A374755 -id:A374755 - 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

A374753 Decimal expansion of the volume of a regular dodecahedron having unit inradius.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 19 2024

The dodecahedral conjecture (proved in 1988 by Thomas C. Hales and Sean McLaughlin, see links) states that, in any packing of unit spheres in the Euclidean 3-space, every Voronoi cell has volume at least equal to this value.

			5.55029102851551026907043211366183924073759821288...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Thomas C. Hales and Sean McLaughlin, A proof of the dodecahedral conjecture, arXiv:math/9811079 [math.MG], 2008.
Thomas C. Hales and Sean McLaughlin, The Dodecaheral Conjecture, Journal of the American Mathematical Society, Vol. 23, No. 2, April 2010, pp. 299-344.
Wikipedia, Dodecahedral conjecture.
Wikipedia, Regular dodecahedron.
Index entries for algebraic numbers, degree 4.

Cf. A019699, A374755 (strong dodecahedral conjecture), A374772 (density).

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

Equals (4/3)*Pi/A374772 = 10*A019699/A374772.
Equals 10*sqrt(130 - 58*sqrt(5)).
Equals A374755/3.

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.

A374838 Decimal expansion of Bezdek and Daróczy-Kiss's lower bound for the surface area of any Voronoi cell in an arbitrary packing of unit balls in the Euclidean 3-space.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 21 2024

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

See A374755 for an improved bound (the strong dodecahedral conjecture).

			16.144502852765379806937602328092933543868922009780...

Paolo Xausa, Table of n, a(n) for n = 2..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, A374837.

First[RealDigits[20*Pi*Tan[Pi/5]/(30*ArcCos[Sqrt[3]/2*Sin[Pi/5]] - 9*Pi), 10, 100]]

Equals 20*Pi*tan(Pi/5)/(30*arccos(sqrt(3)/2*sin(Pi/5)) - 9*Pi).

Equals 4*Pi/A374837.

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.

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A374753 Decimal expansion of the volume of a regular dodecahedron having unit inradius.

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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.

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A374838 Decimal expansion of Bezdek and Daróczy-Kiss's lower bound for the surface area of any Voronoi cell in an arbitrary packing of unit balls in the Euclidean 3-space.

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