A381696 Decimal expansion of the isoperimetric quotient of a snub dodecahedron.

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

Offset: 0

Author

Paolo Xausa, Mar 10 2025

Keywords

• nonn
• cons
• easy

Comments

For the definition of isoperimetric quotient of a solid, references and links, see A381684.

The snub dodecahedron is the Archimedean solid with the highest isoperimetric quotient.

Examples

			0.94699904523421562618454412877087470550479673815077...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• Index entries for transcendental numbers

Crossrefs

Cf. A377804 (surface area), A377805 (volume).

Cf. A000796, A002163, A002194, A381684.

Programs

Mathematica

First[RealDigits[Pi/(20*Sqrt[3] + 3*Sqrt[25 + Sqrt[500]])^3*Root[#^6 - 52845*#^5 + 96583500*#^4 + 22087761875*#^3 + 4747626384375*#^2 - 5059009765293750*# + 187445810737515625 &, 4], 10, 100]]

Formula

Equals 36*Pi*A377805^2/(A377804^3).

Equals Pi/((20*sqrt(3) + 3*sqrt(25 + 10*sqrt(5)))^3)*r = A000796/((20*A002194 + 3*sqrt(25 + 10*A002163))^3)*r, where r is the largest real root of x^6 - 52845*x^5 + 96583500*x^4 + 22087761875*x^3 + 4747626384375*x^2 - 5059009765293750*x + 187445810737515625.