For k = 2 , there are 16 possible rules, and the most complicated pattern obtained is nested like the rule 90 elementary cellular automaton.

With 3 states and 2 colors, the maximum period is 24, and about 0.37% of rules yield non-repetitive behavior, always nested.

Continued fractions The first n terms in the continued fraction representation for a number x can be found from the built-in Mathematica function ContinuedFraction , or from Floor[NestList[1/Mod[#, 1]&, x, n - 1]] A rational approximation to the number x can be reconstructed from the continued fraction using FromContinuedFraction or by Fold[(1/#1 + #2 )&, Last[list], Rest[Reverse[list]]] The pictures below show the digit sequences of successive iterates obtained from NestList[1/Mod[#, 1]&, x, n] for several numbers x . … As discovered by Jeffrey Shallit in 1979, numbers of the form Sum[1/k 2 i , {i, 0, ∞ }] that have nonzero digits in base k only at positions 2 i turn out to have continued fractions with terms of limited size, and with a nested structure that can be found using a substitution system according to {0, k - 1, k + 2, k, k, k - 2, k, k + 2, k - 2, k} 〚 Nest[Flatten[{{1, 2}, {3, 4}, {5, 6}, {7, 8}, {5, 6}, {3, 4}, {9, 10}, {7, 8}, {9, 10}, {3, 4}} 〚 # 〛 ]&, 1, n] 〛 The continued fractions for square roots are always periodic; for higher roots they never appear to show any significant regularities.

With different substitution rules for each type of cell, the structure will in general be nested.

In some cases, the behavior is fairly simple, and the patterns obtained have simple repetitive or nested structures.

And for example in the 1960s early computer enthusiasts tried running various simple programs, and found that in certain cases these programs could succeed in producing nested patterns.

procedure can indeed be used to check that no purely repetitive pattern exists, but as we will see later in this chapter , it does not successfully detect the presence of even certain highly regular nested patterns.

And if one did this what one would find is that many of the rules exhibit obviously simple repetitive or nested behavior.

If the behavior of a system is obviously simple—and is say either repetitive or nested—then it will always be computationally reducible.

Generating repetitive patterns with continuous systems is straightforward, but generating even nested ones is not. Page 147 showed how Sin[x] + Sin[ √ 2 x] has nested features, and these are reflected in the distribution of eigenvalues for ODEs containing such functions.