But remarkably enough there do exist rules for which exactly the same causal network is obtained regardless of what scheme is used.

In each pair of pictures in the upper part of the page , the top one shows the form of the network before the replacement, and the bottom one shows the result after doing the replacement—with the cluster of nodes involved in the replacement being highlighted in both cases.

Part of the answer probably has to do with the fact that outside the piece of the network that corresponds to the particle, the network presumably matches up to yield uniform space in much the same way as without the particle.

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.

But just as one does not need a formal definition of life in order to study biology, so also it has not turned out to be necessary so far in this book to have a formal definition of complexity.

(In the case of rule 225, the width of the overall pattern does not grow at a fixed rate, but instead is on average proportional to the square root of the number of steps.)

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.

And potentially therefore our lack of higher forms of perception and analysis might in the end have nothing to do with any difficulty in implementing such forms, but instead may just be a reflection of the fact that we only have enough context to make descriptions of data useful when these descriptions are fairly close to the ones we get from our own built-in human methods of perception.

Examples of cellular automata that have simple underlying rules but whose overall behavior does not seem to correspond to computations with any kind of simple description in standard mathematical or other terms.

But for each specific computation we wanted to do, we always set up a cellular automaton with a different set of underlying rules.