Each successive base 2 digit in the code number for the rule gives the outcome when the total of the cell and its four neighbors runs from 5 down to 0.

Recursive subdivision [encoding] In one dimension, encoding can be done using Subdivide[a_] := Flatten[ If[Length[a]  2, a, If[Apply[SameQ, a], {1,First[a]}, {0, Map[Subdivide, Partition[a, Length[a]/2]]}]]] In n dimensions, it can be done using Subdivide[a_, n_] := With[{s = Table[1, {n}]}, Flatten[ If[Dimensions[a]  2s, a, If[Apply[SameQ, Flatten[a]], {1, First[Flatten[a]]}, {0, Map[Subdivide[#, n] &, Partition[a, 1/2Length[a] s], {n}]}]]]]

Distribution of behavior [in Turing machines] With 2 possible states and 2 possible colors for each cell, starting from a blank tape, the maximum repetition period obtained is 9 steps, and 12 out of the 4096 possible rules (or about 0.29%) yield non-repetitive behavior. With 3 states and 2 colors, the maximum period is 24, and about 0.37% of rules yield non-repetitive behavior, always nested. … With 2 states and 3 colors, the maximum repetition period is again 24, about 0.65% of rules yield non-repetitive behavior, and the 14 rules discussed on page 709 yield more complex behavior.

The Great Wall of China from 200 BC follows local topography and so is not straight. … All these railroads go through flat empty terrain that previously had only a few nomadic inhabitants—and no settlements to define a route.

Simple case [of three-body problem] The position of the idealized planet in the case shown satisfies the differential equation δ tt z[t]  -z[t]/(z[t] 2 + (1/2 (1 + e Sin[2 π t] ) 2 ) 3/2 where e is the eccentricity of the elliptical orbit of the stars ( e = 0.1 in the picture). … Except when e = 0 , the equation has no solution in terms of standard mathematical functions. … The pictures below show the dependence of z[t] on t and z[0] .

Neighbor-dependent [2D] substitution systems Given a list of individual replacement rules such as {{_, 1}, {0, 1}}  {{1, 0}, {1, 1}} , each step in the evolution shown corresponds to Flatten2D[Partition[list, {2, 2}, 1, -1] /. rule] One can consider rules in which some replacements lead to subdivision of elements but others do not. However, unlike for the 1D case, there will in general in 2D be an arbitrarily large set of different possible neighborhood configurations around any given cell.

Indeed, for period p , the length of blocks required is at most 2 2p (or 2 2 p r for range r rules). … For periods up to 10, examples of such blocks in rule 90 are given by the digits of {0, 40, 24, 2176, 107904, 640, 96, 8421376, 7566031296234863392, 15561286137} For period 1 the possible blocks are and ; for period 2 and . … For p = 2 , rule 18 leaves 20 of the 32 possible length 5 blocks invariant, but these blocks can only be strung together to yield repetitions of {a, b, 0, 0} , where now a and b are not fixed, but in every case can each be either {1} or {0, 1} .

It turns out that if one represents numbers as digit sequences in base 2, then the operation of multiplying by 2 has a very simple effect: it just shifts the digit sequence one place to the left, adding a 0 digit on the right. And as a result, the overall pattern obtained by successive multiplication by 2 has a very simple form. Digit sequences in base 2 of numbers obtained by starting with 1 and then successively adding a constant at each step.

Implementation [of patterning model] Given a 2D array of values a and a list of weights w , each step in the evolution of the system corresponds to WeightedStep[w_List, a_] := Map[If[# > 0, 1, 0]&, Sum[w 〚 1 + i 〛 Apply[Plus, Map[RotateLeft[a, #]&, Layer[i]]], {i, 0, Length[w] - 1}], {2}] Layer[n_] := Layer[n] = Select[Flatten[Table[{i, j}, {i, -n, n}, {j, -n, n}],1], MemberQ[#, n| - n]&]

But given t steps in this sequence as a list of 0's and 1's, the following function will reconstruct the rightmost t digits in the starting value of n : IntegerDigits[First[Fold[{Mod[If[OddQ[#2], 2 First[#1] - 1, 2 First[#1] PowerMod[5, -1, Last[#1]]], Last[#1]], 2 Last[#1]} &, {0, 2}, Reverse[list]]], 2, Length[list]]