[No text on this page] A sequence of totalistic rules involving nearest neighbors and four possible colors for each cell chosen to show transitions between rules with different classes of behavior.

And so at first I looked only at the 32 rules which had left-right symmetry and made blank backgrounds stay unchanged. Among these rules I found examples of repetition and nesting. … Nevertheless, as it happens, the first paper that I published about cellular automata—in 1983—did in fact include a picture of rule 30 from page 27 , as an example of a non-symmetric rule.

So what about cellular automata and other systems with simple rules? … But just as we found that the intuition which suggests that simple rules cannot lead to complex behavior is wrong, so also the intuition that simple rules cannot be universal also turns out to be wrong. … The rules for this cellular automaton itself are always the same.

The rule applies only to this active cell. … Much as for cellular automata, one can enumerate all possible rules of this kind; it turns out that there are 65,536 of them. The pictures at the top of the next page show typical behavior obtained with such rules.

[No text on this page] Universality in arithmetic, illustrated by an integer equation whose solutions in effect emulate the rule 110 universal cellular automaton from Chapter 11 . … In general any statement about the possible behavior of rule 110 can be encoded as a statement in arithmetic about solutions to the equation. … But the universality of rule 110 implies that such statements must in general be undecidable.

A continuous cellular automaton with a slightly more complicated rule. The rule takes the new gray level of each cell to be the fractional part of the average gray level of the cell and its neighbors multiplied by 3/2. The picture shows that starting from a single black cell, this rule yields behavior of considerable complexity.

by computers whose programs in effect implement a huge variety of rules. The programs we use in practice are mostly based on extremely complicated rules specifically designed to perform particular tasks. But a program can in principle follow essentially any definite set of rules.

And in rules like 0R and 90R shown on page 452 the period of repetition is always very short. But for rule 37R it usually turns out to be rather long. … But in rule 37R there is no such ergodicity.

But with most rules—including the one shown on the previous page —there are usually several possible replacements that can be made at each step. … But, again as in mobile automata, there are rules such as (a) in which there is no growth—and effectively no notion of space. … In the case of rules (a) and (b) the result is to update every single element at every step.

The first stripe carries the color of the left-hand neighbor, and causes all cases in the rule where that neighbor does not have the appropriate color to be eliminated. … But the universal cellular automaton is in no way restricted to emulating only rules that involve nearest neighbors. And thus on the facing page , for example, it is shown emulating a rule that involves next-nearest as well as nearest neighbors.