cp's OEIS Frontend

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

%I A195700 #47 Nov 21 2024 07:40:00
%S A195700 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,
%T A195700 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,
%U A195700 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
%N A195700 Decimal expansion of arcsin(sqrt(3/8)) and of arccos(sqrt(5/8)).
%C A195700 arcsin(sqrt(3/8)) = least x>0 satisfying sin(2*x) = 2*sin(4*x).
%C A195700 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
%H A195700 G. C. Greubel, <a href="/A195700/b195700.txt">Table of n, a(n) for n = 0..5000</a>
%H A195700 M. Firsching, <a href="https://arxiv.org/abs/1407.0683">Computing maximal copies of polytopes contained in a polytope</a>, arXiv:1407.0683 [math.MG], 2014.
%H A195700 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A195700 Equals arctan(sqrt(3/5)). - _Amiram Eldar_, Jul 04 2023
%e A195700 0.6590580358264089828728321...
%t A195700 r = Sqrt[3/8];
%t A195700 N[ArcSin[r], 100]
%t A195700 RealDigits[%]  (* this sequence *)
%t A195700 N[ArcCos[r], 100]
%t A195700 RealDigits[%]  (* A195703 *)
%t A195700 N[ArcTan[r], 100]
%t A195700 RealDigits[%]  (* A195705 *)
%t A195700 N[ArcCos[-r], 100]
%t A195700 RealDigits[%]  (* A195706 *)
%o A195700 (PARI) asin(sqrt(3/8)) \\ _G. C. Greubel_, Nov 18 2017
%o A195700 (Magma) [Arcsin(Sqrt(3/8))]; // _G. C. Greubel_, Nov 18 2017
%Y A195700 Cf. A195703, A195705, A195706.
%K A195700 nonn,cons
%O A195700 0,1
%A A195700 _Clark Kimberling_, Sep 23 2011