A374838 -id:A374838 - cp's OEIS frontend

A374772 Decimal expansion of the upper bound of the density of sphere packing in the Euclidean 3-space resulting from the dodecahedral conjecture.

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

Offset: 0

Paolo Xausa, Jul 19 2024

See A374753 for more information on the dodecahedral conjecture.

Also isoperimetric quotient (see A381671 for definition) of a regular dodecahedron. - Paolo Xausa, May 19 2025

			0.7546973993374058303916521059902293313424321921459...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Local Density.
Index entries for transcendental numbers.

Cf. A374753 (dodecahedral conjecture), A374755 (strong dodecahedral conjecture), A374771, A374837, A374838.

Cf. A093825, A019699, A102769, A131595, A381671.

First[RealDigits[Pi*Sqrt[5 + Sqrt[5]]/(15*Sqrt[10]*(Sqrt[5] - 2)), 10, 100]]

Pi*sqrt(5 + sqrt(5))/(15*sqrt(10)*(sqrt(5) - 2)) \\ Charles R Greathouse IV, Feb 07 2025

Equals (4/3)*Pi/A374753 = 10*A019699/A374753.
Equals A374771/A102769.
Equals Pi*sqrt(5 + sqrt(5))/(15*sqrt(10)*(sqrt(5) - 2)).
Equals 4*Pi/A374755.
Equals 36*Pi*A102769^2/(A131595^3). - Paolo Xausa, May 19 2025

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

Original entry on oeis.org

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

Paolo Xausa, Jul 20 2024

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.

			16.6508730855465308072112963409855177222127946386...

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.

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

Cf. A001622, A098317, A131595.

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

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

A374837 Decimal expansion of Bezdek and Daróczy-Kiss's upper bound for the surface area density of a unit ball in any face cone of a Voronoi cell in an arbitrary packing of unit balls in the Euclidean 3-space.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 21 2024

See Theorem 1.1 in Bezdek and Daróczy-Kiss (2005).

See A374772 for an improved bound.

			0.7783683851377739227957671666059435201971163186281...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Károly Bezdek and Endre Daróczy-Kiss, Finding the Best Face on a Voronoi Polyhedron--The Strong Dodecahedral Conjecture Revisited, Monatshefte für Mathematik, Vol. 145, No. 3, July 2005, pp. 191-206.

Cf. A374753 (dodecahedral conjecture), A374755 (strong dodecahedral conjecture), A374772, A374838.

First[RealDigits[(30*ArcCos[Sqrt[3]/2*Sin[Pi/5]] - 9*Pi)/(5*Tan[Pi/5]), 10, 100]]

Equals (30*arccos((sqrt(3)/2)*sin(Pi/5)) - 9*Pi)/(5*tan(Pi/5)).

Equals 4*Pi/A374838.

cp's OEIS Frontend

A374772 Decimal expansion of the upper bound of the density of sphere packing in the Euclidean 3-space resulting from the dodecahedral conjecture.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A374837 Decimal expansion of Bezdek and Daróczy-Kiss's upper bound for the surface area density of a unit ball in any face cone of a Voronoi cell in an arbitrary packing of unit balls in the Euclidean 3-space.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula