So even if a system at some level follows continuous rules it is still possible for the system to exhibit discrete overall behavior. … The situation is quite different from what happens with explicit evolution rules. For if one knows such rules then these rules immediately yield a procedure for working out what behavior will occur.

And indeed, what is perhaps most bizarre about the pictures is just how little trace they ultimately show of the simplicity of the underlying cellular automaton rule that was used to produce them. … Looking just at the original cellular automaton rule one would have no realistic way to foresee all of this. … The Need for a New Intuition The pictures in the previous section plainly show that it takes only very simple rules to produce highly complex behavior.

Even though some of them seem quite complex, it turns out that once again there is a rather simple kind of rule that can account for them. … In both rules shown the cell itself and its nearest neighbors enter with weight 1. Cells at distances 2 and 3 enter with negative weights— -0.4 per cell for the first rule, and -0.2 for the second.

And as a result, the idea of studying cellular automata with simple rules never surfaced, with the result that nothing like the experiments described in this chapter were ever done. In other areas, however, systems that are effectively based on simple rules were quite often studied, and in fact complex behavior was sometimes seen. … The rules for generating primes are simple, yet their distribution seems in many respects random.

But the new kind of science that I describe in this book introduces what are in a sense much more general abstract systems, based on rules of essentially any type whatsoever. … Traditional intuition might suggest that to do more sophisticated computations would always require more sophisticated underlying rules. … But one of the surprising discoveries in this book is that in fact there are systems whose rules are simple enough to describe in just one sentence that are nevertheless universal.

But if the rules can both increase and decrease string length the story is quite different, as the picture on the facing page illustrates. … Or is it just that the rule one has used is somehow inefficient, and there are other rules that generate the short strings much more quickly? Certainly one can take the rules for any multiway system and add transformations that immediately generate particular short strings.

In fact, it turns out that in continuous cellular automata it takes only extremely simple rules to generate behavior of considerable complexity. So as an example the picture below shows a rule that determines the new gray level for a cell by just adding the constant 1/4 to the average gray level for the cell and its immediate neighbors, and then taking the fractional part of the result. … A continuous cellular automaton whose rule adds the constant 1/4 to the average gray level for a cell and its immediate neighbors, and takes the fractional part of the result.

But in fact, as the pictures below show—and as we have seen many times in this book—it is also possible for randomness to arise intrinsically just through the application of simple underlying rules. … The rules in each case take a cell to become black if the specified number of its neighbors (including diagonals) on a square grid are black on the step before. These rules are such that once a cell has become black, corresponding to solid, it never reverts to white again.

(The three dots in the representation of each rule stand for the rest of the elements in the sequence.) The pictures at the top show the first hundred steps in evolution according to various rules starting from a pair of black elements.

Alternative rules [for cryptography] Among elementary rules, rule 45 is the only plausible alternative to rule 30. … (Changes expand about 1.24 cells per step in rule 30, and about 1.17 in rule 45.) Rule 45 shares with rule 30 the property of one-sided additivity.