Rule 225 The width of the pattern after t steps varies between Sqrt[3/2] √ t (achieved when t = 3 × 2 2n + 1 ) and Sqrt[9/2] √ t (achieved when t = 2 2n + 1 ). … Note that with more complicated initial conditions rule 225 often no longer yields a regular nested pattern, as shown on page 951 .

This mechanism can work in several ways; typically it will involve a rotary element that determines which case of the rule to use at each step. Rule (e) from the main text allows a particularly simple supply of new balls.

[Invariance examples in] 2D cellular automata The rule numbers are specified as on page 927 .

ElementaryRule works by converting num into a base 2 digit sequence, padding with zeros on the left so as to make a list of length 8. The scheme for numbering rules works so that if the value of a particular cell is q , the value of its left neighbor is p , and the value of its right neighbor is r , then the element at position 8 - (r + 2(q + 2p)) in the list obtained from ElementaryRule will give the new value of the cell. … The result is that a list is produced which specifies for each cell which element of the rule applies to that cell.

Register machines [emulating Turing machines] Given the rules for a Turing machine in the form used on page 888 , a register machine program to emulate the Turing machine can be obtained by techniques analogous to those used in compilers for practical computer languages. Here TMCompile creates a program segment for each element of the Turing machine rule, and TMToRM resolves addresses and links the segments together. TMToRM[rules_] := Module[{segs, adrs}, segs = Map[TMCompile, rules] ; adrs = Thread[Map[First, rules]  Drop[FoldList[Plus, 1, Map[Length, segs]], -1]]; MapIndexed[(# /.

More general conserved quantities Some rules conserve not total numbers of cells with given colors, but rather total numbers of blocks of cells with given forms—or combinations of these. The pictures below show the simplest quantities of these kinds that end up being conserved by various elementary rules. … Rules that show complicated behavior usually do not seem to have conserved quantities, and this is true for example of rules 30, 90 and 110, at least up to blocks of length 10.

[Structures in] rule 73 on a white background yields a pattern that contains the last structure shown here.

Glider gun [in rule 110] The initial conditions shown correspond to {n, w} = {1339191737336, 41} .

Long halting times [in symbolic systems] Symbolic systems with rules of the form ℯ [x_][y_]  Nest[x, y, r] always evolve to fixed points—though with initial conditions of size n this can take of order Nest[r # &, 0, n] steps (see above ). In general there will be symbolic systems where the number of steps to evolve to a fixed point grows arbitrarily rapidly with n (see page 1145 ), and indeed I suspect that there are even systems with quite simple rules where proving that a fixed point is always reached in a finite number of steps is beyond, for example, the axiom system for arithmetic (see page 1163 ).

Assuming b > a > 0 , the number of zeros from the second family which appear between the n th and (n + 1) th zero from the first family is (Floor[(n + 1) #] - Floor[n #] &)[(b - a)/(a + b)] and as discussed on page 903 this sequence can be obtained by applying a sequence of substitution rules. For Sin[a x] + Sin[b x] a more complicated sequence of substitution rules yields the analogous sequence in which -1/2 is inserted in each Floor .