The underlying rule used here involves eight possible colors for each cell.

And in addition, the rule for a Turing machine can depend on the state of the head, and on the color of the cell at the position of the head, but not on the colors of any neighboring cells.

Continuous generalizations [of additive rules] Functions such as Binomial[t, n] and GegenbauerC[n, -t, -1/2] can immediately be evaluated for continuous t and n .

The pictures below show the results with k = 2 (rule 60) for (a) Fibonacci[n] , (b) Thue–Morse sequence, (c) Fibonacci substitution system, (d) (Prime[n] - 1)/2 , (e) digits of π .

In a few cases—particularly with insects—somewhat complicated forms are seen, but it seems likely that these are actually produced by rather simple local rules like those in aggregation systems (see page 1011 ).

But just as some fraction of cellular automata with very simple rules show immensely complex behavior, so similarly it seems that some fraction of even simple instances of many NP-complete problems also tend to be difficult to solve.

And doing this shows for example that rule 150 conserves the total number of black cells modulo 2.

The 3n+1 problem The system described here is similar to the so-called 3n+1 problem, in which one looks at the rule n  If[EvenQ[n], n/2, (3n + 1)/2] and asks whether for any initial value of n the system eventually evolves to 1 (and thereafter simply repeats the sequence 1, 2, 1, 2, ...). … An alternative formulation is to ask whether for all n FixedPoint[(3#/2^IntegerExponent[#, 2] + 1)/2 &, n]  2 With the rule n  If[EvenQ[n], 5n/2, (n + 1)/2] used in the main text, the sequence produced repeats if n ever reaches 2, 4 or 40 (and possibly higher numbers). … Cases (b) and (c) use somewhat different rules that yield considerably simpler behavior.

But it also known that among the infinite sequences which do this, there are always nested ones (sometimes one has to iterate one substitution rule, then at the end apply once a different substitution rule).

Whether a pattern with all the properties expected in DLA is produced seems to depend in some detail on the rules for the gray cells. … No doubt there are also simpler cellular automaton rules that yield similar results.