A243908 - OEIS

%I #33 Oct 18 2016 05:42:57

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

%T 2,7,4,7,2,6,8,4,2,0,1,4,2,3,9,2,9,7,3,6,9,2,9,0,2,1,8,1,4,1,3,4,8,9,

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

%N Johannes Kepler's polyhedron inscribing constant.

%C Decimal expansion of (5 + 2*sqrt(5))/135.

%C The finite solid analogy to the plane polygon inscribing constant (A085365).

%C The five Platonic solids are the tetrahedron, the hexahedron (or cube), the octahedron, the dodecahedron and the icosahedron.

%C The geometric interpretation is as follows. Begin with a unit sphere. Inscribe a tetrahedron and then inscribe a sphere. Inscribe a cube and then inscribe a sphere. Inscribe an octahedron and then inscribe a sphere. Inscribe a dodecahedron and then inscribe a sphere. Inscribe an icosahedron and then inscribe a sphere. This constant is the radius of this last sphere. Actually, the order in which the five solids are inscribed has no effect on the resulting constant.

%H Chai Wah Wu, <a href="/A243908/b243908.txt">Table of n, a(n) for n = -1..10001</a>

%H Domingo H. A. and Omar E. Pol, <a href="http://gaussianos.com/circunferencias-concentricas-y-poligonos-regulares-inscritos">Circunferencias concéntricas y polígonos regulares inscritos</a>, gaussianos, Nov 16 2007, 18:46, 23:28

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Platonic_solid">Platonic Solids</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Johannes_Kepler">Johannes Kepler</a>

%H Wikipedia, <a href="http://upload.wikimedia.org/wikipedia/commons/1/19/Kepler-solar-system-1.png">Kepler's solar system</a>

%F Equals 1/A211174 = 1/(9*(15 - 6*sqrt(5))).

%e 0.070163970037033921428284054351574462747268420142392973692902181413489198898...

%p Digits:=100: evalf((5+2*sqrt(5))/135); # _Wesley Ivan Hurt_, Sep 05 2014

%t RealDigits[(5 + 2 Sqrt[5])/135, 10, 100][[1]] (* _Wesley Ivan Hurt_, Sep 05 2014 *)

%o (PARI) first(n)=digits(floor(10^(n+1)*(5+2*sqrt(5))/135)) \\ _Edward Jiang_, Sep 05 2014

%Y Cf. A051762, A085365, A211174.

%K nonn,cons

%O -1,1

%A _Omar E. Pol_, Jun 14 2014