For practical reasons, such generators typically keep only, say, the rightmost 31 digits in the numbers at each step. … Keeping only a limited number of digits at each step makes it inevitable that the sequences produced will eventually repeat.

Any model is ultimately an idealization in which only certain aspects of a system are captured, and others are ignored. … But in some fields—especially those where traditional mathematics has been used the most extensively—it has come to be believed that in a sense the only truly objective or scientific way to test a model is to look at certain rather specific details.

And the only way to achieve this is to pick out only certain effects that are important, and to ignore all others.

In the past, the idea of optimization for some sophisticated purpose seemed to be the only conceivable explanation for the level of complexity that is seen in many biological systems. But with the discovery in this book that it takes only a simple program to produce behavior of great complexity, a quite different—and ultimately much more predictive—kind of explanation immediately becomes possible.

And indeed, what I suspect is that in the end natural selection can only operate in a meaningful way on systems or parts of systems whose behavior is in some sense quite simple. … 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.

To get some idea of this one can construct a kind of limit of the array on the next page in which the total number of pictures is in effect infinite, but only a specific infinitesimal region of each picture is shown. … So what this suggests is that it is almost inevitable that features such as the shapes of leaves can sometimes change greatly even when the underlying properties of plants change only slightly.

But in my experience the only way to find phenomena that one does not already know exist is to do very systematic and general experiments, and then to look at the results with as few preconceptions as possible. And while it takes only rather basic

For when one builds a cellular automaton one is in a sense always first setting up an array of cells to represent space itself, and then only subsequently considering the contents of space, as represented by the arrangement of colors assigned to the cells in this array. But if the ultimate model for the universe is to be as simple as possible, then it seems much more plausible that both space and its contents should somehow be made of the same stuff—so that in a sense space becomes the only thing in the universe.

And indeed, the only thing that is defined about each node is what other nodes it is connected to. … With two connections, only very trivial networks can ever be made.

For as I mentioned on page 19 , my very first experiments on cellular automata showed only very simple behavior, and it was only because doing further experiments was technically very easy for me that I persisted.