And, again just like with leaves and shells, it seems likely that among the animals we see are ones that correspond to a fair fraction of the possible choices for relative rates of growth. … From experience with traditional mathematical models, however, one might then assume that this would inevitably imply that all plants and animals would have forms that look quite similar. … And in the past it has usually been assumed that to get the kind of complexity that one sees in such patterns there must be some highly complex underlying mechanism, presumably related to optimization through natural selection.

In a sense it is surprising that so much could be done on the Game of Life without the much simpler one-dimensional cellular automata in this chapter ever being investigated. … It is not uncommon in the history of science that once a general new phenomenon has been identified, one can see that there was already evidence of it much earlier. … It is also one of the ironies of progress in science that results which at one time were so unexpected that they were missed despite many hints eventually come to seem almost obvious.

It leads to the so-called twin paradox in which less time will pass for a member of a twin going at high speed in a spacecraft than one staying at rest. … Once the basic assumptions are established, the derivation of time dilation given here is no different in principle from the original one given in 1905, though I believe it is in many ways considerably clearer. … If these were parallel, one would inevitably get not just pure time dilation, but a mixture of it and length contraction.

And indeed even in traditional general relativity one can try avoiding introducing matter explicitly—for example by imagining that everything we call matter is actually made up of pure gravitational energy, or of something like gravitational waves. But so far as one can tell, the details of this do not work out—so that at the level of general relativity there is no choice but to introduce matter explicitly. … Yet presumably one can still use the Einstein equations on large scales if one introduces matter with appropriate properties as a way to represent small-scale effects in the network.

But as a matter of principle one can ask whether the methods of perception and analysis that we have discussed in a sense cover what is ultimately possible—or whether instead there are higher and fundamentally more powerful forms of perception and analysis that for some reason we do not at present use. … So the point is that these regularities are just not ones that can be detected by our standard methods of perception and analysis. … So might one day some new method of perception and analysis be invented that would in a sense manage to recognize all possible regularities, and thus be able to tell immediately if any particular piece of data could be generated from any kind of simple description?

by executing the appropriate sequence of machine instructions on whatever computer system one is using. … For one knows that given a single fixed underlying language, it is possible to describe an almost arbitrarily wide range of things. … And knowing this, one might conclude that any system which was universal must include direct analogs of these specific elements.

Despite the fact that mobile automata update only one cell at a time, it is thus still possible for them to produce behavior of great complexity. … The basic idea of such generalized mobile automata is to allow more than one cell to be active at a time. … If one chooses generalized mobile automata at random, most of them will produce simple behavior, as shown in the first few pictures on the facing page .

So if one makes a list of all possible axiom systems—say starting with the simplest—where in such a list should one expect to see axiom systems that correspond to traditional areas of mathematics? … How complicated an axiom system does one need for this? Textbook discussions of logic mostly use axiom systems at least as complicated as the first one on page 773 .

Substitution Systems One of the features that cellular automata, mobile automata and Turing machines all have in common is that at the lowest level they consist of a fixed array of cells. … In the typical case illustrated below, one has a sequence of elements—each colored say black or white—and at each step each one of these elements is replaced by a new block of elements. … But at least for these kinds of rules, one can make clearer pictures by thinking of each step not as replacing every element by a sequence of elements that are drawn the same size, but rather of subdividing each element into several that are drawn smaller.

Large-scale fractures It is remarkable to what extent very large-scale fractures can look like small-scale ones. … But when one looks at geological systems, for example, the smallest relevant scales for the cracks one sees are certainly no smaller than particles of soil. And as a result, one needs a more general mechanism, not just one that just relates to atoms and molecules.