For the top picture the pattern is what would be generated by an additive cellular automaton following rule 90; for the bottom picture it is what would be generated by one following rule 150.

Operator systems One can generalize symbolic systems by having rules that define transformations for any Mathematica pattern. … The rule x_  x ∘ x then for example generates a balanced binary tree. The pictures below show the patterns of opening and closing parentheses obtained from operator system evolution rules in a few cases.

And the analog of this for networks is that by applying the rules shown below at appropriate locations it is possible to transform any network into any other. These rules correspond to the moves invented by James Alexander in 1923 in connection with transforming one knot into another. (Note that the first two rules suffice for all planar networks, and are sometimes called respectively T2 and T1.)

Implementation [of tag systems] With the rules for case (a) on page 94 given for example by {2, {{0, 0}  {1, 1}, {1, 0}  {}, {0, 1}  {1, 0}, {1, 1}  {0, 0, 0}}} the evolution of a tag system can be obtained from TSEvolveList[{n_, rule_}, init_, t_] := NestList[If[Length[#] < n, {}, Join[Drop[#, n], Take[#, n] /. rule]]&, init, t] An alternative implementation is based on applying to the list at each step rules such as {{0, 0, s___}  {s, 1, 1}, {1, 0, s___}  {s}, {0, 1, s___}  {s, 1, 0}, {1, 1, s___}  {s, 0, 0, 0}} There are a total of ((k r + 1 - 1)/(k - 1)) k n possible rules if blocks up to length r can be added at each step and k colors are allowed.

Implementation [of sequential substitution systems] Sequential substitution systems can be implemented quite directly by using Mathematica's standard mechanism for applying transformation rules to symbolic expressions. … The rule on page 82 can then be given simply as s[1, 0]  s[0, 1, 0] while the rule on page 85 becomes {s[0, 1, 0]  s[0, 0, 1], s[0]  s[0, 1, 0]} The Flat attribute of s makes these rules apply not only for example to the whole sequence s[1, 0, 1, 0] but also to any subsequence such as s[1, 0] . … And with this setup, t steps of evolution can be found with SSSEvolveList[rule_, init_s, t_Integer] := NestList[(# /. rule)&, init, t] Note that as an alternative to having s be Flat , one can explicitly set up rules based on patterns such as s[x___, 1, 0, y___]  s[x, 0, 1, 0, y] .

Density oscillations in rule 73 Although there are always some fluctuations, most rules yield densities that converge more or less uniformly to their final values. One exception is rule 73, which yields densities that continue to oscillate with a period of 3 steps forever. The origin of this phenomenon is that with completely random initial conditions rule 73 evolves to a collection of independent regions, as in the picture below, and many of these regions contain patterns that repeat with period 3.

The concept of rules Processes based on rules occur in a great many areas of human endeavor. Sometimes the rules serve mainly as a constraint. … Rule-based pictures.

Results of starting with the number 1, then applying the following rule: if the number at a particular step is even, multiply by 3/2; otherwise, add 1, then multiply by 3/2. … The system here can be represented by the rule n  If[EvenQ[n], 3n/2, 3(n + 1)/2] , while the one on page 100 follows the rule n  If[EvenQ[n], 3n/2, (3n + 1)/2] .

The pictures below show what happens when the local curvature—which is essentially the local rate of folding—is taken to vary according to several simple rules as one goes along a curve. … And it takes only very simple rules to generate shapes that look like the villi and other corrugated structures one often sees in animals. … If one looks at the sequence of events that take place in a Curves obtained by varying the local curvature according to definite rules as one goes from one end to the other.

But while this means that it might be possible for there to be arbitrariness in the causal network for the universe, it still tends to be my suspicion that there is not—and that in fact the particular rules followed by the universe do in the end have the property that they always yield the same causal network. … The most straightforward possibility is that it could work much like the substitution systems that we have discussed in the past few sections [ 9 , 10 , 11 ]—and that at each step some piece or pieces of the network could be replaced by others according to some fixed rule. … Starting with a network whose connections are like the edges of a tetrahedron, both the rules shown work by replacing each node at each step by a certain fixed cluster of nodes.