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A385445 Decimal expansion of (-1 + 3*phi)*sqrt(3 - phi), with the golden section phi = A001622.

%I A385445 #16 Jul 07 2025 05:23:13
%S A385445 4,5,3,0,7,6,8,5,9,3,1,8,5,9,7,5,1,7,4,3,6,1,2,2,4,0,9,0,9,9,8,1,4,7,
%T A385445 3,2,3,2,3,8,8,8,6,9,2,9,4,6,8,2,0,9,3,5,2,5,3,9,2,8,8,9,0,5,0,6,6,3,
%U A385445 6,2,0,7,2,1,8,6,4,5,7,0,9,5,2,9
%N A385445 Decimal expansion of (-1 + 3*phi)*sqrt(3 - phi), with the golden section phi = A001622.
%C A385445 This constant c gives the real part of -2*11*Z = (c + d*i), where Z is the (finite) fixed point of a complex function w (of the loxodromic type) mapping iteratively the vertices of golden triangles, starting with vertices (D_1, D_2, D_3), circumscribed by the unit circle with center at the origin, and D_1 = i, D_2 = (s - phi*i)/2 and D_3 = (-s - phi*i)/2. This function is w(z) = a*z + b, with a = (-1 + phi) * exp(-(3*Pi/5)*i) = -((2 - phi) + s*i)/2 and b = (1 - phi)*i, where s = sqrt(3 - phi) = A182007 (the length of the base (D_2, D_3) of the first triangle, also called s_1).
%C A385445 The imaginary part of -2*11*Z is d = -7 + 10*phi = A385446.
%C A385445 If the fixed point Z = -(0.20594... + 0.41728...*i) is chosen as origin then the loxodromic map is W(z') = a*z' (where z' = z - Z and W(z') = w(z'+Z) - Z).
%C A385445 For details see the linked paper, eqs.(5a,b) for w(z), eq.(6) for Z and eq.(7) for W(z'). (In eq.(5b) the i is missing in the exponent.) The nesting of golden triangles as shown in Fig. 1 of the link leads to the fixed point Z.
%C A385445 The vertices of the nested golden triangles can be connected by a spiral built of circular sections with angle 108 degrees, centered at vertices D_{n+3} and shrinking radii s_n =(- 1 + phi)^(n-1)*s. Note that the curvature of this spiral is not continuous.
%C A385445 The length(Z, D_n) =: rho_n of the spokes of the spiral is (-1 + phi)^(n-1)*rho_1, with sqrt(11)*rho_1 = sqrt(8 + 9*phi) = sqrt(5 + 7*phi)*s = A385447.
%C A385445 For the length ratio rho_1/s see A385448.
%C A385445 The logarithmic spiral connecting the vertices D_{n+1} is given in polar coordinates by rho(Phi) = rho_1 * exp((-(5/(3*Pi)) * log(phi)*Phi), with the vertices obtained in polar coordinates for Phi = Phi_n = (3*Pi/5)*n, namely rho(Phi_n) = rho_{n+1}, for n >= 0. For log(phi) see A002390. Note that the nonnegative x-axis is now along Z, D_1. The angle(Z, D1, D4) =: gamma is given by arctan((18 - 11*phi)/s) = arcsin(rho_4 / 2) = 0.169860704... or 9.7323... degrees. See Fig. 3 of the linked paper.
%H A385445 Wolfdieter Lang, <a href="/A385445/a385445.pdf">On a Conformal Mapping of Golden Triangles</a>, Papua New Guinea Journal of Maths. 1991, Vol.2, No:2, pp. 12 - 18 (with corrections).
%F A385445 Equals (-1 + 3*phi)*sqrt(3 - phi) = (A090550 - 2)*A182007.
%F A385445 Equals sqrt(27 - 4*phi).
%e A385445 4.5307685931859751743612240909981473232388869294682093...
%t A385445 RealDigits[Sqrt[27 - 4*GoldenRatio], 10, 120][[1]] (* _Amiram Eldar_, Jul 02 2025 *)
%Y A385445 Cf. A001622, A002390, A090550, A182007, A385446, A385447, A385448.
%K A385445 nonn,cons,easy
%O A385445 1,1
%A A385445 _Wolfdieter Lang_, Jul 01 2025