Examples of rules that yield patterns which seem to be on the edge between growth and extinction.

One might think that adding rules to a system could never reduce its computational sophistication. And this is correct if with suitable input one can always avoid the new rules. But often these rules will allow transformations that in effect short-circuit any sophisticated computation.

Even in the simplest case, however, with only four neighbors involved there are already (4k) k 5 possible rules, or nearly 10 29 even for k = 2 .

Multidimensional multiway systems As a generalization of multiway systems based on 1D strings one can consider systems in which rules operate on arbitrary blocks of elements in an array in any number of dimensions.

But the results so far in this book have shown that such intuition is far from correct, and that in reality even systems with extremely simple rules can give rise to behavior of great complexity. … For it implies that there is never an immediate reason to go beyond studying systems with rather simple underlying rules.

But in one respect all these systems have ultimately been set up in the same basic way: they are all based on explicit rules that specify how the system evolves from step to step. In traditional science, however, it is common to consider systems that are set up in a rather different way: instead of having explicit rules for evolution, the systems are just given constraints to satisfy.

The underlying rules in such systems involve a parameter that can vary smoothly from 0 to 1. … But since continuous cellular automata have underlying rules based on a continuous parameter, one can ask what happens if one smoothly varies this parameter—and in particular one can ask what sequence of classes of behavior one ends up seeing.

Of the elementary rules, however, only rule 45 seems to yield periods that always stay close to the maximum of 2 n .

For the patterns we saw are in effect built according to very simple plans—that just tell us to start with a single black cell, and then repeatedly to apply a simple cellular automaton rule. … But because the only situations in which we are routinely aware both of underlying rules and overall behavior are ones in which we are building things or doing engineering, we never normally get any intuition about systems like the ones at the end of the previous section .

this book even systems with very simple underlying rules can still perform computations that are as sophisticated as in any system. … (The rule used is the same as on page 740 .)