Understanding nature In Greek times it was noted that simple geometrical rules could explain many features of astronomy—the most obvious being the apparent revolution of the stars and the circular shapes of the Sun and Moon. But it was noted that with few exceptions—like beehives—natural objects that occur terrestrially did not appear to follow any simple geometrical rules. … What rules for natural objects might in effect have been tried in the Judeo-Christian tradition is less clear—though for example the Book of Job does comment on the difficulty of "numbering the clouds by wisdom".

[Nesting in] phase transitions Nesting in systems like rule 184 (see page 273 ) is closely related to the phenomenon of scaling studied in phase transitions and critical phenomena since the 1960s. As discussed on page 983 ordinary equilibrium statistical mechanics effectively samples configurations of systems like rule 184 after large numbers of steps of evolution. But the point is that when the initial number of black and white cells is exactly equal—corresponding to a phase transition point—a typical configuration of rule 184 will contain domains with a nested distribution of sizes.

For case (b) on pages 83 and 84 , the rule that gives the color of the next branch in terms of the color of the current branch and the next digit is {{0, 0}  0, {0, 1}  1, {1, 0}  1, {1, 1}  0} . In terms of this rule, the color of the element at position n is given by Fold[Replace[{#1, #2}, rule]&, 1, IntegerDigits[n - 1, 2]] The rule used here can be thought of as a finite automaton with two states. … Note that if the rule for the finite automaton is represented for example as {{1, 2}, {2, 1}} where each sublist corresponds to a particular state, and the elements of the sublist give the successor states with inputs Range[0, k - 1] , then the n th element in the output sequence can be obtained from Fold[rule 〚 #1, #2 〛 &, 1, IntegerDigits[n - 1, k] + 1] - 1 while the first k m elements can be obtained from Nest[Flatten[rule 〚 # 〛 ] &, 1, m] - 1 To treat examples such as case (c) where elements can subdivide into blocks of several different lengths one must generalize the notion of digit sequences.

Rules with more than two colors will sometimes be appropriate. For rugs, it is typically desirable to have each cell correspond to more than one tuft, since otherwise with most rules the rug looks too busy.

But explicit construction, based on correspondence with one-dimensional cellular automata, leads to the example shown at the top of the facing page : a system with 56 allowed templates in which the only pattern satisfying the constraint is a complex and largely random one, derived from the rule 30 cellular automaton. … The system shown was specifically constructed in correspondence with the rule 60 elementary one-dimensional cellular automaton.

This time the rule specifies that a cell should be black when either its left neighbor or its right neighbor—but not both—were black on the step before. And again this rule is undeniably quite simple.

So what happens if one uses a more complicated rule for generating an encrypting sequence from a key? … Encryption using the rule 60 additive cellular automaton.

One-element-dependence tag systems [emulating TMs] Writing the rule {3, {{0, _, _}  {0, 0}, {1, _, _}  {1, 1, 0, 1}}} from page 895 as {3, {0  {0, 0}, 1  {1, 1, 0, 1}}} the evolution of a tag system that depends only on its first element is obtained from TS1EvolveList[rule_, init_, t_] := NestList[TS1Step[rule, #] &, init, t] TS1Step[{n_, subs_}, {}] = {} TS1Step[{n_, subs_}, list_] := Drop[Join[list, First[list] /. subs], n] Given a Turing machine in the form used on page 888 the following will construct a tag system that emulates it: TMToTS1[rules_] := {2, Union[Flatten[rules /.

Properties [of example multiway systems] The second rule shown has the property that black elements always appear before white, so that strings can be specified just by the number of elements of each color that they contain—making the rule one of the sorted type discussed on page 937 , based on the difference vector {{2,-1}, {-1,3}, {-4,-1}} .

Growth [2D cellular automaton] rules The pictures below show examples of rules in which a cell becomes black if it has exactly the specified numbers of black neighbors (the initial conditions used have the minimal number of black cells for growth).