A374755 Decimal expansion of the surface area of a regular dodecahedron having unit inradius.

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

Offset: 2

Author

Paolo Xausa, Jul 20 2024

Keywords

• nonn
• cons
• changed

Comments

Bezdek's strong dodecahedral conjecture (proved by Hales, see links) states that, in any packing of unit spheres in the Euclidean 3-space, the surface area of every bounded Voronoi cell is at least this value.

Examples

			16.6508730855465308072112963409855177222127946386...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 2..10000
• Károly Bezdek, On a stronger form of Rogers' lemma and the minimum surface area of Voronoi cells in unit ball packings, Journal für die reine und angewandte Mathematik, No. 518, 2000, pp. 131-143.
• Thomas C. Hales, The Strong Dodecahedral Conjecture and Fejes Toth's Conjecture on Sphere Packings with Kissing Number Twelve, arXiv:1110.0402 [math.MG], 2012.
• Wikipedia, Regular dodecahedron.

Crossrefs

Cf. A374753 (dodecahedral conjecture), A374772, A374837, A374838.

Cf. A001622, A098317, A131595.

Programs

Mathematica

First[RealDigits[30*Sqrt[130 - 58*Sqrt[5]], 10, 100]]

Formula

Equals 30*sqrt(130 - 58*sqrt(5)).

Equals 60*sqrt(3 - A001622)/A098317.

Equals 4*Pi/A374772.

Equals 3*A374753.

Minimal polynomial: x^4 - 234000*x^2 + 64800000. - Stefano Spezia, Sep 03 2025