A211174 Johannes Kepler's polyhedron circumscribing constant.

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

Offset: 2

Author

William H. Richardson and Robert G. Wilson v, Feb 01 2013

Keywords

• cons
• nonn

Comments

The finite solid analogy to the plane polygon circumscribing constant (A051762).

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

The geometric interpretation is as follows. Begin with a unit sphere. Circumscribe a tetrahedron and then circumscribe a sphere. Circumscribe a cube and then circumscribe a sphere. Circumscribe an octahedron and then circumscribe a sphere. Circumscribe a dodecahedron and then a sphere. Circumscribe an icosahedron and then a sphere. The constant is the radius of this last sphere. In actuality, it makes no difference the order of the five solids.

Examples

			14.25232921501135639390462188851108328620660858097761088937154874783...
		

Links

• Rüdiger Appel, 3Quarks: Platonic Solids (September 2010).
• David A. Fontaine, The Five Platonic Solids.
• George W. Hart, Virtual Polyhedra, 1998, Johannes Kepler's Polyhedra.
• Omar E. Pol, Circunferencias concéntricas y polígonos regulares inscritos, gaussianos, Nov 17 2007, 13:17
• Wikipedia, Platonic Solids.
• Wikipedia, Johannes Kepler.
• Wikipedia, Kepler solar system.

Crossrefs

Cf. A051762.

Programs

Mathematica

RealDigits[ 9(15 - 6 * Sqrt[5]), 10, 111][[1]]

Formula

= 9*(15 - 6*sqrt(5)).

Extensions

Offset corrected by Rick L. Shepherd, Dec 31 2013