%I #17 Feb 25 2023 11:46:08
%S 6,0,5,4,6,1,3,8,2,9,1,2,5,2,5,5,8,3,3,8,6,2,6,5,2,0,5,1,2,8,0,4,4,4,
%T 9,0,3,0,0,8,4,5,4,0,8,8,0,1,4,2,8,8,9,3,3,2,0,0,9,3,5,0,0,0,8,3,8,2,
%U 9,5,6,8,3,8,2,0,7,2,7,2,7,8,5,3,6,2,4,2,6,2,5,9,6,8,8,1,3,0,5,1,9,3,2,4,1
%N Decimal expansion of sqrt(10 + 2*sqrt(5))/(2*Pi).
%C The ratio of the volume of a regular icosahedron to the volume of the circumscribed sphere (with circumradius a*sqrt(10 + 2*sqrt(5))/4 = a*A019881, where a is the icosahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165952, and A165953. A063723 shows the order of these by size.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Icosahedron.html">Icosahedron</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F sqrt(10 + 2*sqrt(5))/(2*Pi) = sqrt(10 + 2*A002163)/(2*A000796) = 2*sin(2*Pi/5)/Pi = 2*sin(A019694)/A000796 = 2*sin(72 deg)/Pi = 2*A019881/A000796 = 2*A019881*A049541 = (2/Pi)*sin(72 deg) = A060294*A019881.
%e 0.6054613829125255833862652051280444903008454088014288933200935000838295683...
%t RealDigits[Sqrt[10+2Sqrt[5]]/(2Pi),10,120][[1]] (* _Harvey P. Dale_, Aug 27 2013 *)
%o (PARI) sqrt(10+2*sqrt(5))/(2*Pi)
%Y Cf. A000796, A002163, A165922, A049541, A165952, A165953, A063723, A019881, A019694, A060294.
%K cons,nonn
%O 0,1
%A _Rick L. Shepherd_, Oct 04 2009