For in these cases complex and seemingly random results are obtained even on the first of the previous two pages [ 150 , 151 ]—when the original number has a very simple digit sequence. And the point is that these maps actually do intrinsically generate complexity and randomness; they do not just transcribe it when it is inserted in their initial conditions.

But while the first arrangement of colors shown below looks somewhat random, the last two are simple and purely repetitive. … Some, like the first arrangement above, look quite random.

Fourth, if random mutations can only, say, increase or decrease a length, then even if one mutation goes in the wrong direction, it is easy for another mutation to recover by going in the opposite direction. … And finally, as the results in Chapter 7 suggest, for anything beyond the very simplest forms of behavior, iterative random searches rapidly tend to get stuck, and make at best excruciatingly slow progress towards any kind of global optimum.

My guess is that it has almost nothing to do with optimality, and that instead it is essentially just a consequence of strings of random mutations that happened to add more and more features without introducing fatal flaws. … My guess is that as in other situations, its main systematic contribution is to make things simpler, and that insofar as things do end up getting more complicated, this is almost always the result of essentially random An example of a basic pattern that is produced in several variants by a wide range of simple programs.

In both cases, starting from a random distribution of black and white elements there quickly emerge definite patterns—in the first case a collection of spots, and in the second case a maze-like or labyrinthine structure. … Starting from random initial conditions, both rules quickly evolve to stationary states that look very much like pigmentation patterns seen in animals.

But just as in elementary cellular automata, there are some rules that yield behavior that seems in many respects random. … Beyond fairly uniform random behavior, there are also cases similar to elementary rule 110 in which definite structures are produced that interact in complicated ways.

And most often the stated reason for this would be that they do not seem to fit into any general framework of mathematical results, but instead just seem like isolated random mathematical facts. … And indeed I suspect that by looking at issues such as how easy a given theorem makes it to get from one part of a network to another it will be possible to formalize many intuitive notions about the practice of mathematics—much as earlier in this book we were able to formalize notions of everyday experience such as complexity and randomness.

[Causal networks for] 2D mobile automata As in 2D random walks, active cells in 2D mobile automata often do not return to positions they have visited before, with the result that no causal connections end up being created.

[No text on this page] Behavior of a two-dimensional cellular automaton starting from a random initial condition.

Examples of patterns produced by the evolution of each of the simplest possible symmetrical one-dimensional cellular automaton rules, starting from a random initial condition.