%I #11 Sep 02 2024 10:04:02
%S 7,9,4,6,5,4,4,7,2,2,9,1,7,6,6,1,2,2,9,5,5,5,3,0,9,2,8,3,2,7,5,9,4,0,
%T 4,2,0,2,6,5,9,0,5,8,8,3,0,9,2,6,4,8,0,1,7,5,4,9,5,5,7,7,5,0,0,8,4,3,
%U 8,6,4,4,9,7,1,7,3,7,1,1,6,7,9,3,0,2,7,2,9,9,4,8,4,8,7,0,8,7,1,3,7,8,5,2,8
%N Decimal expansion of radius of inscribed sphere of an icosahedron with radius of circumscribed sphere = 1.
%C Icosahedron: A three-dimensional figure with 20 equilateral triangle faces, 12 vertices, and 30 edges.
%H Mathforum, <a href="http://mathforum.org/dr.math/faq/formulas/faq.polyhedron.html#icosahedron">Icosahedron</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Icosahedron">Icosahedron</a>
%H MathWorld, <a href="http://mathworld.wolfram.com/Icosahedron.html">Icosahedron</a>
%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%F Sqrt(75 + 30*sqrt(5))/15.
%e 0.794654472291766122955530928327594042026590588309264801754955775008438...
%t R=1;RealDigits[N[(Sqrt[75+30*Sqrt[5]]/15)*R,175]]
%o (PARI) polrootsreal(45*x^4-30*x^2+1)[4] \\ _Charles R Greathouse IV_, Sep 02 2024
%o (PARI) sqrt(6/sqrt(5)+3)/3 \\ _Charles R Greathouse IV_, Sep 02 2024
%Y Cf. A010527, A102208, A179290.
%K nonn,cons,easy
%O 0,1
%A _Vladimir Joseph Stephan Orlovsky_, Jul 09 2010
%E Offset corrected, keyword:cons added by _R. J. Mathar_, Jul 11 2010