The point is that these initial conditions in effect contain only blocks for which rule 126 behaves like rule 90. … In every example in this book where nested patterns like those from rule 90 are obtained it turns out that the underlying rules that are responsible can be set up to behave exactly like rule 90. … So what about rule 90 itself?

And among other things this explains why it is that with simple initial conditions rule 126 produces exactly the same kind of nested pattern as rule 90. Two examples of the fact that with special initial conditions rule 126 behaves exactly like rule 90. … This correspondence is the basic reason that rule 126 produces the same kind of nested patterns as rule 90 when it is started from simple initial conditions.

No single rule is able to emulate many others—and the rules that are emulated tend to be rather simple. An example of a slight surprise is that rule 45 ends up being able to emulate rule 90. But at least with blocks up to length 25, rule 30 for example is not able to emulate any non-trivial rules at all.

But what about the rules for the universe? … I suspect that the same will be true of the basic rule for the universe. The particular rule shown here is the elementary cellular automaton with rule number 94, and with initial condition .

The rules used are of the so-called totalistic type described on page 60 . With two possible colors, just 4 cases need to be specified in such rules, and there are 16 possible rules in all. … With three colors, there are 7 cases to be specified, and 2187 possible rules; with five colors, there are 13 cases to be specified, and 1,220,703,125 possible rules.

rules turn out to be far from minimal. … Different criteria for optimality could lead to different rules, but usually they will be rules like those on the facing page —and sometimes rules with quite complex behavior. … Its rule number is 5407067979.

TEST A cellular automaton with a slightly different rule. … In the numbering scheme of Chapter 3 , this is cellular automaton rule 250. … In the numbering scheme of Chapter 3 , it is cellular automaton rule 90.

[No text on this page] The behavior of all 256 possible cellular automata with rules involving two colors and nearest neighbors. … Note that some of the rules are related just by interchange of left and right or black and white (e.g. rules 2 and 16 or rules 126 and 129). There are 88 fundamentally inequivalent such elementary rules.

[No text on this page] The behavior of all 256 possible cellular automata with rules involving two colors and nearest neighbors. … Note that some of the rules are related just by interchange of left and right or black and white (e.g. rules 2 and 16 or rules 126 and 129). There are 88 fundamentally inequivalent such elementary rules.

So what about other underlying rules? The picture on the facing page shows for several different underlying rules which of the 256 possible elementary rules can successfully be emulated with successively wider blocks. In cases where the underlying rules have only rather simple behavior—as with rules 90 and 184—it turns out that it is never possible to emulate more than a Illustrations of how rule 30 can be set up to emulate a single step in the evolution of all elementary cellular automata.