The repetitive structure of picture (a) implies that to reproduce this picture all we need do is to specify the colors in a 49×2 block, and then say that this block should be repeated an appropriate number of times. Similarly, the nested structure of picture (b) implies that to reproduce this picture, all we need do is to specify the colors in a 3×3 block, and then say that as in a two-dimensional substitution system each black cell should repeatedly be replaced by this block.

In general, there are k nk n possible rules for block cellular automata with k colors and blocks of size n . … In general, a block cellular automaton is reversible only if its rule simply permutes the k n possible blocks. Compressing each block into a single cell, and n steps into one, any block cellular automaton with k colors and block size n can be translated directly into an ordinary cellular automaton with k n colors and range r = n/2 .

Excluded blocks [in cellular automaton evolution] As the evolution of a cellular automaton proceeds, the set of sequences that can appear typically shrinks, with progressively more blocks being excluded. … And if there are going to be an infinite number of excluded blocks, there must be additional excluded blocks with lengths between n and 2n . In rule 126, the lengths of the shortest newly excluded blocks on successive steps are 0, 3, 12, 13, 14, 14, 17, 15.

The background pattern consists of blocks of 14 cells that repeat every 7 steps.

In every block of 20 cells in the universal cellular automaton, these rules are encoded in a very straightforward way, by listing in order the outcomes for each of the 8 possible cases. To update the color of the cell represented by a particular block, what the universal cellular automaton must then do is to determine which of the 8 cases applies to that cell. … There are three basic stages, visible in the pictures as three stripes moving to the left across each block.

But from the network, one finds that now an infinite collection of other blocks are forbidden, beginning with the length 12 block .

But if one zooms out, and looks at average motion of increasingly large blocks of particles—as in pictures (b) and (c)—then what begins to emerge is behavior that seems smooth and continuous—just like one expects to see in a fluid. … Picture (a) shows the configuration of individual particles; pictures (b) and (c) show total velocities of successively larger blocks of particles.

One can imagine finding the outcome of evolution more efficiently by adding rules that specify what happens to larger blocks of cells after more steps. And as a practical matter, one can look up different blocks using a method like hashing. … Note that dealing with blocks of different sizes requires going beyond an ordinary cellular automaton rule.

The only difference in initial conditions from the picture on the previous page is that each block now encodes rule 90 instead of rule 254.

The next page shows the kinds of persistent structures that can be generated in rule 110 from blocks less than 40 cells wide. And just like in other class 4 rules, there are stationary structures and moving structures—as well as structures that can be extended by repeating blocks they contain. … But if one looks at blocks of width 41, then such structures do eventually show up, as the picture on page 293 demonstrates.