And one can then assume that in the visual cortex there is a corresponding array of cells, with each cell receiving input from, say, a 2×2 block of squares, and following the rule that it responds whenever the colors of these squares form some particular pattern. … And with the specific choice of rule used here, what effectively happens is that the vertical black edges in the original image get picked out. … The sample images used here are ones generated by the evolution of elementary one-dimensional cellular automata with rules 60 and 124 respectively.

For the rules of the universal cellular automaton in this section are quite complicated—involving 19 possible colors for each cell, and next-nearest as well as nearest neighbors. … But what we will discover later in this chapter is that such complication in underlying rules is in fact not needed. … But the existence of universal cellular automata with such simple underlying rules makes it clear that the basic results we have obtained in this section are potentially of very broad significance.

Previous examples of systems that are known to be universal have typically had rules that are far too complicated to see this with any clarity. … The basic rules for combinators are given below. … But with initial condition (e) of length 8 the pictures show Rules for symbolic systems known as combinators, first introduced in 1920, and proved universal by the mid-1930s.

And the underlying rule is then typically set up so that under certain circumstances an active cell can split in two, or can disappear entirely. … The rule given above is applied to every cell that is active at a particular step. In many cases, the rule specifies just that the active cell should move to the left or right.

And with traditional intuition it has normally been assumed that the only way to create systems that show a higher degree of complexity is somehow to build this complexity into their underlying rules. But one of the central discoveries of this book is that this is not the case, and that in fact it is perfectly possible for systems even with extremely simple underlying rules to produce behavior that has immense complexity—and that looks like what one sees in nature. … There are some places where just the abstract ability to produce complexity from simple rules is already important.

But there is always a small fraction of rules in which the creation and destruction of elements is almost perfectly balanced. … And as it turns out, among substitution systems with the same type of rules, all those which yield slow growth also seem to produce only such simple repetitive patterns. … Two views of a substitution system whose rules allow both creation and destruction of elements.

The array for rule 60 is then 1/(1- (1 + x) y) , for rule 90 1/(1 - (1/x + x) y) , for rule 150 1/(1 - (1/x + 1 + x) y) and for second-order reversible rule 150 (see page 439 ) 1/(1 - (1/x + 1 + x) y - y 2 ) .

In a rule 30 system of infinite size, it turns out that at 45° clockwise from vertical all possible sequences can occur on any two adjacent lines, probably making cryptanalysis more difficult in this case. (Note that directional sampling is always equivalent to looking at a vertical column in the evolution of a cellular automaton whose basic rule has been composed with an appropriate shift rule.)

For rule 150R, however, there is no black cell on the first initial step. The pattern generated by rule 150R has fractal dimension Log[2, 3 + Sqrt[17]] - 1 or about 1.83. In rule 154R, each diagonal stripe is followed by at least one 0; otherwise, the positions of the stripes appear to be quite random, with a density around 0.44.

The basic reason is that at every step the rules for the substitution system simply replace each black square with several smaller black squares. And on subsequent steps, each of these new black squares is then in turn replaced in exactly the A two-dimensional substitution system in which each square is replaced by four smaller squares at every step according to the rule shown on the left.