Examples of images obtained by keeping only certain fractions of the complete set of basic forms.

This leaves only one remaining type of system from Chapter 3 : register machines.

One can extend the set of rules one considers by allowing not only the color of the active cell itself but also the colors of its immediate neighbors to be updated at each step.

Basic theory [of cryptography] As was recognized in the 1920s the only way to make a completely secure cryptographic system is to use a so-called one-time pad and to have a key that is as long as the message, and is chosen completely at random separately for each message. … And as Claude Shannon argued in the 1940s, the length of message needed to be reasonably certain that only one key will satisfy this criterion is equal to the length of the key divided by the redundancy of the language in which the message is written—equal to about 0.5 for English (see below ).

And so at first I looked only at the 32 rules which had left-right symmetry and made blank backgrounds stay unchanged. … But the picture showed only 20 steps of evolution, and at the time I did not look carefully at it, and certainly did not appreciate its significance.

A few percent exhibit repetitive behavior, while only one in several million exhibit more complex behavior.

But vastly more common in practice is instability only at specific critical points—say bifurcation points—combined with either intrinsic randomness generation or randomness from the environment.

Note that rules of the kind discussed on page 508 which involve replacing clusters of nodes can only apply when cycles in the cluster match those in the network.

The pictures below show the smallest few symmetric graphs with 3 connections at each node (with up to 100 nodes there are still only 37 such graphs; compare page 1029 ).

But searching all 4 billion or so possible such systems with 2 × 2 blocks and up to four colors one finds not a single case in which a nested pattern is forced to occur. … One starts from the substitution system with rules {1  {{3}}, 2  {{13, 1}, {4, 10}}, 3  {{15, 1}, {4, 12}}, 4  {{14, 1}, {2, 9}}, 5  {{13, 1}, {4, 12}}, 6  {{13, 1}, {8, 9}}, 7  {{15, 1}, {4, 10}}, 8  {{14, 1}, {6, 10}}, 9  {{14}, {2}}, 10  {{16}, {7}}, 11  {{13}, {8}}, 12  {{16}, {3}}, 13  {{5, 11}}, 14  {{2, 9}}, 15  {{3, 11}}, 16  {{6, 10}}} This yields the nested pattern below which contains only 51 of the 65,536 possible 2 × 2 blocks of cells with 16 colors. It then turns out that with the constraint that the only 2 × 2 arrangements of colors that can occur are ones that match these 51 blocks, one is forced to get the nested pattern below.