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.

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

Author

Paolo Xausa, Jul 25 2024

Keywords

• nonn
• cons

Comments

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

Examples

			5.4184829626607232941445725209324645278183095589982...
		

Links

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

Crossrefs

Cf. A374771, A374753, A374956 (density).

Programs

Mathematica

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]]]

Formula

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.