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

0,1

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

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.

Formula

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

Example

0.6590580358264089828728321...

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 *)

Prog

(PARI) asin(sqrt(3/8)) \\ G. C. Greubel, Nov 18 2017
(Magma) [Arcsin(Sqrt(3/8))]; // G. C. Greubel, Nov 18 2017

Crossrefs

Cf. A195703, A195705, A195706.

Sequence in context: A048236 A193719 A364931 * A275110 A011284 A196760

Adjacent sequences: A195697 A195698 A195699 * A195701 A195702 A195703

Keyword

nonn,cons

Author

Clark Kimberling, Sep 23 2011

Status

approved