translation between the objects one is describing—usually in effect just by renaming the operators used to manipulate them. … And if one does this it is indeed perfectly possible, say, to program arithmetic to reproduce any proof in set theory. … For given almost any general property that one can pick out in axiom systems like those on pages 773 and 774 there typically seem to be all sorts of operator and multiway systems—often including some rather simple ones—that share the exact same property.

And in a sense this means that if one tries directly to produce specific pieces of technology one can potentially always get stuck. … In chemistry for example one might start by studying the basic science of how all sorts of different substances behave. … In general one can think of technology as trying to take systems that exist in nature or elsewhere and harness them to achieve human purposes.

Yet if one sets up elements on a grid it is straightforward to allow the replacements for a given element to depend on its neighbors, as in the picture at the top of the next page . And if one does this, one immediately gets all sorts of fairly complicated patterns that are often not just purely nested—as illustrated in the pictures on the next page . In Chapter 3 we discussed both ordinary one-dimensional substitution systems, in which every element is replaced at each step, and sequential substitution systems, in which just a single block of elements are replaced at each step.

But now we must consider how such networks are transformed from one step in evolution to the next. … But to see the effect of any such rules, one must first find a uniform way of displaying the networks that can be produced. The pictures at the top of the next page show one possible approach based on always arranging the nodes in each network in a line across the page.

But what if one was just presented with the raw pattern of connections for some network? How could one see whether the network could correspond to ordinary space of a particular dimension? … So how then can one proceed?

And indeed in the course of this chapter we have seen that in every single one of the general kinds of systems that we have discussed, it ultimately takes only very simple rules to produce behavior of great complexity. One might nevertheless have thought that if one were to increase the complexity of the rules, then the behavior one would get would also become correspondingly more complex. … One observation that can be made from the examples in this chapter is that when the behavior of a system does not look complex, it tends to be dominated by either repetition or nesting.

The pictures on the facing page show what happens if one starts with a single circle, then successively adds new circles in such a way that the center of each one is as close to the center of the first circle as possible. … Traditional intuition might have made one assume that there must be a direct correspondence between the complexity of observed behavior and the complexity of underlying rules. But one of the central discoveries of this book is that in fact there is not.

The Breaking of Materials In everyday life one of the most familiar ways to generate randomness is to break a solid object. … At first one might think that it must be a reflection of random small-scale irregularities within the material. … The answer is that except in a few special cases the pattern of fracture one gets seems to look just as random as in other materials.

How can one now tell that such systems are reversible? … 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. … One such class is illustrated below.

But one can certainly imagine that space could work very differently. … At first, one might think that this would be completely inconsistent with everyday observations. For even though the individual cells in the array might be extremely small, one might still imagine that one would for example see all sorts of signs of the overall orientation of the array.