Additive [continuous cellular automaton] rules In the case a = 0 the systems on page 159 are purely additive. A simpler example is the rule Mod[RotateLeft[list] + RotateRight[list], 1] With a single nonzero initial cell with value 1/k the pattern produced is just Pascal's triangle modulo k .

Multiway systems based on numbers One can consider for example the rule n  {n + 1, 2 n} implemented by NestList[Union[Flatten[{# + 1, 2 #}]] &, {0}, t] In this case there are Fibonacci[t + 2] distinct numbers obtained at step t . In general, rules based on simple arithmetic operations yield only simple nested structures.

For I no longer control the basic rules of the systems I am studying, and instead I must just try to deduce these rules from observation—with the potential that despite my best efforts my deductions could simply be incorrect.

Occurrences of progressively longer blocks in the pattern generated by rule 30 starting from a single black cell.

Alternating colors The pictures below show rules 45 and 73 with the colors of cells on alternate steps reversed.

Rule 22 With more complicated initial conditions the pattern is often no longer nested, as shown on page 263 .

The point is that of the 32 5-cell neighborhoods involved in the 2D cellular automaton rule, only some subset will have the property that the center cell remains unchanged after applying the rule.

Then in 1970 Roger Banks managed to show that the 2-state 5-neighbor symmetric 2D rule 4005091440 was able to reproduce all the same logical elements. (This system, like rule 110, requires an infinite repetitive background in order to support universality.) … If one considers rules with more than two colors, it becomes straightforward to emulate standard logic circuits.

Constructions of groups with undecidable word problems have been based on setting up relations that correspond to the rules in a universal Turing machine. … From the results in this book it seems likely that there are still much simpler examples—some of which could perhaps be found by setting up groups to emulate rule 110. … For ordinary multiway (semi-Thue) systems, an example with an undecidable word problem is known with 2 types of elements and 5 very complicated rules—but I am quite certain that much simpler examples are possible. (1-rule multiway systems always have decidable word problems.)

The picture below shows a simple rule by which such primes can be obtained. … Given the simplicity of this rule, one might imagine that the sequence of primes it generates would also be correspondingly simple.