[No text on this page] An example of evolution according to rule 37R from an initial condition containing a fairly random region.

Even with various additional optimizations, it is remarkable how much slower it is to grow a cluster with a model that requires external random input than to generate similar patterns with models such as cellular automata that intrinsically generate their own randomness. The implementation above is a so-called type B Eden model in which one first selects a cell in the cluster, then randomly selects one of its neighbors. One gets extremely similar results with a type A Eden model in which one just randomly selects a cell from all the ones adjacent to the cluster.

[No text on this page] A cellular automaton whose behavior seems neither highly regular nor completely random.

But how is such randomness produced? … Despite their simple definition, all these sequences seem for practical purposes random.

And if one assumes that the program for each new offspring involves small random mutations then this means that over the course of many generations biological evolution will in effect carry out a random search for programs that maximize the fitness of an organism. … And what we found there is that for sufficiently simple constraints—particularly continuous ones—iterative random searches can converge fairly quickly to an optimal solution.

In the course of biological evolution random mutations will in effect cause a whole sequence of programs to be tried. … And if this is so, then in the course of random mutation, the chances are that the first program encountered that is successful enough to survive will also, quite coincidentally, exhibit complex behavior. … But what I believe instead is that the vast majority of the complexity we see in biological systems actually has its origin in the purely abstract fact that among randomly chosen programs many give rise to complex behavior.

certain kinds of loss of randomness are prevented—sometimes by explicit suspension of trading. But why is there randomness in markets in the first place? Practical experience suggests that particularly on short timescales much of the randomness that one sees is purely a consequence of internal dynamics in the market, and has little if anything to do with the nature or value of what is being traded.

The rules for generating primes are simple, yet their distribution seems in many respects random. But almost without exception mathematical work on primes has concentrated not on this randomness, but rather on proving the presence of various regularities in the distribution. … By the 1700s more than a hundred digits of π had been computed, and they appeared quite random.

It is often useful in practical computing to produce sequences of numbers that seem random. … But perhaps because these procedures always seemed quite ad hoc, no general conclusions about randomness and complexity were drawn from them. Along similar lines, systems not unlike the cellular automata discussed in this chapter were studied in the late 1950s for generating random sequences to be used in cryptography.

Talking about a random background affecting processes in the universe immediately tends to suggest certain definite relations between probabilities for different processes. … Nevertheless, a potentially important point is that it is in some ways misleading to think of particles in a network as just interacting according to some definite rule, and being perturbed by what is in essence a random background. For this suggests that there is in effect a unique history to every particle interaction—determined by the initial conditions and the configuration that exists in the random background.