[No text on this page] An example of evolution according to rule 37R from an initial condition containing a fairly random region.

Note that the detailed pattern of evolution—though not any final fixed point reached—can depend on the fact that the combinator rules are applied at each step in Mathematica /. order.

Note that a given rule can yield very different growth rates with different initial conditions. … With most rules, states that appear at one step can disappear at later steps. … The rule for the system can then be stated in terms of a difference vector—which for {"AB"  "BBB", "ABB"  "AAAB"} is {{-1, 2}, {2, -1}} .

[Generalized substitution system] rule (b) The maximum number of steps for which the rule can be applied occurs with initial conditions consisting of a white element followed by n black elements, and in this case the number of steps is 2 n + n .

Multidimensional additive rules The 2D analog of rule 90 yields the patterns shown below. … The 2D analog of rule 150 yields the patterns below; the fractal dimension of the structure in this case is Log[2, (1 + Sqrt[1 + 4/d]) d] .

Recognition of art One bizarre possibility is that forms like those from rule 30 could have been created as art long ago but not be recognized now. For while it is easy to tell that a cave painting of an animal is a piece of purposeful art, dots carved into a rock in an approximate rule 30 pattern might not even be noticed as something of human origin. But although there are many seemingly random painted patterns in caves from perhaps 30,000 BC, I would be amazed if any of them were actually produced by definite simple rules.

[No text on this page] All patterns that repeat in 10 or less steps under evolution according to rule 30.

And seeing this one might conclude that it must be essentially impossible to produce good randomness with any kind of system that has reasonably simple rules. But the rule 30 cellular automaton that we discussed above demonstrates that in fact this is absolutely not the case.

In previous chapters, I have argued that the basic mechanisms responsible for many processes that occur in nature can be captured by simple computer programs based on simple rules. … But what I will argue in this chapter is that the evidence we have suggests that the basic mechanisms at work can once again successfully be captured by simple programs based on simple rules.

The first stage, illustrated in the top picture on the next page , is to get a cyclic tag system to emulate an ordinary tag system with the property that its rules depend only on the very first element that appears at each step. Symbolic systems set up to emulate cellular automata that have rules 90 and 30.