A237603 - OEIS

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

(list; constant; graph; refs; listen; history; text; internal format)

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).