But among slightly larger clusters there turn out to be many that do not overlap themselves—and indeed this becomes common as soon as there are at least two connections between each dangling one. … One feature of the various rules I showed earlier is that they all maintain planarity of networks—so that if one starts with a network that can be laid out in the plane without any lines crossing, then every subsequent network one gets will also have this property. … So in the end, if one manages to find the ultimate rules for the universe, my expectation is that they will give rise to networks that on a small scale look largely random.

In an extreme but in practice common case one might happen to know what certain parts of the original message were—perhaps standardized greetings or some such—and by comparing the original and encrypted forms of these parts one can immediately deduce what the corresponding parts of the encrypting sequence must have been. And even if all one knows is that the original message was in some definite language this is still typically good enough. … So this means that just by looking at the distribution of spacings between repeats one can expect to determine the repetition period of the encrypting sequence.

But if one looks just at the absolute maximum number of steps for any given length of input one finds an exactly linear increase with this length. … One can readily enumerate all 4096 possible Turing machines with 2 states and 2 colors. … It turns out that there are 351 different functions that can be computed by one or more of the 4096 Turing machines with 2 states Examples of the behavior of a simple Turing machine that does the computation of adding 1 to a number.

But one of the core ideas of this book is to consider the more general scientific question of what arbitrary computational systems do. And much of what I have found is vastly different from what one might expect on the basis of existing computer science. … And by thinking about this in computational terms one develops a new intuition about the very nature of computation.

The pictures at the top of the facing page show what happens if one uses as the initial conditions for this system two numbers whose sizes differ by just one part in a billion billion. … And at least if one looks only at the sizes of numbers, this seems rather mysterious. But as soon as one looks at digit sequences, it immediately becomes much clearer.

purely in terms of size, one should make no distinction between numbers that are sufficiently close in size. … But it turns out that if one picks a number at random subject only to the constraint that its size be in a certain range, then it is overwhelmingly likely that the number one gets will have a digit sequence that is essentially random. And if one then uses this number as the initial condition for a shift map, the results will also be correspondingly random—just like those on the previous page .

For if one knows such rules then these rules immediately yield a procedure for working out what behavior will occur. Yet if one only knows constraints then such constraints do not on their own immediately yield any specific procedure for working out what behavior will occur. In principle one could imagine looking at every possible pattern, and then picking out the ones that satisfy the constraints.

So how then can one even expect to tell whether a particular program is a correct model for the universe? … So if there is indeed a definite ultimate model for the universe, how might one set about finding it? … So how then could one ever expect to find the underlying rule in such a case?

So one possibility is to define randomness so that something is considered random only if no short description whatsoever exists of it. … One might imagine that one could always just try running all programs with progressively longer descriptions, and see whether any of them ever generate what one wants. But the problem is that one can never in general tell in

One might have thought that such systems would be too diverse for meaningful general statements to be made about them. … But one of the surprising discoveries in this book is that in fact there are systems whose rules are simple enough to describe in just one sentence that are nevertheless universal. … And this one very basic principle has a quite unprecedented array of implications for science and scientific thinking.