But by doing Nest[s, Range[52], 26] one ends up with a simple reversal of the original deck, as in the pictures below.

The idea is to start out on the top line with all possible numbers. Then on the second line, one removes all numbers larger than 2 that are divisible by 2. … One starts on the top line with all numbers between 1 and 100.

The pictures below show in stacked form (as on page 208 ) all sequences generated at various steps of evolution. … Different initial conditions for this multiway system lead to behavior that either dies out (as for "ABA" ), or grows exponentially forever (as for "ABAABABA" ).

With three possible states, only repetitive and nested patterns are ever ultimately produced, at least starting with all cells white. … The top set of pictures show the first 150 steps of evolution according to various different rules, starting with the head in the first state (arrow pointing up), and all cells white.

In practical situations this kind of approximate result can sometimes be useful, but the pictures at the top of the facing page show that the actual patterns obtained do not look much at all like the exact results that we saw for this system in Chapter 5 . … The top picture shows one particular run of this procedure. … In all cases 10×10 patterns are used.

The picture at the top of the facing page shows an example of a procedure for generating the base 2 digit sequence for the square root of a given number n . … Digit sequences for various square roots, given at the top in base 10 and at the bottom in base 2. Despite their simple definition, all these sequences seem for practical purposes random.

The actual numbers of functions which require 0, 1, 2, ... terms is for n = 2 : {1, 9, 6} ; for n = 3 : {1, 27, 130, 88, 10} , and for n = 4 : {1, 81, 1804, 13472, 28904, 17032, 3704, 512, 26} .

But in fact, in the fifteen years or so since I first emphasized the importance of cellular automata all sorts of traditional mathematical work has actually been done on them. … The top statement on the right makes the assertion that the outcome after one step of evolution from a single black cell has a particular form. … All the statements in the top block above can be proved true from the axiom system.

whether, say, a particular pattern would ever die out in the evolution of a given cellular automaton. … For while simple infinite quantities like 1/0 or the total number of integers can readily be summarized in finite ways—often just by using symbols like ∞ and ℵ 0 —the same is not in general true of all infinite processes.

If the rate of such disappearances is too large, then almost any pattern will quickly die out. … 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.