[No text on this page] All patterns that repeat in 10 or less steps under evolution according to rule 30.

the idea is to do all possible replacements at each step—and then to keep all the possible different sequences that are generated.

And indeed with this thought in mind all sorts of elaborate combinations of linear congruential and other generators have been proposed. … In all cases, the complete pattern has a triangular form, but except in the first case, it is truncated on the left here.

With two possible colors and blocks of size two the only kinds of block cellular automata that conserve the total number of black cells are the ones shown in the second set of pictures—and all of these exhibit rather trivial behavior. … It so happens that all but the second of the rules shown here not only conserve the total number of black cells but also turn out to be reversible.

The pictures below show three different cellular automata, all set up on the same two-dimensional grid. … In all cases the patterns produced follow at least to some extent the orientation of the initial block.

The key unifying idea that has allowed me to formulate the Principle of Computational Equivalence is a simple but immensely powerful one: that all processes, whether they are produced by human effort or occur spontaneously in nature, can be viewed as computations. … And as I discussed in the previous chapter the process of evolution of a system like a cellular automaton can for example perfectly well be viewed as a computation, even though in a sense all the computation does is generate the behavior of the system.

There will normally be many different possible choices of spacelike slices, but in all cases a particular such slice is set up to represent what can consistently be thought of as all of space at a given time. … Given a spacelike slice in a causal network, it is always possible to construct another such slice by finding all those points whose immediate predecessors are all included either in the original slice or its predecessors.

For when one deals with systems in nature it is normally impossible to set up or measure them with perfect precision—and indeed it can be a challenge even to make a traditional experiment be at all repeatable. … For in almost all cases they involve programs whose rules and initial conditions can be specified with perfect precision—so that they work exactly the same whenever and wherever they are run. … Yet as with all types of experiments it requires considerable skill and judgement to know how to set up a computer experiment that will yield meaningful results.

After all, between n = 2000 and 2001 there is only a 0.05% change in the size of n . … And in case (c), the fluctuations are determined by the total number of 1's that occur in the digit sequences of all numbers less than n . … But in general one suspects that all these rules can be thought of as being like simple computer programs that take some representation of n as their input.

So why then do higher organisms exist at all? … At first there is then rapid expansion, with many new species trying out all sorts of possibilities that have been opened up. … So what is the role of natural selection in all of this?