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A302708
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Constant of a logarithmic spiral interpolating the centers of regular hexagons: (-6/Pi)*log(-1 + sqrt(3)).
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0
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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, 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, 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
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OFFSET
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0,1
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COMMENTS
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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).
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LINKS
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FORMULA
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Equals -(6/Pi)*log(-1 + sqrt(3)) = -(6/Pi)*log(A160390).
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EXAMPLE
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0.59569535437899341987896613377536017371231315458288726686676607503292533487083...
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MATHEMATICA
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RealDigits[6*Log[Sqrt[3] - 1]/Pi, 10, 120][[1]] (* Amiram Eldar, Jun 12 2023 *)
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PROG
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(PARI) default(realprecision, 120); -(6/Pi)*log(-1 + sqrt(3)) \\ Georg Fischer, Jul 18 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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