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

2,2

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.

Links

Table of n, a(n) for n=2..112.

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.

Formula

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

Example

14.25232921501135639390462188851108328620660858097761088937154874783...

Mathematica

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

Crossrefs

Cf. A051762.

Sequence in context: A275927 A072907 A250719 * A367574 A059833 A123152

Adjacent sequences: A211171 A211172 A211173 * A211175 A211176 A211177

Keyword

cons,nonn

Author

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

Extensions

Offset corrected by Rick L. Shepherd, Dec 31 2013

Status

approved