Block rules [examples] These pictures show the behavior of rule (c) starting from some special initial conditions. … Note that even in rule (b) wraparound phenomena can lead to repetition periods that increase rapidly with n (e.g. 4820 for n = 20 , q = 15 ), but presumably not exponentially. In rule (d), the repetition periods can typically be larger than in rule (c): e.g. 803,780 for n = 20 , q = 13 .

For every single cellular automaton after all ultimately has a different underlying rule, with different properties and potentially different consequences. … And while it is indeed true that for almost every rule the specific pattern produced is at least somewhat different, when one looks at all the rules together, one sees something quite remarkable: that even though each pattern is different in detail, the number of fundamentally different types of patterns is very limited.

The Notion of Reversibility At any particular step in the evolution of a system like a cellular automaton the underlying rule for the system tells one how to proceed to the next step. … Rule 51 is reversible, so that it preserves enough information to allow one to go backwards from any particular step as well as forwards. Rule 254 is not reversible, since it always evolves to uniform black and preserves no information about the arrangement of cells on earlier steps.

The main reason is that it potentially allows one to discuss in a unified way systems that have completely different underlying rules. … And by thinking in terms of such computations, it then becomes possible to imagine formulating principles that apply to a very wide variety of different systems—quite independent of the detailed structure of their underlying rules. … The cellular automaton follows elementary rule 132, as shown on the bottom left.

And then by setting up appropriate rules and choosing initial conditions that contain only one darker cell, one can produce in the cellular automaton an exact emulation of every step in the evolution of a mobile automaton—as in the picture below. … The rules for the mobile automaton and the cellular automaton are shown above. In the rules for the cellular automaton, indicates a cell of any color.

And what this suggests is that it is possible to think of any process that follows definite rules as being a computation—regardless of the kinds of elements it involves. … And indeed in the end the only unfamiliar aspect of this is that the rules such processes follow are defined not by some computer program that we as humans construct but rather by the basic laws of nature. But whatever the details of the rules involved the crucial point is that it is possible to view every process that occurs in nature or elsewhere as a computation.

[No text on this page] A thousand steps in the evolution of a system with the same rule as on the previous page , but now starting with the number 512.

[No text on this page] Examples of class 4 cellular automata with totalistic rules involving nearest neighbors and three possible colors for each cell.

[No text on this page] Examples of sequential substitution systems whose rules involve three possible replacements.

And as a first example, the picture on the facing page shows a rule that produces a pattern whose surface has seemingly random irregularities, at least on a small scale. … The pictures on pages 179 – 181 show one rule, for example, that does not. … In order to get any kind of growth with this rule one must start with at least three black cells.