A374771 -id:A374771 - 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

A237603 Decimal expansion of the inscribed sphere radius in a regular dodecahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Feb 25 2014

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)).

			1.1135163644116067351943750394869493758831503698864877726012080...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A182007, A019863, A019863, A019952, A374771 (sphere volume).

Cf. Platonic solids inradii: A020781 (tetrahedron), A020763 (octahedron), A179294 (icosahedron).

RealDigits[ Cos[Pi/5]^2 / Sin[Pi/5], 10, 111][[1]] (* Or *)
RealDigits[ Sqrt[5/8 + 11/(8 Sqrt[5])], 10, 111][[1]] (* Robert G. Wilson v, Feb 28 2014 *)

PARI
```
sqrt(250+110*sqrt(5))/20
```

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.

A374955 Decimal expansion of Muder's 1993 lower bound for the volume of any Voronoi polyhedron defined by a packing of unit spheres in the Euclidean 3-space.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 25 2024

See A374753 (the dodecahedral conjecture) for an improved bound.

			5.4184829626607232941445725209324645278183095589982...

Douglas J. Muder, A New Bound on the Local Density of Sphere Packings, Discrete & Computational Geometry, Vol. 10, 1993, pp. 351-375.

Cf. A374771, A374753, A374956 (density).

Module[{beta, r, s},
  s[p_] := Pi - 5*ArcTan[Sqrt[(1 - 2*r^2)/(p*r^2)]];
  beta = 5*r*Sqrt[1 - 2*r^2]/(3*Sqrt[2]) + s[2]/6;
  r = SolveValues[4/13*Pi == 2*s[3] - Sqrt[8/3]*s[2] && r > 0, r, Reals];
  RealDigits[13*beta, 10, 100][[1,1]]]

Equals 13*beta, where beta = 5*r*sqrt(1-2*r^2)/(3*sqrt(2)) + (1/6)*(Pi - 5*arctan(sqrt((1 - 2*r^2)/(2*r^2)))) and r is the positive solution to (4/13)*Pi = 2*(Pi - 5*arctan(sqrt((1 - 2*r^2)/(3*r^2)))) - sqrt(8/3)*(Pi - 5*arctan(sqrt((1 - 2*r^2)/(2*r^2)))). See Theorem in Muder (1993), p. 352.

Equals (4/3)*Pi/A374956.

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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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A237603 Decimal expansion of the inscribed sphere radius in a regular dodecahedron with unit edge.

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A374955 Decimal expansion of Muder's 1993 lower bound for the volume of any Voronoi polyhedron defined by a packing of unit spheres in the Euclidean 3-space.

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