Here different elements in the system do interact, but the result is still that all of them evolve to the same state. So far all the mechanisms for uniformity I have mentioned involve behavior that is in a sense simple at every level.

But in fact, in almost all cases, rather little turns out to be known, and indeed at any fundamental level the behavior that is observed has often in the past seemed quite mysterious. … It should be said at the outset that it is not my purpose to explain every detail of all the various kinds of systems that I discuss.

And what one sees is that even though the same very simple underlying model is used, there are all sorts of visually very different geometrical forms that can nevertheless be produced. … All shells produced by adding material according to fixed rules of the kind shown here have the property that throughout their growth they maintain the same overall shape.

the last two images on each row below all clumps of squares of the same color, and then all lines of squares of the same color, have explicitly been removed.

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.

Charge quantization It is an observed fact that the electric and other charges of all particles are simple rational multiples of each other. … But as soon as different particles are related by a non-Abelian symmetry group, then the discreteness of the representations of such a group immediately implies that all charges must be rational multiples of each other.

The symmetry of all the patterns is a consequence of the basic structure of totalistic rules.

Numbers of [cellular automaton] rules Allowing k possible colors for each cell and considering r neighbors on each side, there are k k 2r + 1 possible cellular automaton rules in all, of which k 1/2 k r + 1 (1 + k r ) are symmetric, and k 1 + (k - 1)(2r + 1) are totalistic. … For k = 2 , r = 2 there are 4,294,967,296 rules in all, of which 64 are totalistic. And for k = 3 , r = 1 there are 7,625,597,484,987 rules in all, with 2187 totalistic ones.)

And while ordinary integers still satisfy all the constraints, the system is sufficiently incomplete that all sorts of other objects with quite different properties also do. … Yet in doing group theory in practice one normally adds axioms that in effect constrain one to be dealing say with a specific group rather than with all possible groups.

And almost all the arguments in the book—while often not conceptually simple—require no specialized scientific or other knowledge to follow. … And while I hope that all the effort I have put into presentation in this book will make it easier for others, I do not expect it to be a quick process. … For though there are connections of all sorts, this book is first and foremost about a fundamentally new intellectual structure, that needs to be understood in its own terms, and cannot reasonably be fit into any existing framework.