For the vast majority of rules written down at random, such problems do indeed occur. But it is possible to find rules in which they do not, and the pictures on the previous two pages [ 129 , 130 ] show a few examples I have found of such rules. … There are no such simple relationships for the other rules shown on the facing page .

The pictures on the next page show the array of possible forms that can be produced by rules in which each stem splits into exactly two new stems at each step. … But even in these cases the pictures show that comparatively small changes in underlying rules can lead to much more complex patterns. And so if in the course of biological evolution gradual changes occur in the rules, it is almost inevitable that complex patterns will sometimes be seen.

With great effort one might perhaps come up with some immensely complex rule that would work in most cases. … It is already difficult enough to work out from an underlying rule what behavior it will produce. … So how then could one ever expect to find the underlying rule in such a case?

In detail, some of the patterns are definitely more complicated than those seen in elementary rules. … And in the case of nested patterns even the specific structures seen are usually the same as for elementary rules. … The only new structure not already seen in elementary rules is the one in code 420—but this occurs only quite rarely.

Cellular automaton rules equivalent to multiplication of digit sequences in various bases. The left part of the picture shows the explicit form of the rule for base 6 and multiplier 3. The arrays of numbers summarize the rule for this case and other cases.

Then the point is that because the rules depend only on the color of a particular branch, and not on the colors of any neighboring branches, the subtrees that are generated from all the branches of the same color must have exactly the same structure, as in the pictures below. … Each rule yields a different sequence of elements, but all of them ultimately have simple nested forms. … Starting from the trunk at the bottom, the rules specify that at each step every branch of a particular color should split into smaller branches in the same way.

The most obvious exceptions are cases like rule 0R and rule 90R, where the behavior that is produced has only a very simple fixed or repetitive form. … The picture on the next page , however, shows the behavior of rule 37R over the course of many steps. … The picture on page 456 shows what happens if one starts rule 37R with a single small region of randomness.

But one of the discoveries of this book is that it is actually quite possible to generate what appears to be almost perfect randomness just by following definite underlying rules. … And in fact my guess is that the only way to show this with any certainty would be actually to find a specific set of multiway system rules with the property that regardless of the path that gets followed these rules would always yield behavior that agrees with the various observed features of our universe. … And indeed, in any multiway system with a limited set of rules, such sequences must necessarily be subject to all sorts of constraints.

One approach often used in practice is to form combinations of rules of the kind described above, and then to hope that the complexity of such rules will somehow have the effect of making cryptanalysis difficult. But as we have seen many times in this book, more complicated rules do not necessarily produce behavior that is fundamentally any more complicated. … Another consequence of additivity: the correspondence between colors of cells on rows and columns in the rule 60 cellular automaton.

But one can get more complicated behavior if one uses rules that involve more than just one possible replacement. … The picture on the next page shows a sequential substitution system with rule {ABA  AAB, A  ABA} involving two possible replacements. … At each step, the rule then has the effect of adding an A inside the string.