[Nesting in] random walks It is a consequence of the Central Limit Theorem that the pattern of any random walk with steps of bounded length (see page 977 ) must have a certain nested or self-similar structure, in the sense that rescaled averages of different numbers of steps will always yield patterns that look qualitatively the same. As emphasized by Benoit Mandelbrot in connection with a variety of systems in nature, the same is also true for random walks whose step lengths follow a power-law distribution, but are unbounded.

Boolean networks Several lines of development from the cybernetics movement (notably in immunology, genetics and management science) led in the 1960s to a study of random Boolean networks—notably by Stuart Kauffman and Crayton Walker . Such systems are like cellular automata on networks, except for the fact that when they are set up each node has a rule that is randomly chosen from all 2 2 s possible ones with s inputs. … But for s > 2 , the behavior one sees quickly approaches what is typical for a random mapping in which the network representing the evolution of the 2 m states of the m underlying nodes is itself connected essentially randomly (see page 963 ).

And instead what I believe is that such differences are in essence just reflections of completely random changes in underlying genetic programs, with no systematic effects from natural selection. … And this observation fits precisely with the idea that complexity is easy to get by randomly sampling simple programs, but is hard for natural selection to handle in any kind of systematic way. … But the point is that what I have discovered in this book shows that in fact if one just chooses programs at random, then it is easy to get behavior of great complexity.

For the more superficial aspects of organisms—such as pigmentation patterns—it seems likely that among programs sampled at random a fair fraction will produce results that are not disastrous for the organism. But when one is dealing with the basic structure of organisms, the vast majority of programs sampled at random will no doubt have immediate disastrous consequences. … And indeed it is my strong suspicion that for essentially all purposes the only reasonable model for important new features of organisms is that they come from programs selected purely at random.

But in many cases, even with simple initial conditions, the patterns produced are highly complex, and seem in many respects random. … Most often, what happens is that a system which starts in a fairly regular or organized state becomes progressively more and more random and disorganized. … But over the course of time the picture shows that the arrangement of particles becomes progressively more random.

And at the lowest level what I expect is that even though the rules being applied are perfectly definite, the overall pattern of connections that will exist in the network corresponding to our universe will continually be rearranged in ways complicated enough to seem effectively random. Yet on a slightly larger scale such randomness will then lead to a certain average uniformity. … But superimposed on this effectively random background will then presumably also be some definite structures that persist through many updatings of the network.

It turns out that square roots are certainly not alone in having apparently random digit sequences. … And so far as one can tell, almost all these kinds of numbers also have apparently random digit sequences.

In neither case is it clear what the final outcome will be—whether apparent randomness will take over, or whether a simple repetitive form will emerge.

In neither case is it clear what the final outcome will be—whether apparent randomness will take over, or whether a simple repetitive form will emerge.

[No text on this page] A mobile automaton in which the position of the active cell moves in a seemingly random way.