cp's OEIS Frontend

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

%I A179290 #77 Aug 29 2025 13:00:23
%S A179290 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,
%T A179290 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,
%U A179290 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
%N A179290 Decimal expansion of length of edge of a regular icosahedron with radius of circumscribed sphere = 1.
%C A179290 Regular icosahedron: A three-dimensional figure with 20 congruent equilateral triangle faces, 12 vertices, and 30 edges.
%C A179290 Shorter diagonal of golden rhombus with unit edge length. - _Eric W. Weisstein_, Dec 11 2018
%C A179290 The length of the shorter side of a golden rectangle inscribed in a unit circle. - _Michal Paulovic_, Sep 01 2022
%C A179290 The side length of a square inscribed within a golden ellipse with a unit semi-major axis. - _Amiram Eldar_, Oct 02 2022
%C A179290 (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
%H A179290 Muniru A Asiru, <a href="/A179290/b179290.txt">Table of n, a(n) for n = 1..1000</a>
%H A179290 J. Brandts, S. Korotov, M. Krizek, and J. Solc, <a href="http://dx.doi.org/10.1137/060669073">On nonobtuse simplicial partitions</a>, Siam Rev. 51 (2) (2009) 317-335.
%H A179290 Dr. Math, <a href="http://mathforum.org/dr.math/faq/formulas/faq.polyhedron.html#icosahedron">Regular Polyhedra: Formulas</a>.
%H A179290 Hugo Pfoertner, <a href="http://www.randomwalk.de/sphere/volmax/images/32side.png">Visualization of the 32-edge polyhedron with conjecturally maximum volume</a>, (2021).
%H A179290 Hugo Pfoertner, <a href="http://www.randomwalk.de/sphere/volmax/videos/32c.mp4">Number of edges incident with the vertices of the 32-edge polyhedron</a>, video (2021).
%H A179290 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRhombus.html">Golden Rhombus</a>.
%H A179290 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Icosahedron.html">Icosahedron</a>.
%H A179290 Wikipedia, <a href="http://en.wikipedia.org/wiki/Icosahedron">Icosahedron</a>.
%H A179290 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%F A179290 Equals sqrt(50-10*sqrt(5))/5.
%F A179290 Equals csc(2*Pi/5). - _Eric W. Weisstein_, Dec 11 2018
%F A179290 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
%F A179290 Equals 1/A019881. - _R. J. Mathar_, Jan 17 2021
%F A179290 From _Antonio Graciá Llorente_, Mar 15 2024: (Start)
%F A179290 Equals Product_{k >= 1} ((10*k - 1)*(10*k + 1))/((10*k - 2)*(10*k + 2)).
%F A179290 Equals Product_{k >= 1} 1/(1 - 1/(25*(2*k - 1)^2)). (End)
%F A179290 Equals Product_{k>=1} (1 - (-1)^k/A090773(k)). - _Amiram Eldar_, Nov 23 2024
%F A179290 A root of 5*x^4 - 20*x^2 + 16=0 (see A121570). - _R. J. Mathar_, Aug 29 2025
%e A179290 1.051462224238267212051338169695753214570995864486683563057871046482422...
%p A179290 evalf[120](csc(2*Pi/5)); # _Muniru A Asiru_, Dec 11 2018
%t A179290 RealDigits[Csc[2 Pi/5], 10, 110][[1]] (* _Eric W. Weisstein_, Dec 11 2018 *)
%o A179290 (Python)
%o A179290 from decimal import *
%o A179290 getcontext().prec = 110
%o A179290 c = Decimal.sqrt(2 - 2 / Decimal.sqrt(Decimal(5)))
%o A179290 print([int(i) for i in str(c) if i != '.'])
%o A179290 # _Karl V. Keller, Jr._, Jul 10 2020
%o A179290 (PARI) sqrt(50-10*sqrt(5))/5 \\ _Charles R Greathouse IV_, Jan 22 2024
%Y A179290 Cf. A010527, A019881, A090773, A102208.
%Y A179290 Cf. A179290 (longer golden rhombus diagonal).
%K A179290 nonn,cons,easy,changed
%O A179290 1,3
%A A179290 _Vladimir Joseph Stephan Orlovsky_, Jul 09 2010
%E A179290 Partially rewritten by _Charles R Greathouse IV_, Feb 02 2011