A179290 Decimal expansion of length of edge of a regular icosahedron with radius of circumscribed sphere = 1.

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

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jul 09 2010

Keywords

• nonn
• cons
• easy
• changed

Comments

Regular icosahedron: A three-dimensional figure with 20 congruent equilateral triangle faces, 12 vertices, and 30 edges.

Shorter diagonal of golden rhombus with unit edge length. - Eric W. Weisstein, Dec 11 2018

The length of the shorter side of a golden rectangle inscribed in a unit circle. - Michal Paulovic, Sep 01 2022

The side length of a square inscribed within a golden ellipse with a unit semi-major axis. - Amiram Eldar, Oct 02 2022

(10/3)*(this constant)=3.504874080794224... is the volume of the polyhedron with 32 edges with conjectured maximum volume inscribed in a sphere of radius 1. It has 60 congruent triangular faces and the symmetry group of the regular icosahedron. See Pfoertner links for visualizations. - Hugo Pfoertner, Aug 02 2025

Examples

			1.051462224238267212051338169695753214570995864486683563057871046482422...
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..1000
• J. Brandts, S. Korotov, M. Krizek, and J. Solc, On nonobtuse simplicial partitions, Siam Rev. 51 (2) (2009) 317-335.
• Dr. Math, Regular Polyhedra: Formulas.
• Hugo Pfoertner, Visualization of the 32-edge polyhedron with conjecturally maximum volume, (2021).
• Hugo Pfoertner, Number of edges incident with the vertices of the 32-edge polyhedron, video (2021).
• Eric Weisstein's World of Mathematics, Golden Rhombus.
• Eric Weisstein's World of Mathematics, Icosahedron.
• Wikipedia, Icosahedron.
• Index entries for algebraic numbers, degree 4.

Crossrefs

Cf. A010527, A019881, A090773, A102208.

Cf. A179290 (longer golden rhombus diagonal).

Programs

Maple

evalf[120](csc(2*Pi/5)); # Muniru A Asiru, Dec 11 2018

Mathematica

RealDigits[Csc[2 Pi/5], 10, 110][[1]] (* Eric W. Weisstein, Dec 11 2018 *)

PARI

sqrt(50-10*sqrt(5))/5 \\ Charles R Greathouse IV, Jan 22 2024

Python

from decimal import *
getcontext().prec = 110
c = Decimal.sqrt(2 - 2 / Decimal.sqrt(Decimal(5)))
print([int(i) for i in str(c) if i != '.'])
# Karl V. Keller, Jr., Jul 10 2020

Formula

Equals sqrt(50-10*sqrt(5))/5.

Equals csc(2*Pi/5). - Eric W. Weisstein, Dec 11 2018

Equals 1/Im(e^(3*i*Pi/5)) = 1/Im(e^(3*i*Pi/5) - 1) = sqrt(2 - 2/sqrt(5)). - Karl V. Keller, Jr., Jun 11 2020

Equals 1/A019881. - R. J. Mathar, Jan 17 2021

From Antonio Graciá Llorente, Mar 15 2024: (Start)

Equals Product_{k >= 1} ((10*k - 1)*(10*k + 1))/((10*k - 2)*(10*k + 2)).

Equals Product_{k >= 1} 1/(1 - 1/(25*(2*k - 1)^2)). (End)

Equals Product_{k>=1} (1 - (-1)^k/A090773(k)). - Amiram Eldar, Nov 23 2024

A root of 5*x^4 - 20*x^2 + 16=0 (see A121570). - R. J. Mathar, Aug 29 2025

Extensions

Partially rewritten by Charles R Greathouse IV, Feb 02 2011