But in fact it turns out to be fairly straightforward to do so, as illustrated at the bottom of the facing page .

And since there are many kinds of behavior that do not return to such predictable forms after any limited number of steps, one must conclude that additive rules cannot be universal.

Out of about 6 billion humans, however, it is notable that only extremely few choose, say, to explore life in the depths of the oceans—though perhaps this is just because technology has not yet made it easy to do. … But as a practical matter, it seems likely that there will be vastly more room to do more extensive computations by using smaller components than by trading and collaborating with even millions of other civilizations. … And what such entities might do has to some extent been considered in the context of the notion of heaven in theology and art.

Instead, once the threshold for complex behavior has been reached, what one usually finds is that adding complexity to the underlying rules does not lead to any perceptible increase at all in the overall complexity of the behavior that is produced. … 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.

Sensitive dependence on initial conditions thus does not in and of itself imply that a system will behave in a random way. Indeed, all it does is to cause digits which make an arbitrarily small contribution to the size of numbers in the initial conditions eventually to have a significant effect.

With intrinsic randomness generation, however, there is no such limit: in the cellular automaton above, for example, all one need do to get a longer random sequence is to run the cellular automaton for more steps. … The basic reason is that intrinsic randomness generation in a sense puts all the components in a system to work in producing new randomness, while getting randomness from the environment does not.

But in most of their obvious structural features animals do not typically look much like plants at all. … And an immediate consequence of this is that the kind of branching that one sees in plants does not normally occur in horns.

But one of the crucial points discovered in this book is that more complex phenomena do not always require more complex models. … But just knowing the underlying program does not mean that one can immediately deduce every aspect of how the universe will behave.

For at least as far as the computations that it can perform are concerned, it does not matter how sophisticated the underlying rules for the system are, or even whether the system is a cellular automaton, a Turing machine, or something else. … But how complicated do the underlying rules need to be in a specific case in order actually to achieve universality?

For other systems will tend to perform computations that are just as sophisticated as those we can do, even with all our mathematics and computers. … And at some level the Principle of Computational Equivalence does this as well.