Note (b) for Chaos Theory and Randomness from Initial Conditions…Recognizing chaos Any system that depends sensitively on digits in its initial conditions must necessarily be able to show behavior that is not purely repetitive (compare page 955 ). … To give evidence that this is not merely a reflection of continual injection of randomness from the environment what is normally done is to show that at least some aspect of the behavior of the system can be fit by a definite simple iterated map or differential equation. But inevitably the fit will only be approximate, so there will always be room for effects from randomness in the environment.

Implementation [of operators from axioms] Given an axiom system in the form {f[a, f[a, a]]  a, f[a, b]  f[b, a]} one can find rule numbers for the operators f[x, y] with k values for each variable that are consistent with the axiom system by using Module[{c, v}, c = Apply[Function, {v = Union[Level[axioms, {-1}]], Apply[And, axioms]}]; Select[Range[0, k k 2 - 1], With[{u = IntegerDigits[#, k, k 2 ]}, Block[{f}, f[x_, y_] := u 〚 -1 - k x - y 〛 ; Array[c, Table[k, {Length[v]}], 0, And]]] &]] For k = 4 this involves checking nearly 16 4 or 4 billion cases, though many of these can often be avoided, for example by using analogs of the so-called Davis–Putnam rules.

The cellular automaton rule then corresponds to a continuous mapping of this Cantor set to itself (continuity follows from the locality of the rule). … (Periodic boundary conditions are used, so that the a[t, i] can be viewed as corresponding precisely to digits of rational numbers.)

Corresponding to the result on page 870 for rule 90, the number of black cells at row t in the pattern from rule 150 is given by Apply[Times, Map[(2 # + 2 - (-1) # + 2 )/3 &, Cases[Split[IntegerDigits[t, 2]], k:{1 ..}  Length[k]]]] There are a total of 2 m Fibonacci[m+2] black cells in the pattern obtained up to step 2 m , implying fractal dimension Log[2, 1 + Sqrt[5]] .

The number of nodes at distance up to r from a given node is at most 1 + Sum[c[i] + c[i - 1], {i, n}] where c[i_] := 2^DigitCount[i, 2] .

The maximum halting times for the first few sizes n are {5, 159, 161, 1021, 5419, 315391, 1978213883, 1978213885, 3018415453261} These occur for inputs {1, 2, 5, 10, 26, 34, 106, 213, 426} and correspond to outputs (each themselves maximal for given n ) 2^{3, 23, 24, 63, 148, 1148, 91148, 91149, 3560523} - 1 Such maxima often seem to occur when the input x has the form (20 4 s - 2)/3 (and so has digits {1, 1, 0, 1, 0, … , 1, 0} ). … But if IntegerDigits[x, 2] involves no consecutive 0's then for example f[x] can be obtained from 2^(b[Join[{1, 1}, #], Length[#]] &)[IntegerDigits[x, 2]] - 1 a[{l_, _}, r_] := ({l + (5r - 3#)/2, #} &)[Mod[r, 2]] a[{l_, 0}, 0] := {l + 1, 0} a[{l_, 1}, 0] := ({(13 + #(5/2)^IntegerExponent[#, 2])/6, 0} &[6l + 2] b[list_, i_] := First[Fold[a, {Apply[Plus, Drop[list, -i]], 0}, Apply[Plus, Split[Take[list, -i], #1  #2 ≠ 0 &], 1]]] (The corresponding expression for t[x] is more complicated.)

Fibonacci[n] can be obtained in many ways: • (GoldenRatio n - (-GoldenRatio) -n )/ √ 5 • Round[GoldenRatio n / √ 5 ] • 2 1 - n Coefficient[(1 + √ 5 ) n , √ 5 ] • MatrixPower[{{1, 1}, {1, 0}}, n - 1] 〚 1, 1 〛 • Numerator[NestList[1/(1 + #)&, 1, n]] • Coefficient[Series[1/(1 - t - t 2 ), {t, 0, n}], t n - 1 ] • Sum[Binomial[n - i - 1, i], {i, 0, (n - 1)/2}] • 2 n - 2 - Count[IntegerDigits[Range[0, 2 n - 2 ], 2], {___, 1, 1, ___}] A fast method for evaluating Fibonacci[n] is First[Fold[f, {1, 0, -1}, Rest[IntegerDigits[n, 2]]]] f[{a_, b_, s_}, 0] = {a (a + 2b), s + a (2a - b), 1} f[{a_, b_, s_}, 1] = {-s + (a + b) (a + 2b), a (a + 2b), -1} Fibonacci numbers appear to have first arisen in perhaps 200 BC in work by Pingala on enumerating possible patterns of poetry formed from syllables of two lengths. … Their recurrence relation appears to have been understood from the early 1600s, but it has only been in the past very few decades that they have in general become widely discussed. … In addition: • GoldenRatio is the solution to x  1 + 1/x or x 2  x + 1 • The right-hand rectangle in is similar to the whole rectangle when the aspect ratio is GoldenRatio • Cos[ π /5]  Cos[36 ° ]  GoldenRatio/2 • The ratio of the length of the diagonal to the length of a side in a regular pentagon is GoldenRatio • The corners of an icosahedron are at coordinates Flatten[Array[NestList[RotateRight, {0, (-1) #1 GoldenRatio, (-1) #2 }, 3]&, {2, 2}], 2] • 1 + FixedPoint[N[1/(1 + #), k] &, 1] approximates GoldenRatio to k digits, as does FixedPoint[N[Sqrt[1 + #],k]&, 1] • A successive angle difference of GoldenRatio radians yields points maximally separated around a circle (see page 1006 ).

And so, as one example, it might appear from the pictures on the previous page that (c), (d) and (e) always stay systematically above the axis. … Some of this complexity can be understood if we look at each number not in terms of its overall size, but rather in terms of its digit sequence or set of possible divisors.

(Any cellular automaton rule with an n -cell neighborhood corresponds to such a function; digit sequences in rule numbers correspond to explicit tables of values.) … As in the note on multilevel formulas below, one can also in effect introduce intermediate variables to get recursive formulas of larger depth, somewhat analogous to results from Collect .

Apart from a glitch that gets narrower with increasing k (the so-called Gibbs phenomenon), the result has a simple triangular form. … The curves obtained in this case show a definite nested structure, in which the value at a point x is essentially determined directly from the base 2 digit sequence of x .