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%I #65 Mar 20 2024 04:26:02
%S 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 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 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 Decimal expansion of length of edge of a regular icosahedron with radius of circumscribed sphere = 1.
%C Regular icosahedron: A three-dimensional figure with 20 congruent equilateral triangle faces, 12 vertices, and 30 edges.
%C Shorter diagonal of golden rhombus with unit edge length. - _Eric W. Weisstein_, Dec 11 2018
%C The length of the shorter side of a golden rectangle inscribed in a unit circle. - _Michal Paulovic_, Sep 01 2022
%C The side length of a square inscribed within a golden ellipse with a unit semi-major axis. - _Amiram Eldar_, Oct 02 2022
%H Muniru A Asiru, <a href="/A179290/b179290.txt">Table of n, a(n) for n = 1..1000</a>
%H 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 Dr. Math, <a href="http://mathforum.org/dr.math/faq/formulas/faq.polyhedron.html#icosahedron">Regular Polyhedra: Formulas</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldenRhombus.html">Golden Rhombus</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Icosahedron.html">Icosahedron</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Icosahedron">Icosahedron</a>.
%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%F Equals sqrt(50-10*sqrt(5))/5.
%F Equals csc(2*Pi/5). - _Eric W. Weisstein_, Dec 11 2018
%F 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 Equals 1/A019881. - _R. J. Mathar_, Jan 17 2021
%F From _Antonio Graciá Llorente_, Mar 15 2024: (Start)
%F Equals Product_{k >= 1} ((10*k - 1)*(10*k + 1))/((10*k - 2)*(10*k + 2)).
%F Equals Product_{k >= 1} 1/(1 - 1/(25*(2*k - 1)^2)). (End)
%e 1.051462224238267212051338169695753214570995864486683563057871046482422...
%p evalf[120](csc(2*Pi/5)); # _Muniru A Asiru_, Dec 11 2018
%t RealDigits[Csc[2 Pi/5], 10, 110][[1]] (* _Eric W. Weisstein_, Dec 11 2018 *)
%o (Python)
%o from decimal import *
%o getcontext().prec = 110
%o c = Decimal.sqrt(2 - 2 / Decimal.sqrt(Decimal(5)))
%o print([int(i) for i in str(c) if i != '.'])
%o # _Karl V. Keller, Jr._, Jul 10 2020
%o (PARI) sqrt(50-10*sqrt(5))/5 \\ _Charles R Greathouse IV_, Jan 22 2024
%Y Cf. A010527, A019881, A102208.
%Y Cf. A179290 (longer golden rhombus diagonal).
%K nonn,cons,easy
%O 1,3
%A _Vladimir Joseph Stephan Orlovsky_, Jul 09 2010
%E Partially rewritten by _Charles R Greathouse IV_, Feb 02 2011
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