Of all possible such networks, most large ones end up being connected. … If one requires the networks to be planar the numbers are {0, 1, 1, 3, 9, 32, 133, 681, 3893, 24809, 169206} . If one looks at subnetworks with dangling connections, the number of these up to size 10 is {2, 5, 7, 22, 43, 141, 373, 1270, 4053, 14671} , or {1, 1, 2, 6, 10, 29, 64, 194, 531, 1733} if no self or multiple connections are allowed (see also page 1039 ).

(Note that from the axioms of ring theory one can only expect to prove results that hold for any ring; to get most results in number theory, for example, one needs to use the axioms of arithmetic, which are intended to be specific to ordinary integers.)

But one can also imagine using other operations to combine such values. … One can also consider combining values of blocks by the multiplication operation in a group—and seeing whether the conjugacy class of the result is conserved.

It turns out that the system described here is closely related to one that arose in studying the register machine shown on page 100 . The system here can be represented by the rule n  If[EvenQ[n], 3n/2, 3(n + 1)/2] , while the one on page 100 follows the rule n  If[EvenQ[n], 3n/2, (3n + 1)/2] .

But most of the scientific work that was done ended up being based only on my earliest discoveries, and being very much within the framework of one or another of the existing sciences—with the result that it managed to make very little progress on any general and fundamental issues. One feature of the new kind of science that I describe in this book is that it finally makes possible the development of a basic understanding of the general phenomenon of complexity, and its origins.

One can update the gray level of each cell by using rules that are in a sense a cross between the totalistic cellular automaton rules that we discussed at the beginning of the last chapter and the iterated maps that we just discussed in the previous section . … The picture below shows a very simple case in which the new gray level of each cell is exactly the average of the one for that cell and its immediate neighbors.

And with an appropriate form for these blocks what one finds is that the configuration of blocks evolves exactly according to rule 90. … One consequence of this is that the patterns produced by rule 90 have a nested or self-similar form.

But since one realistically cannot keep track of all these things, the ocean will inevitably seem in many respects unpredictable and random. This same basic effect can be even more pronounced when one looks at smaller-scale systems.

But given the spectrum, one can immediately tell how we will perceive the sound. When the spectrum is dominated by just one large peak, we hear a definite tone.

But in general the problem of working out what model is most appropriate for any given set of data is an extremely difficult one. … For typically it restricts itself to very specific classes of models—and usually ones which even by the standards of this book are extremely simple.