[No text on this page] Details of how the universal cellular automaton emulates rule 254. Each of the blocks in the universal cellular automaton represents a single cell in rule 254, and encodes both the current color of the cell and the form of the rule used to update it.

[No text on this page] Emulating the rule 110 cellular automaton using combinators. The rule 110 combinator from the previous page is applied once for each step of rule 110 evolution.

[No text on this page] An example of how a cellular automaton with three possible colors and nearest-neighbor rules can be emulated by a cellular automaton with only two possible colors but a larger number of neighbors (in this case five on each side). The basic idea is to represent each cell in the three-color rule by a block of three cells in the two-color rule, according to the correspondence given on the left. The three-color rule illustrated here is totalistic code 1599 from page 70 .

In rule (a), black and gray cells remain in localized regions. In rule (b), they move in fairly simple ways, and in rules (c) and (d), they move in a seemingly somewhat random way. The rules shown here are reversible, although their behavior is similar to that of non-reversible rules, at least after a few steps.

And out of the 256 rules discussed here, it turns out that 10 yield such apparent randomness. … Rule 22—like rule 90 from page 26 —gives a pattern with fractal dimension Log[2,3] ≃ 1.58 ; rule 150 gives one with fractal dimension Log[2, 1+Sqrt[5]] ≃ 1.69 . The width of the pattern obtained from rule 225 increases like the square root of the number of steps.

In addition, even when one can tell rather little from a single rule, it is often the case that rules which occur next to each other in some sequence have similar behavior. … The top row of rules all have class 1 behavior. … And after that, the remainder of the rules are mostly class 3.

[No text on this page] The universal cellular automaton emulating elementary rule 254. Each cell in rule 254 is represented by a block of 20 cells in the universal cellular automaton. Each of these blocks encodes both the color of the cell it represents, and the rule for updating this color.

With simple initial conditions of the type we have used so far this rule will always produce essentially trivial behavior. … But for rule 184, an appropriate choice of nested initial conditions yields the highly regular pattern shown below. The pattern produced by rule 184 (shown at left) evolving from a nested initial condition.

And with this extension, there are a total of 4,294,967,296 possible rules. If one samples these rules at random, one finds that more than 99% of them just yield simple repetitive behavior. … A mobile automaton with slightly more complicated rules that yields a nested pattern.

The rule used—that I call rule 30—is of exactly the same kind as before, and can be described as follows. … A cellular automaton with a simple rule that generates a pattern which seems in many respects random. … In the numbering scheme of Chapter 3 , the cellular automaton shown here is rule 30.