In what follows I consider only non-degenerate networks, in which there is at most one connection between any two nodes.

And if one starts off with a network like the one in the bottom picture that can only be drawn with lines crossing, then what will happen is that the non-planarity of the network will be preserved.

And given only the underlying rule for a substitution system, it turns out to be fairly difficult to tell even roughly what the spectrum will be like.

For as the picture below demonstrates, in an additive cellular automaton like the one considered here the underlying rule is such that it allows one not only to deduce the form of a particular row from the row above it, but also to deduce the form of a particular column from the column to its right.

But if only every other cell in a column is given, almost nothing definite follows about the colors of other cells.

But as a practical matter one can say that not only have direct attempts to find easy ways to deduce the key in rule 30 failed, but also—despite some considerable effort—little progress has been made in solving any of various problems that turn out to be equivalent to this one. … The procedure takes the x and y coordinates of the square, and computes their remainders after division by 3 and 2 respectively.

For as I will discuss in Chapter 12 it seems far from inconceivable that some of the extraterrestrial radio and other signals that we pick up and assume to be random noise could in fact be meaningful messages—but just encoded in a way that can be recognized only by higher forms of perception and analysis than those we have so far applied to them.

In the past it has tended to be assumed that universality is somehow a rare and special quality, usually possessed only by systems that are specifically constructed to have it.

But while this is already a remarkable result, it represents only a first step in the direction of the Principle of Computational

But what happens if one assumes that one can set up a system that not only finds values of such functions but also finds solutions to arbitrary equations involving them? … The top row of pictures start from the repetitive base 2 digit sequence of x=3/5 ; the bottom row of pictures from x=π/4 .