cp's OEIS Frontend

A302708 Constant of a logarithmic spiral interpolating the centers of regular hexagons: (-6/Pi)*log(-1 + sqrt(3)).

%I A302708 #22 Jun 12 2023 02:55:44
%S A302708 5,9,5,6,9,5,3,5,4,3,7,8,9,9,3,4,1,9,8,7,8,9,6,6,1,3,3,7,7,5,3,6,0,1,
%T A302708 7,3,7,1,2,3,1,3,1,5,4,5,8,2,8,8,7,2,6,6,8,6,6,7,6,6,0,7,5,0,3,2,9,2,
%U A302708 5,3,3,4,8,7,0,8,3,0,2,9,0,5,7,8,5,2,4,7,9,8,3,7,4,7,9,2,4,0,8,6,5,9,5
%N A302708 Constant of a logarithmic spiral interpolating the centers of regular hexagons: (-6/Pi)*log(-1 + sqrt(3)).
%C A302708 For the sequence of regular hexagons H_k with centers 0_k, for integers k, see the link. These centers form a discrete spiral which is interpolated by a logarithmic spiral r(phi) = exp(-kappa*phi) with origin S = (0, 1) if the hexagon H_0 has center 0_0 = (0, 0), inscribed in a circle of radius 1 length unit, and a vertex V_0(0) = (1, 0). In the link this coordinate system is called (x_0, y_0). The constant of the logarithmic spiral is kappa = (-6/Pi)*log(-1 + sqrt(3)). For -1 + sqrt(3) (the scaling factor for the hexagons called sigma in the linked paper) see A160390.
%C A302708 The constant angle between the radial direction of a spiral point and the tangent is given by arccot(kappa) approximately 1.033548019, corresponding to an angle of about 59.218 degrees (complementary to 120.782 degrees).
%H A302708 Wolfdieter Lang, <a href="/A300068/a300068_1.pdf">On a Conformal Mapping of Regular Hexagons and the Spiral of its Centers</a>.
%F A302708 Equals -(6/Pi)*log(-1 + sqrt(3)) = -(6/Pi)*log(A160390).
%e A302708 0.59569535437899341987896613377536017371231315458288726686676607503292533487083...
%t A302708 RealDigits[6*Log[Sqrt[3] - 1]/Pi, 10, 120][[1]] (* _Amiram Eldar_, Jun 12 2023 *)
%o A302708 (PARI) default(realprecision,120); -(6/Pi)*log(-1 + sqrt(3)) \\ _Georg Fischer_, Jul 18 2021
%Y A302708 Cf. A160390.
%K A302708 nonn,cons,easy
%O A302708 0,1
%A A302708 _Wolfdieter Lang_, Apr 14 2018
%E A302708 a(102) corrected by _Georg Fischer_, Jul 18 2021