A195700 Decimal expansion of arcsin(sqrt(3/8)) and of arccos(sqrt(5/8)).

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

Offset: 0

Author

Clark Kimberling, Sep 23 2011

Keywords

• nonn
• cons

Comments

arcsin(sqrt(3/8)) = least x>0 satisfying sin(2*x) = 2*sin(4*x).

This number apparently also represents the angle, in radians, by which a regular dodecahedron (centered at the origin and having vertices at both the points (0, phi, 1/phi) and (1,1,1)) must be rotated about the axis y=x=z to optimally fit in a cube, also centered at the origin, aligned with the unit axes. A dodecahedron rotated by this amount can fit in the smallest possible cube. See Firsching (2018) and the graphic provided in it. This result comes from spherical geometry: If one pentagon of a regular dodecahedron is projected onto a sphere, this value is the angle between a line from a pentagon vertex to the midpoint of the farthest (opposite) edge and another line from the same vertex to the midpoint of either edge adjacent to the first. After being rotated, the dodecahedron still has a point at (1,1,1) with six edges aligning exactly with the faces of the cube. - Jonah D. Vanke, Oct 22 2023

Examples

			0.6590580358264089828728321...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• M. Firsching, Computing maximal copies of polytopes contained in a polytope, arXiv:1407.0683 [math.MG], 2014.
• Index entries for transcendental numbers

Crossrefs

Cf. A195703, A195705, A195706.

Programs

Magma

[Arcsin(Sqrt(3/8))]; // G. C. Greubel, Nov 18 2017

Mathematica

r = Sqrt[3/8];
N[ArcSin[r], 100]
RealDigits[%]  (* this sequence *)
N[ArcCos[r], 100]
RealDigits[%]  (* A195703 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A195705 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195706 *)

PARI

asin(sqrt(3/8)) \\ G. C. Greubel, Nov 18 2017

Formula

Equals arctan(sqrt(3/5)). - Amiram Eldar, Jul 04 2023