Indeed, as was noticed around 1980, generalizations of additive rules involving cells in different relative locations can produce textures with similar statistics, but different visual appearance, as shown below.

Korec showed that finding elements in the nested pattern produced by the k = 3 cellular automaton with rule {{1, 1, 3}, {2, 2, 1}, {3, 3, 2}} 〚 #1, #2 〛 & (compare page 886 ) was also enough.

Properties [of network systems] Random behavior seems to occur in a few out of every thousand randomly selected rules of the kind shown here.

And indeed this phenomenon has been seen in other systems with randomness in their underlying rules.

Multiway systems are reversible whenever both a  b and b  a are present as rules, so that the system corresponds mathematically to a semigroup.

And indeed, as we discuss in Chapter 12 , it seems likely that above a fairly low threshold the vast majority of underlying rules can in fact in some way or another support arbitrarily complex computations—potentially allowing something one might call intelligence in a vast range of very different universes.

Many nested patterns (such as the one made by rule 90, for example) contain large areas of uniform white, and it is typically difficult to prevent pure repetition of that area. … One starts from the substitution system with rules {1  {{3}}, 2  {{13, 1}, {4, 10}}, 3  {{15, 1}, {4, 12}}, 4  {{14, 1}, {2, 9}}, 5  {{13, 1}, {4, 12}}, 6  {{13, 1}, {8, 9}}, 7  {{15, 1}, {4, 10}}, 8  {{14, 1}, {6, 10}}, 9  {{14}, {2}}, 10  {{16}, {7}}, 11  {{13}, {8}}, 12  {{16}, {3}}, 13  {{5, 11}}, 14  {{2, 9}}, 15  {{3, 11}}, 16  {{6, 10}}} This yields the nested pattern below which contains only 51 of the 65,536 possible 2 × 2 blocks of cells with 16 colors.

The particular rule used here eventually generates all states beginning with a white cell.

And in general, in any system with definite rules that only ever visits a limited number of states, it is With each cell at each step having a gray level that is the average of its predecessor and its two neighbors the total amount of black is conserved, but eventually becomes spread uniformly throughout the system.

But just as in so many other situations that we have seen in this book, what we have seen is that in fact a very simple rule is all that is in the end needed.