For in most cases all that one studies are rather simple features that can readily be extracted by very straightforward processes—and which can for example be described by just a few numbers or by a simple mathematical formula. … At the level of raw data the picture at the top of the facing page , for example, can be thought of as consisting of many thousands of individual black and white cells.

In cellular automata like the ones at the top of the facing page some information can be transmitted over larger distances. … These cellular automata are necessarily all class 1 or class 2 systems.

And so for example all rules that lie in the first two columns on page 232 can be shown to be unable ever to produce anything besides class 1 or class 2 behavior. … The top row of rules all have class 1 behavior.

For all one need do is to repeat the process that was used for encryption, and reverse the color of every square in (c) for which the corresponding square in (b) is black. … The picture at the top of the facing page shows an extremely simple approach that was widely used in practical cryptography until less than a century ago. … This scheme is the basis for essentially all practical stream ciphers.

And in the past there have certainly been several instances when new algorithms have suddenly allowed all sorts of computations to be done much more efficiently than had ever been thought possible before. … But what my discoveries have shown is that in fact even very small programs can be quite capable of doing all sorts of sophisticated computations. As a first example—based on a rather simple computation—the picture at the top of the facing page shows a Turing machine set up to add 1 to any number.

The basic idea of a block cellular automaton is illustrated at the top of the next page .

And in fact it turns out that for all the two-dimensional cellular automata shown on the last few pages [ 173 , 174 , 175 ], these shapes are always very regular. … But it is certainly not the case that all two-dimensional cellular automata produce only simple overall shapes. … The picture at the top of page 179 shows what happens with various numbers of black cells.

If one makes a fairly drastic perturbation, such as changing the colors of cells all the way from white to black, then the sequence will indeed often change, as illustrated in the pictures at the top of the next page .

But when the temperature reaches 100°C, a discrete transition occurs, and all the water evaporates into steam. … The pictures at the top of the next page show a simple example based on a one-dimensional cellular automaton.

But one can still check that starting with the specific configuration of cells at the bottom of each picture, one can evolve backwards to get to the top of the picture. … The rule for the system remains unchanged if all its elements are turned upside-down—effectively interchanging the roles of past and future.