A237603 Decimal expansion of the inscribed sphere radius in a regular dodecahedron with unit edge.

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

Offset

1,4

References

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Links

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Polytope Wiki, Dodecahedron.

Eric Weisstein's World of Mathematics, Regular Dodecahedron.

Wikipedia, Regular dodecahedron.

Index entries for algebraic numbers, degree 4.

Formula

Equals phi^2/(2*xi), where phi is the golden ratio (A001622, 2*cos(Pi/5)) and xi is its associate (A182007, 2*sin(Pi/5)).

Equals A001622^2/A182007 = (cos(Pi/5))^2/sin(Pi/5) = A019863^2/A019845 = cos(Pi/5)*cotan(Pi/5) = A019863*A019952 = 1/sin(Pi/5) - sin(Pi/5) = A019845^(-1) - A019845 = sqrt(250+110*sqrt(5))/20.

Minimal polynomial: 80*x^4 - 100*x^2 + 1. - Amiram Eldar, May 18 2026

Example

1.1135163644116067351943750394869493758831503698864877726012080...

Mathematica

RealDigits[ Cos[Pi/5]^2 / Sin[Pi/5], 10, 111][[1]]
(* Alternative: *)
RealDigits[ Sqrt[5/8 + 11/(8 Sqrt[5])], 10, 111][[1]] (* Robert G. Wilson v, Feb 28 2014 *)

Prog

(PARI) sqrt(250+110*sqrt(5))/20

Crossrefs

Cf. A001622, A182007, A019863, A019863, A019952, A374771 (sphere volume).

Cf. Platonic solids inradii: A020781 (tetrahedron), A020763 (octahedron), A179294 (icosahedron).

Sequence in context: A344076 A333336 A061649 * A073365 A316152 A302204

Adjacent sequences: A237600 A237601 A237602 * A237604 A237605 A237606

Keyword

nonn,cons,easy

Author

Stanislav Sykora, Feb 25 2014

Status

approved