But starting in the late 1970s, the field of fractal geometry emphasized the importance of nested shapes that contain arbitrarily intricate pieces, and argued that such shapes are common in nature. In this book we will encounter a fair number of systems that produce such nested shapes. But we will also find many systems that produce shapes which are much more complex, and have no nested structure.

In cases (c) and (d), the fluctuations have a regular nested form, and turn out to be directly related to the base 2 digit sequence of n .

Examples of three-color totalistic rules which yield nested patterns.

In case (c), it is nested—the size of the network at step t is related to the number of 1's in the base 2 digit sequence of t .

Complex maps Many kinds of nonlinear transformations on complex numbers yield nested patterns. … Transformations of the form z  {Sqrt[z - c], -Sqrt[z - c]} yield so-called Julia sets which form nested patterns for many values of c (see note below ). In fact, a fair fraction of all possible transformations based on algebraic functions will yield nested patterns.

And among other things this explains why it is that with simple initial conditions rule 126 produces exactly the same kind of nested pattern as rule 90. … This correspondence is the basic reason that rule 126 produces the same kind of nested patterns as rule 90 when it is started from simple initial conditions.

And as the first set of pictures below illustrates, nested patterns are generated very directly in substitution systems by each element successively splitting explicitly into blocks of smaller and smaller elements. … Nested patterns generated by simple branching processes.

But while some of these give behavior that looks slightly more complicated in detail, as in cases (a) and (b) on the next page , all ultimately turn out to yield just repetitive or nested patterns—at least if they are started with all cells white. … Repetitive and nested patterns are seen, but nothing more complicated ever occurs.

In the first example, the pattern obtained still has a simple nested structure. But in the second example, the behavior is more complicated, and there is no obvious nested structure.

Nested digit sequences The number obtained from the substitution system {1  {1, 0}, 0  {0, 1}} is approximately 0.587545966 in base 10. It is certainly conceivable that a quantity such as Feigenbaum's constant (approximately 4.6692016091) could have a digit sequence with this kind of nested structure. … The fact that nested digit sequences do not correspond to algebraic numbers follows from work by Alfred van der Poorten and others in the early 1980s.