OFFSET
0,1
COMMENTS
This constant is the packing density of a regular tetrahedron. That is, let S be a regular tetrahedron of edge length 2 and let B the part of S that lies within distance 1 of some vertex. Then this constant is the ratio of the volume of B to the volume of S.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
Thomas C. Hales, A proof of the Kepler conjecture, Annals of Mathematics, Vol. 162, No. 3 (2005), pp. 1065-1185.
Carl A. Rogers, The packing of equal spheres, Proceedings of the London Mathematical Society, Vol. 3, No. 4 (1958), pp. 609-620.
FORMULA
Equals sqrt(8)*arctan(sqrt(2)/5).
Equals 3*sqrt(2)*(arccos(1/3) - Pi/3). - Amiram Eldar, Jun 26 2021
EXAMPLE
0.77963557004425293782545082499322903001111595395254...
MATHEMATICA
First@ RealDigits@ N[Sqrt[8] ArcTan[Sqrt[2]/5], 120] (* Michael De Vlieger, Jan 09 2016 *)
PROG
(PARI) sqrt(8)*atan(sqrt(2)/5) \\ Michel Marcus, Jan 09 2016
(Magma) SetDefaultRealField(RealField(100)); Sqrt(8)*Arctan(Sqrt(2)/5); // G. C. Greubel, Aug 19 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Martin Renner, Jan 09 2016
STATUS
approved