OFFSET
1,1
COMMENTS
See A374753 (the dodecahedral conjecture) for an improved bound.
LINKS
Douglas J. Muder, A New Bound on the Local Density of Sphere Packings, Discrete & Computational Geometry, Vol. 10, 1993, pp. 351-375.
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.
EXAMPLE
5.4184829626607232941445725209324645278183095589982...
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]]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Paolo Xausa, Jul 25 2024
STATUS
approved