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

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Offset: 0

Author

Wolfdieter Lang, Apr 14 2018

Keywords

• nonn
• cons
• easy

Comments

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.

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).

Examples

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Links

• Wolfdieter Lang, On a Conformal Mapping of Regular Hexagons and the Spiral of its Centers.

Crossrefs

Cf. A160390.

Programs

Mathematica

RealDigits[6*Log[Sqrt[3] - 1]/Pi, 10, 120][[1]] (* Amiram Eldar, Jun 12 2023 *)

PARI

default(realprecision,120); -(6/Pi)*log(-1 + sqrt(3)) \\ Georg Fischer, Jul 18 2021

Formula

Equals -(6/Pi)*log(-1 + sqrt(3)) = -(6/Pi)*log(A160390).

Extensions

a(102) corrected by Georg Fischer, Jul 18 2021