A361618 Decimal expansion of the mean of the distribution of the least of the nine acute angles between pairs of edges of two randomly disoriented cubes (in radians).
4, 0, 4, 2, 8, 3, 3, 4, 5, 0, 4, 4, 8, 9, 3, 5, 8, 5
Offset: 0
Examples
0.404283345044893585...
Links
- Amiram Eldar, Mathematica code for A361618 and A361619.
- J. K. Mackenzie, Second Paper on Statistics Associated with the Random Disorientation of Cubes, Biometrika, Vol. 45, No. 1-2 (1958), pp. 229-240.
- J. K. Mackenzie and M. J. Thomson, Some Statistics Associated with the Random Disorientation of Cubes, Biometrika, Vol. 44, No. 1-2 (1957), pp. 205-210.
- Wikipedia, Misorientation.
Programs
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Mathematica
(* See the program in the links section. *)
Formula
Equals Integral_{t=0..arccos(2/3)} t * P(t) dt, where P(t) = P1(t) if 0 <= t <= Pi/4, and P1(t) + P2(t) if Pi/4 < t <= arccos(2/3), and where
P1(t) = (288/Pi^2) * sin(t) * (Pi^2/32 - Integral_{x=0..Pi/4} arcsin(tan(t/2)*cos(x)) dx),
P2(t) = -(288/Pi^2) * sin(t) * (Pi*g(t)/4 - Integral_{x=Pi/4-g(t)..Pi/8+g(t)} arcsin(tan(t/2)*cos(x)) dx),
g(t) = arcsin(sqrt(((sqrt(2) + 1)^2 * tan(t/2)^2 - 1)/(4 * sqrt(2) * tan(t/2)^2))).
Comments