A238238 Decimal expansion of the polar angle, in radians, of a cone which makes a golden-ratio cut of the full solid angle.

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

Offset: 1

Author

Stanislav Sykora, Feb 20 2014

Keywords

• nonn
• cons
• easy

Comments

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

Examples

			1.3324788649850305102080097919555854413349802774518956856629476856...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..2000
• Duane W. DeTemple, The Triangle of Smallest Perimeter which Circumscribes a Semicircle, The Fibonacci Quarterly, Vol. 30, No. 3 (1992), p. 274.
• Wikipedia, Solid angle.

Crossrefs

Cf. A001622, A019670, A137914, A238239.

Programs

Mathematica

RealDigits[ArcCos[2/GoldenRatio  -1],10,120][[1]] (* Harvey P. Dale, Jul 05 2019 *)

PARI

acos(4/(1+sqrt(5))-1)

Formula

arccos(1-2*(1-1/phi)) = arccos(2/phi-1), with phi = A001622.