The cellular automaton effectively does its computation using the standard sieve of Eratosthenes method.

The active cell in these cases does not move in a strictly repetitive way, but instead sweeps backwards and forwards, going progressively further every time.

Consequences of models Given a program it is always possible to run the program to find out what it will do.

And for the new kind of science in this book this is in a sense unprecedentedly easy, for all it requires is a standard computer on which to do computer experiments. … Would the system go on doing what one sees in the book, or might it start doing something quite different? … How long it will take a given individual to get to the point of being able to do something specific with the new kind of science in this book will depend greatly on their background and particular goals.

Do these also exhibit the same four classes of behavior that we have seen in one dimension? … Much as in one dimension this does not seem to happen with the very simplest possible kinds of rules.

So does the notion of time that emerges actually have the familiar features of time as we know it? … The other rules shown do not, however, suffer from this problem: in all of them progressively more points are reached in space as time goes on.

For there is often no obvious pattern to which equations ultimately have solutions and which do not. And even when equations do have solutions, the integers involved can be quite large.

And while ordinary integers still satisfy all the constraints, the system is sufficiently incomplete that all sorts of other objects with quite different properties also do. … Yet in doing group theory in practice one normally adds axioms that in effect constrain one to be dealing say with a specific group rather than with all possible groups.

Yet if any axiom system with just a single axiom is going to be able to do this, the axiom must be of the form {…  a} . … But it turns out that there are all sorts of other axioms that also do this.

Cellular automata whose behavior does and does not give the impression of being purposeful.