Beyond uniformity and repetition, the one further type of simple behavior that we have often encountered in this book is nesting. … Nesting in one- and two-dimensional neighbor-independent substitution systems in which each element breaks into a block of smaller elements at each step.

But in general nesting need not just arise from larger elements being broken down into smaller ones: for as we have discovered in this book it can also arise when larger elements are built up from smaller ones—and indeed I suspect that this is its more common origin in nature.

The majority of the systems I consider are quite familiar from everyday life, and at first one might assume that the origins of their behavior would long ago have been discovered. … And in fact, to do this for even just one kind of system would most likely take at least another whole book, if not much more.

The pictures below show spectra obtained from nested sequences produced by various simple one-dimensional substitution systems. … Frequency spectra of nested sequences generated by one-dimensional neighbor-independent substitution systems.

So what happens if one considers more steps? … And given this, one can imagine finding for any particular rule the formula that involves the smallest number of Nand functions.

But as we saw in the previous chapter , such analysis tends to be useful only when the overall behavior one is studying is fairly simple. So what can one do when the behavior is more complex?

In traditional mathematics it is normally assumed that once one has an explicit formula involving standard mathematical functions then one can in effect always evaluate this formula immediately.

But in almost all cases, one imagines constructing examples to perform particular tasks, with a huge number of possible states and a huge number of possible colors for each cell. … Like a mobile automaton, the Turing machine has one active cell or "head", but now the head has several possible states, indicated by the directions of the arrows in this picture.

Inevitable regularities and Ramsey theory One might have thought that there could be no meaningful type of regularity that would be present in all possible data of a given kind. … As one example, consider looking for runs of m equally spaced squares of the same color embedded in sequences of black and white squares of length n . … But it turns out that for n ≥ 9 every single possible sequence contains at least one run of length 3.

If there is only one inequivalent model the axiom system is said to be categorical—a notion discussed for example by Richard Dedekind in 1887. … So this means that even if one tries to set up an axiom system to describe an uncountable set—such as real numbers—there will inevitably always be extra countable models. Any axiom system that is incomplete must always allow more than one model.