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.

Gegenbauer functions Introduced by Leopold Gegenbauer in 1893 GegenbauerC[n, m, z] is a polynomial in z with integer coefficients for all integer n and m .

At a given time all whales in a group typically sing almost the same song, which gradually changes.

Like essentially all forms of science, however, what I do in this book is done in a rational tradition—with limited relation to the more mystical traditions of Eastern thinking.

Each repeating block of digits typically seems quite random, and has properties such as all possible subblocks of digits up to a certain length appearing (see page 1084 ).

As an alternative one can consider starting with very simple initial conditions, such as all cells white, and then at each step randomly changing the color of the center cell.

Thus for example {(0 | 1) ...} corresponds to all possible sequences of 0 's and 1 's, while {1, 1, (1) ..., 0, (0) ...} ... corresponds to the sequences that can occur after 2 steps in rule 126 and {(0) ..., 1, {0, (0) ..., 1, 1} | {1, (1) ..., 0}} ... to those that can occur after 2 steps in rule 110 (see page 279 ).

No doubt the methods of this book will in the future lead to all sorts of algorithms based on much more complex patterns of behavior.

Properties [of substitution systems] The examples shown here all appear in quite a number of different contexts in this book. … The complete pattern formed by looking at all the steps together has the same structure as the arrangement of base 2 digits in successive numbers shown on page 117 . … All four possible pairs of successive elements occur, though not with equal frequency.

After 180 steps, it turns out that all that survives are a few objects that one can view as localized structures.