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A238238
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Decimal expansion of the polar angle, in radians, of a cone which makes a golden-ratio cut of the full solid angle.
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4
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1, 3, 3, 2, 4, 7, 8, 8, 6, 4, 9, 8, 5, 0, 3, 0, 5, 1, 0, 2, 0, 8, 0, 0, 9, 7, 9, 1, 9, 5, 5, 5, 8, 5, 4, 4, 1, 3, 3, 4, 9, 8, 0, 2, 7, 7, 4, 5, 1, 8, 9, 5, 6, 8, 5, 6, 6, 2, 9, 4, 7, 6, 8, 5, 6, 0, 7, 9, 5, 7, 9, 7, 8, 7, 5, 8, 1, 1, 8, 5, 6, 3, 4, 1, 5, 8, 1
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OFFSET
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1,2
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COMMENTS
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The polar angle (or apex angle) of a cone which cuts a fraction f of the full solid angle (i.e., subtends a solid angle of 4*Pi*f steradians) is given by arccos(1-2*f). For a golden cut of the sphere surface by a cone with apex in its center, set f = 1-1/phi, phi being the golden ratio A001622. This value is in radians, its equivalent in degrees is A238239.
The apex angle of the isosceles triangle of smallest perimeter which circumscribes a semicircle (DeTemple, 1992). - Amiram Eldar, Jan 22 2022
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LINKS
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FORMULA
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arccos(1-2*(1-1/phi)) = arccos(2/phi-1), with phi = A001622.
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EXAMPLE
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1.3324788649850305102080097919555854413349802774518956856629476856...
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MATHEMATICA
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RealDigits[ArcCos[2/GoldenRatio -1], 10, 120][[1]] (* Harvey P. Dale, Jul 05 2019 *)
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PROG
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(PARI) acos(4/(1+sqrt(5))-1)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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