A231987 Decimal expansion of the side length (in radians) of the spherical square whose solid angle is exactly one steradian.

1, 0, 4, 1, 1, 9, 1, 8, 0, 3, 6, 0, 6, 8, 7, 3, 3, 4, 0, 2, 3, 4, 6, 0, 7, 5, 3, 3, 5, 9, 2, 5, 6, 8, 7, 8, 8, 9, 0, 0, 6, 9, 6, 6, 7, 6, 0, 0, 6, 0, 8, 7, 1, 3, 4, 9, 1, 5, 2, 3, 0, 2, 8, 1, 3, 1, 2, 9, 9, 7, 1, 9, 7, 0, 4, 8, 2, 2, 3, 8, 5, 8, 9, 2, 8, 9, 5, 5, 5, 8, 8, 7, 1, 8, 8, 6, 4, 4, 3, 0, 7, 2, 7, 5, 9

Offset: 1

Author

Stanislav Sykora, Nov 17 2013

Keywords

• nonn
• cons
• easy

Comments

This is an inverse problem (but not an inverse value) to the one leading to A231986: what is the side s of a spherical square (in radians, rad) if it covers a given solid angle (in steradians, sr)? The solution (inverse of the formula in A231896) is s = 2*arcsin(sqrt(sin(Omega/4))). In this particular case, Omega = 1.

Examples

			1.041191803606873340234607533592568788900696676006087134915230281312997...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..2000
• Wikipedia, Solid angle, Section 3.3 (Pyramid).
• Wikipedia, Steradian.

Crossrefs

Cf. A072097 (rad/deg), A019685 (deg/rad), A231981 (sr/deg^2), A231982 (deg^2/sr), A231986 (inverse problem), A231896.

Programs

Mathematica

RealDigits[2*ArcSin[Sqrt[Sin[1/4]]], 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

PARI

default(realprecision, 120);
2*asin(sqrt(sin(1/4))) \\ or
solve(x = 1, 2, 4*asin((sin(x/2))^2) - 1) \\ least positive solution - Rick L. Shepherd, Jan 28 2014

Formula

Equals 2*arcsin(sqrt(sin(1/4))).

Extensions

Formula and comment corrected by Rick L. Shepherd, Jan 28 2014