Yet looking at these patterns one notices a remarkable similarity to patterns that we have seen many times before in this book—generated by simple one-dimensional cellular automata. … In each close-up the pattern grows from top to bottom, just like in a one-dimensional cellular automaton. … Most of the shells are between one and four inches long; the one on the bottom right is nine inches long.

So how can one force more complex patterns to occur? The basic answer is that one must extend at least slightly the kinds of constraints that one considers. … So how can one find constraints that force more complex patterns?

Four Classes of Behavior In the previous section we saw what a number of specific cellular automata do if one starts them from random initial conditions. … One might at first assume that such a general question could never have a useful answer. … These classes are conveniently numbered in order of increasing complexity, and each one has certain immediate distinctive features.

But what if one wants to go backwards? … For any cell that has one color at a particular step must always have had the opposite color on the step before. But the second cellular automaton works differently, and does not allow one to go backwards.

First, those aspects of data that are not relevant for whatever purpose one has can simply be ignored. And second, one can avoid explicitly having to specify every element in the data by making use of regularities that one sees. … Such a summary is important whenever one wants to store or communicate data efficiently.

For even in the very best case any block of cells in the input can never be compressed to less than one cell in the output. So how can one achieve greater compression? … Each section of output starts with an element which indicates whether what follows is a new sequence, or a pointer to a previous one.

Turing machines (b), (c) and (d) are ones that always compute the same function. … Nevertheless, if one looks at the maximum number of steps needed for any given length of input one finds that this still always increases exactly linearly—just as for the Turing machines that add 1 shown at the top of this page. … And it turns out that at least among 2-state 2-color Turing machines this is the only one that computes the function it computes—so that at least if one wants to use a program this simple there is no faster way to do the computation.

So can one speed this up? The more one knows about a particular system, the more one can invent tricks that work for that system. … But if one starts these rules from random initial conditions, one typically never gets the pattern of page 211 .

incomplete, so that there must be more than one kind of object that can satisfy its constraints. Yet it is rather close to being complete—since as we saw earlier one has to go through at least millions of statements before finding ones that it cannot prove true or false. … Yet in doing group theory in practice one normally adds axioms that in effect constrain one to be dealing say with a specific group rather than with all possible groups.

Traditional mathematics and the existing theoretical sciences would have suggested using a basic methodology in which one starts from whatever behavior one wants to study, then tries to construct examples that show this behavior. … And in much the same way, one can set up a program on a computer and then watch how it behaves. … And this is what makes it possible to discover genuinely new phenomena that one did not expect.