The pictures at the bottom of the facing page show all the possible symmetrical rules that involve two colors and nearest neighbors. … Mollusc shells are almost unique in having patterns that are built up one line at a time; much more common is for patterns to develop all at once all over a surface.

Underneath the laws of physics as we know them today it could be that there lies a very simple program from which all the known laws—and ultimately all the complexity we see in the universe—emerges. … But from what I have found so far I am extremely optimistic that by using the ideas of this book the most fundamental problem of physics—and one of the ultimate problems of all of science—may finally be within sight of being solved.

And certainly this allows us with rather modest effort to do quite well in handling all sorts of data that we choose to interact with in everyday life. … There are definitely some limitations, since after all, if general human thinking could easily find simple descriptions of, for example, all the various pictures in this book, then we would never have considered any of them complex.

In the past it was often thought that logic might be an appropriate idealization for all of human thinking. … But it turns out that almost all of the major languages in use today are based on setting up procedures that are in essence direct analogs of step-by-step logical arguments. … For at the core of Mathematica is the notion of storing collections of rules in which each rule specifies how to transform all pieces of data that are similar enough to match a single Mathematica pattern.

But already in Chapter 3 we discovered that at the level of overall behavior, all of them had certain features in common. … The main point, as the previous two sections [ 5 , 6 ] have demonstrated, is that essentially all of these various kinds of systems—despite their great differences in underlying structure—can ultimately be made to emulate each other. … But it should be emphasized that among systems of any particular type—say cellular automata—not all possible underlying rules are capable of supporting the same kinds of computations.

even though the underlying laws for this system are perfectly definite, its overall behavior ends up being sufficiently complicated that many aspects of it seem to follow no obvious laws at all. … For if one looks at the individual cells in the cellular automaton one can plainly see that they just follow definite rules, with absolutely no freedom at all. … And it is in this separation, I believe, that the basic origin of the apparent freedom we see in all sorts of systems lies—whether those systems are abstract cellular automata or actual living brains.

At some rather abstract level one can immediately recognize a basic similarity between nature and mathematics: for in nature one knows that fairly simple underlying laws somehow lead to the rich and complex behavior we see, while in mathematics the whole field is in a sense based on the notion that fairly simple axioms like those on the facing page can lead to all sorts of rich and complex results. … Yet at some level there is still all sorts of complexity in mathematics. … And then the point is that all these computations follow the Principle of Computational Equivalence, so

but the simplest results can get much longer—making it in practice often difficult to tell whether a given result can actually even be proved at all. … But one might have assumed that to achieve their universality these axiom systems would have to be specially set up with all sorts of specific sophisticated features. … After all, what it means for an axiom system to be universal is that by setting up a suitable encoding it must in principle be possible to make that axiom system reproduce any other possible axiom system.

All the rules shown start with f[1]=f[2]=1 .

Capabilities [of sequential substitution systems] Even with the single rule {s[1, 0]  s[0, 1]} , a sequential substitution system can sort its initial conditions so that all 0's occur before all 1's.