A211174 - OEIS

A211174

Decimal expansion of 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

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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, Johannes Kepler's Polyhedra, Virtual Polyhedra, 1998.

Omar E. Pol, Circunferencias concéntricas y polígonos regulares inscritos, gaussianos, Nov 17 2007, 13:17.

Wikipedia, Johannes Kepler.

Wikipedia, Kepler solar system.

Wikipedia, Platonic Solids.

Index entries for algebraic numbers, degree 2.

FORMULA

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