Comparison of [periods for cellular automaton] rules Rules 45, 30 and 60, together with their conjugates and reflections, yield the longest repetition periods of all elementary rules (see page 1087 ).

[No text on this page] The path traced out by the head of the two-dimensional Turing machine with rule (e) from the previous page .

[No text on this page] Examples of reversible cellular automata with various rules.

[Structures in] rule 41 Various rules like rule 41 below can perhaps be viewed as having localized structures—though ones that apparently always travel in the same direction at the same speed. None of the first million initial conditions for rule 41 yield unbounded growth, though some can still generate fairly wide patterns, as in the pictures below.

2D [discrete] transitions [in cellular automata] The simplest symmetrical rules (such as 4-neighbor totalistic code 56) which make the new color of a cell be the same as the majority of the cells in its neighborhood do not exhibit the discrete transition phenomenon, but instead lead to fixed regions of black and white. The 4-neighbor rule with totalistic code 52 can be used as an alternative to the second rule shown here. A probabilistic version of the first rule shown here was discussed by Andrei Toom in 1980.

Doubling rules [cellular automata] Rule (a) is {{0, 2, _}  5, {5, 3, _}  5, {5, _, _}  1, {_, 5, _}  1, {_, 2, _}  3, {_, 3, 2}  2, {_, 1, 2}  4, {_, 4, _}  3, {4, 3, _}  4, {4, 0, _}  2, {_, x_, _}  x} and takes 2n 2 + n steps to yield Table[1, {2n}] given input Append[Table[1, {n - 1}], 2] . Rule (b) is {{_, 2, _}  3, {_, 1, 2}  2, {3, 0, _}  1, {3, _, _}  3, {_, 3, _}  1, {_, x_, _}  x} and takes 3n steps. Rule (c) is k = 3 , r = 1 rule 5407067979 and takes 3n - 1 steps.

Cyclic tag systems which allow any value for each element can be obtained by adding the rule CTStep[{{r_, s___}, {n_, a___}}] := {{s, r}, Flatten[{a, Table[r, {n}]}]} The leading elements in this case can be obtained using CTListStep[{rules_, list_}] := {RotateLeft[rules, Length[list]], With[{n = Length[rules]}, Flatten[Apply[Table[#1, {#2}] &, Map[Transpose[ {rules, #}] &, Partition[list, n, n, 1, 0]], {2}]]]}

Tag systems [emulating cellular automata] Given the rules for an elementary cellular automaton in the form used on page 867 , the following will construct a tag system which emulates it: CAToTS[rules_] := {2, {{s[x_], s[y_]}  {d[x, y], d[x, y]}, {d[w_, x_], d[y_, z_]}  {s[{w, x, y} /. rules], s[{x, y, z} /. rules]}, {s[x_], d[y_, z_]}  {s[0], s[0], s[{0, y, z} /. rules]}, {d[x_, y_], s[z_]}  {s[{x, y, 0} /. rules], s[0], s[0]}}} The initial condition for the tag system that corresponds to a single black cell in the cellular automaton is {s[0], s[0], s[1], s[0], s[0]} .

Among the 4,294,967,296 rules that depend on next-nearest neighbors, there are a handful of examples, including rules with numbers 4196304428, 4262364716, 4268278316 and 4266296876. The behavior obtained with the first of these rules is shown below. … It has rule number 294869764523995749814890097794812493824.

In each case what is shown is the pattern obtained after five steps of evolution according to the rules on the right, starting with a single black square.