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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).
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%I #8 Mar 19 2023 08:29:36

%S 4,0,4,2,8,3,3,4,5,0,4,4,8,9,3,5,8,5

%N 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).

%C The angle in degrees is 23.1637293985...

%H Amiram Eldar, <a href="/A361618/a361618.txt">Mathematica code for A361618 and A361619</a>.

%H J. K. Mackenzie, <a href="http://www.jstor.org/stable/2333059">Second Paper on Statistics Associated with the Random Disorientation of Cubes</a>, Biometrika, Vol. 45, No. 1-2 (1958), pp. 229-240.

%H J. K. Mackenzie and M. J. Thomson, <a href="http://www.jstor.org/stable/2333253">Some Statistics Associated with the Random Disorientation of Cubes</a>, Biometrika, Vol. 44, No. 1-2 (1957), pp. 205-210.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Misorientation">Misorientation</a>.

%F 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

%F P1(t) = (288/Pi^2) * sin(t) * (Pi^2/32 - Integral_{x=0..Pi/4} arcsin(tan(t/2)*cos(x)) dx),

%F 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),

%F g(t) = arcsin(sqrt(((sqrt(2) + 1)^2 * tan(t/2)^2 - 1)/(4 * sqrt(2) * tan(t/2)^2))).

%e 0.404283345044893585...

%t See the program in the links section.

%Y Cf. A228496, A361601, A361619, A361620, A361621.

%K nonn,cons,more

%O 0,1

%A _Amiram Eldar_, Mar 18 2023