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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).
5
4, 0, 4, 2, 8, 3, 3, 4, 5, 0, 4, 4, 8, 9, 3, 5, 8, 5
OFFSET
0,1
COMMENTS
The angle in degrees is 23.1637293985...
LINKS
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.
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))).
EXAMPLE
0.404283345044893585...
MATHEMATICA
See the program in the links section.
CROSSREFS
KEYWORD
nonn,cons,more
AUTHOR
Amiram Eldar, Mar 18 2023
STATUS
approved