[No text on this page] Examples of three-color totalistic rules that yield patterns with seemingly random features.

For any such differences provide clear evidence that the data cannot in fact be considered random. … And the point is that if the sequences used as initial conditions for the other pictures are to be considered random then the behavior they yield should be similar. … And what this means is that in such cases statistical analysis based on simple cellular automata succeeds in recognizing that the sequences are not in fact random.

But the point is that these signals typically seem to us quite random. … Yet in the course of this book we have seen a great many examples where data that appears to us quite random can in fact be produced by very simple underlying rules. … But even if at some point we were to find that some of the seemingly random radio noise that we detect can be generated by simple rules, what would this mean about extraterrestrial intelligence?

Random walks In one dimension, a random walk with t steps of length 1 starting at position 0 can be generated from NestList[(# + (-1)^Random[Integer])&, 0, t] or equivalently FoldList[Plus, 0, Table[(-1)^Random[Integer], {t}]] A generalization to d dimensions is then FoldList[Plus, Table[0, {d}], Table[RotateLeft[PadLeft[ {(-1)^Random[Integer]}, d], Random[Integer, d - 1]], {t}]] A fundamental property of random walks is that after t steps the root mean square displacement from the starting position is proportional to √ t . … The character of random walks changes somewhat in different numbers of dimensions. … There are nevertheless some properties of random walks that are not isotropic.

Despite the simplicity of the rule on the facing page that generates the primes, the actual sequence of primes that is obtained seems in many respects remarkably random.

But as the results at the bottom of the facing page illustrate, the sequence of which numbers are even and which are odd seems to be completely random. Despite this randomness, however, the overall sizes of the numbers obtained still grow in a rather regular way. … Starting for example with the number 6, the sizes of the numbers obtained on successive steps show a generally increasing trend, but there are considerable fluctuations, and these fluctuations seem to be essentially random.

In the other cases, the patterns show many features of randomness.

[No text on this page] The same iterated maps as on the facing page , but now started from the initial condition π/4 —a number with a seemingly random digit sequence. … But looking at digit sequences one realizes that this complexity is actually just a direct transcription of complexity introduced by giving an initial condition with a seemingly random digit sequence.

[No text on this page] Examples of cellular automata that produce patterns with many apparently random features.

And in this picture one immediately sees some apparently random variation in the instructions that are executed. … The results appear to show considerable randomness. … With three registers, for example, seemingly random behavior can be obtained with a program that is seven rather than eight instructions long.