A sequence of identical digits d then corresponds to the number (1 + Sqrt[4d + 1])/2 . (Note that Nest[Sqrt[# + 2] &, 0, n]  2 Cos[ π /2 n + 1 ] .) … For random x , digits 0, 1, 2 appear to occur with limiting frequencies Sqrt[2 + d] - Sqrt[1 + d] .

To find Sqrt[n] one starts by setting r=n and s=0 . … The result is that the digits of s in base 2 turn out to correspond exactly to the digits of Sqrt[n] .

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 ).

For large n , the average number of distinct cycles in all such networks is Sqrt[ π /2] Log[n] , and the average length of these cycles is Sqrt[ π /8 n] .

Transformations of the form z  {Sqrt[z - c], -Sqrt[z - c]} yield so-called Julia sets which form nested patterns for many values of c (see note below ).

Sqrt[1 - 4 x]/2 yields a sequence with 1's at positions 2 m , as essentially obtained from the substitution system {2  {2, 1}, 1  {1, 0}, 0  {0, 0}} . Sqrt[(1 - 3 x)/(1 + x)]/2 yields sequence (a) on page 84 . (1 + Sqrt[(1 - 3 x)/(1 + x)])/(2(1 + x)) (see page 890 ) yields the Thue–Morse sequence.

Rule 22—like rule 90 from page 26 —gives a pattern with fractal dimension Log[2,3] ≃ 1.58 ; rule 150 gives one with fractal dimension Log[2, 1+Sqrt[5]] ≃ 1.69 .

In the first 200 billion digits, the frequencies of 0 through 9 differ from 20 billion by {30841, -85289, 136978, 69393, -78309, -82947, -118485, -32406, 291044, -130820} An early approximation to π was 4 Sum[(-1) k /(2k + 1), {k, 0, m}] 30 digits were obtained with 2 Apply[Times, 2/Rest[NestList[Sqrt[2 + #]&, 0, m]]] An efficient way to compute π to n digits of precision is (# 〚 2 〛 2 /# 〚 3 〛 )& [NestWhile[Apply[Function[{a, b, c, d}, {(a + b)/2, Sqrt[a b], c - d (a - b) 2 , 2 d}], #]&, {1, 1/Sqrt[N[2, n]], 1/4, 1/4}, # 〚 2 〛 ≠ # 〚 2 〛 &]] This requires about Log[2, n] steps, or a total of roughly n Log[n] 2 operations (see page 1134 ).

Equation for the background [in my PDEs] If u[t, x] is independent of x , as it is sufficiently far away from the main pattern, then the partial differential equation on page 165 reduces to the ordinary differential equation u''[t]  (1 - u[t] 2 )(1 + a u[t]) u[0]  u'[0]  0 For a = 0 , the solution to this equation can be written in terms of Jacobi elliptic functions as ( √ 3 JacobiSN[t/3 1/4 , 1/2] 2 ) / (1 + JacobiCN[t/3 1/4 , 1/2] 2 ) In general the solution is (b d JacobiSN[r t, s] 2 )/(b - d JacobiCN[r t, s] 2 ) where r = -Sqrt[1/8 a c (b - d)] s = (d (c - b))/(c (d - b)) and b , c , d are determined by the equation (x - b)(x - c)(x - d)  -(12 + 6 a x - 4 x 2 - 3 a x 3 )/(3a) In all cases (except when -8/3 < a < -1/ √ 6 ), the solution is periodic and non-singular. … For a = 8/3 , the solution can be written without Jacobi elliptic functions, and is given by 3 Sin[Sqrt[5/6] t] 2 /(2 + 3 Cos[Sqrt[5/6] t] 2 )

The necessary transformation is the so-called Lorentz transformation {t, x}  {t - v x/c 2 , x - v t}/Sqrt[1 - v 2 /c 2 ] And from this the time dilation factor 1/Sqrt[1 - v 2 /c 2 ] shown on page 524 follows, as well as the length contraction factor Sqrt[1 - v 2 /c 2 ] .