The particular rules shown are ones that lead to slow growth in the total number of elements. … Note that the pattern in case (a) does eventually repeat, while the one in case (b) eventually shows a nested structure.

motion, in which one takes a small grain, say of pollen, puts it in a liquid, and then looks at its motion under a microscope. What one finds is that the grain jumps around in an apparently random way. … But to observe random Brownian motion, one needs a microscope.

And indeed, in a modern semiconductor diode, for example, a breakdown event can be initiated by the motion of just one electron. … For one might think that microscopic physical processes would always produce the best possible randomness. … One might think that such effects could be avoided by allowing a certain "dead time" between successive events.

And it turns out that one of these is related to the method used since the late 1940s for generating random numbers in almost all practical computer systems. The pictures on the facing page show what happens if one successively multiplies a number by various constant factors, and then looks at the digit sequences of the numbers that result. … It has then often been assumed that having maximal repetition period will somehow imply maximum randomness in all aspects of the sequence one gets.

Over the past century or so it has become almost universally believed that at some level these programs must end up being the ones that maximize the fitness of the organism, and the number of viable offspring it produces. … But how successful can one expect such a search to be? … 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.

Yet one of the main conclusions of this book is that even given a particular program, it can be very difficult to see what the behavior of the program will be. … When one does engineering, one normally operates under the constraint that the systems one builds must behave in a way that is readily predictable and understandable. And in order to achieve this one typically

So to what extent does the actual history of biological evolution reflect the kinds of simple characteristics that I have argued one should expect from natural selection? If one looks at species that exist today, and at the fossil record of past species, then one of the most striking features is just how much is in common across vast ranges of different organisms. … One effect, to be discussed at greater length in the next section , is essentially just a matter of geometry.

One reason is just that they were not in the mainstream of any existing field of science or mathematics. … And perhaps one day some Babylonian artifact created using the rule 30 cellular automaton from page 27 will be unearthed. … For I tend to think that if pictures like the one on page 27 had ever in fact been seen in ancient times then science would have been led down a very different path from the one it actually took.

So should one conclude from this that the universe is in fact a giant cellular automaton with rules like those of case (c)? … For there are immediately simple issues like what one imagines happens at the edges of the cellular automaton array. … 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.

For one thing, there is no need to distinguish between incoming and outgoing connections, or indeed to associate any direction with each connection. … But if one uses three connections, a vast range of networks immediately become possible. One might think that one could get a