule 184, the nested behavior seen here is most obvious when the density of black and white cells in the initial conditions is equal.

The picture is obtained by applying the simple rule shown for a total of 150 steps, starting with a single black cell. Note that the particular rule used here yields a pattern that expands on the left but not on the right. In the scheme defined in Chapter 3 , the rule is number 110.

In both the rules shown on the facing page , the only replacement specified is for the block . … In a system like a cellular automaton, the same underlying rule is in a sense always applied in exact synchrony to every cell at every step. … Simply by choosing the appropriate underlying rules it is possible to ensure that any sequence of events consistent with these rules will yield the same causal network and thus in effect the same perceived history for the universe.