Conditions for convergence [in string rewriting] One way to guarantee that there is convergence after one step is to require as in the previous section that blocks to be replaced cannot overlap with themselves or each other. And of the 196 possible rules involving two colors and blocks of length at most three, 112 have this property.

In the way that the pictures are drawn below, the blocks that appear in each rule are encoded in the pattern of lines coming in from the left edge of the picture.

The bottom row shows all localized structures involving up to 25 cells supported by rule 30 on repetitive backgrounds with blocks of up to 25 cells.

Implementation [of finite automata for nested patterns] Given the rules for a substitution system in the form used on page 931 a finite automaton (as on page 957 ) which yields the color of each cell from the digit sequences of its position is Map[Flatten[MapIndexed[#2 - 1  Position[rules, #1  _] 〚 1, 1 〛 &, Last[#], {-1}]] &, rules] This works in any number of dimensions so long as each replacement yields a block of the same cuboidal form.

Practical empirical mathematics In looking for formulas to describe behavior seen in this book I have in practice typically taken associated sequences of numbers and then tested whether obvious regularities are revealed by combinations of such operations as: computing successive differences (see note below ), computing running totals, looking for repeated blocks, picking out running maxima, picking out numbers with particular modular residues, and looking at positions of particular values, and at the forms of the digit sequences of these positions.

With a background consisting of repetitions of the block , insertion of a single initial white cell yields a largely random pattern that expands by one cell per step.

Projections from 3D [cellular automata] Looking from above, with closer cells shown darker, the following show patterns generated after 30 steps, by (a) the rule at the top of page 183 , (b) the rule at the bottom of page 183 , (c) the rule where a cell becomes black if exactly 3 out of 26 neighbors were black and (d) the same as (c), but with a 3×3×1 rather than a 3×1×1 initial block of black cells:

In the substitution systems for strings discussed in previous sections , the rules that are given can involve replacing any block of elements by any other.

A black element in the tag system is set up to correspond to a block of four cells in the Turing machine, while a white element corresponds to a single cell.

From the discussion of page 1024 one can reproduce the 1D diffusion equation with a continuous block cellular automaton in which the new value of each block is given by {{1 - ξ , ξ }, { ξ , 1 - ξ }} . … So in the case of quantum mechanics one can consider having each new block be given by {{Cos[ θ ],  Sin[ θ ]}, {  Sin[ θ ], Cos[ θ ]}} .