But just as for random walks, it appears once again that the details of the underlying rules for the system do not have much effect on the main features of the behavior that is seen. … Despite the difference in underlying rules, the same basic overall shape of pattern is eventually produced.

The underlying rule for the cellular automaton shown takes the new color of a cell to be the color of its right neighbor if the cell is black and its left neighbor if the cell is white. (This corresponds to rule 184 in the scheme described on page 53 .)

Patterns generated by rules of the type shown on the previous page , with a range of choices for the weights of cells at distances 2 and 3. … Examples of rules in which cells in the horizontal and vertical directions are weighted differently.

The underlying rules for the cellular automaton used here are reversible, and conserve the total number of particles. The specific rules are based on 2×2 blocks—a two-dimensional generalization of the block cellular automata to be discussed in the next section .

Because the underlying rule is reversible, however, the details with different initial conditions are always at least slightly different—otherwise it would not be possible to go backwards in a unique way. The rule used here is 122R.

Such systems in general take a string of elements and at each step replace blocks of these elements with other elements according to some definite rule. … The substitution system works by replacing blocks of elements at each step according to the rule shown.

The first example corresponds to elementary cellular automaton rule 60. Note that any cellular automaton rule can be reproduced by some appropriate combination of bitwise and arithmetic operations.

Operator systems One can represent the possible values of expressions like f[f[p, q], p] by rule numbers analogous to those used for cellular automata. Specifying an operator f (taken in general to have n arguments with k possible values) by giving the rule number u for f[p, q, …] , the rule number for an expression with variables vars can be obtained from With[{m = Length[vars]}, FromDigits[ Block[{f = Reverse[IntegerDigits[u, k, k n ]] 〚 FromDigits[ {##}, k] + 1 〛 &}, Apply[Function[Evaluate[vars], expr], Reverse[Array[IntegerDigits[# - 1, k, m] &, k m ]], {1}]], k]]

The paths obtained at successive steps for rule (b) above are shown below. The pictures below show paths obtained with the rule {1  {1}, 0  {0, 0, 1}} , starting from {0} . … But in a case like the rule {1  {0, 0, 1}, 0  {1, 0}} starting with {1} , the presence of many crossings tends to hide such regularity, as in the pictures below.

Intrinsic synchronization in cellular automata Taking the rules for an ordinary cellular automaton and applying them sequentially will normally yield very different results. But it turns out that there are variants on cellular automata in which the rules can be applied in any order and the overall behavior obtained—or at least the causal network—is always the same. … Note that the rules can be thought of as replacements such as "A>" for blocks of length 4 with 4 colors.