An obvious observation in mathematics is that proofs can be difficult to do.

And just as in cellular automata, adding more complexity to the underlying rules does not yield behavior that is ultimately any more complex.

What I do here is simply to divide the whole width of the picture equally among all elements that appear at each step.

When the paths do not cross themselves, nested structure is evident.

And any configuration which does not change must involve only these subsets.

Often I was sure that some feature of the behavior of a system must be built into the underlying model—yet I could see no simple way to do it.

And doing this shows for example that rule 150 conserves the total number of black cells modulo 2.

For among other things the new kind of science in this book does not rely on elaborate abstract concepts from traditional mathematics; instead it is based mostly just on pictures, and on ideas that have become increasingly familiar from practical use of computers. And in fact, in my experience, with good presentation, surprisingly young children are able to grasp many key ideas in this book—even if their knowledge of mathematics does not go beyond the simplest operations on numbers. … They involve some of the same kinds of precise thinking, but do not rely on abstract concepts that are potentially very difficult to communicate.

(An example is the set of initial conditions for which a Turing machines does not halt.) … (Showing that a statement with n ≥ 1 is undecidable does not establish that it is always true or always false.)

Pattern-avoiding sequences As another form of constraint one can require, say, that no pair of identical blocks ever appear together in a sequence, so that the sequence does not match {___, x__, x__, ___} . … But it also known that among the infinite sequences which do this, there are always nested ones (sometimes one has to iterate one substitution rule, then at the end apply once a different substitution rule).