For additive rules such as 60 and 90, and for partially additive rules such as 30 and 45, any possible sequence can occur if an appropriate initial condition is given. For rule 18, it appears that any sequence can occur that never contains more than one adjacent black cell.

The pictures below show examples of behavior obtained with this rule. … (Note that all rules based on matrices are additive, reflecting the usual assumption of linearity at the level of amplitudes in quantum mechanics. Non-additive unitary rules can also be found.

And what I say about the irreducibility of processes in nature to short formal rules may seem to fit with Taoism.

3n+1 problem as cellular automaton If one writes the digits of n in base 6, then the rule for updating the digit sequence is a cellular automaton with 7 possible colors (color 6 works as an end marker that appears to the left and right of the actual digit sequence): {a_, b_, c_}  If[b  6, If[EvenQ[a], 6, 4], 3 Mod[a, 2] + Quotient[b, 2] /. 0  6 /; a  6] The 3n+1 problem can then be viewed as a question about the existence of persistent structure in this cellular automaton.

Continual injection of randomness [in cellular automata] In the main text we discuss what happens when one starts from random initial conditions and then evolves according to a definite cellular automaton rule.

In 1D, localized structures sometimes arise as defects in largely repetitive behavior, or more generally as boundaries between states with different properties—such as the different phases of the repetitive background in rule 110.

Directed network systems If one adds directionality to the connections in a network it becomes particularly easy to set up rules for clusters of nodes that cannot overlap.

And indeed, of the 4096 symmetric 5-neighbor rules, only identity and complement conserve e[s] . Of the 2 32 general 5-neighbor rules 34 conserve e[s] —but all have only very simple behavior. (Compositions of several such rules can nevertheless yield complex behavior.

And indeed, it seems that just like so many other examples that we have discussed in this book, the procedure for generating square roots is based on simple rules but nevertheless yields behavior of great complexity.

The first set of pictures below show what happens with some very simple rules.