A342571 Decimal expansion of the surface area of a golden ellipsoid with semi-axes lengths 1, 1 and phi (A001622).
1, 7, 9, 8, 0, 7, 9, 7, 4, 3, 4, 1, 0, 4, 7, 7, 3, 4, 2, 1, 5, 2, 4, 5, 4, 9, 5, 9, 0, 4, 3, 9, 6, 3, 8, 8, 2, 0, 4, 2, 6, 5, 9, 3, 5, 0, 6, 0, 0, 7, 3, 9, 8, 3, 9, 3, 1, 0, 3, 2, 3, 4, 8, 7, 8, 1, 2, 8, 3, 0, 6, 7, 3, 4, 6, 6, 7, 3, 3, 5, 5, 7, 3, 3, 3, 9, 2
Offset: 2
Examples
17.9807974341047734215245495904396388204265935060073...
Links
- Kenneth Brecher, The "PhiTOP": A Golden Ellipsoid, Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture, 2015, pp. 371-374.
- Kenneth Brecher and Rod Cross, Physics of the PhiTOP, The Physics Teacher, Vol. 57, No. 2 (2019), pp. 74-75.
- Eric Weisstein's World of Mathematics, Ellipsoid.
- Wikipedia, Ellipsoid.
Programs
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Mathematica
RealDigits[SurfaceArea[Ellipsoid[{0,0,0},{1,1,GoldenRatio}]], 10, 100][[1]] (* requires Mathematica 12+, or *) RealDigits[2*Pi*(1 + GoldenRatio/Sinc[ArcCos[1/GoldenRatio]]), 10, 100][[1]]
Formula
Equals 2*Pi*(1 + phi*c/sin(c)), where c = arccos(1/phi) (A195692).
Equals 2*Pi*(1 + sqrt(2+sqrt(5))*arcsec(phi)).