This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A361602 #12 Apr 13 2025 03:19:25
%S A361602 7,1,0,9,7,4,6,0,7,6,8,6,0,5,9,1,1,9,1,6,4,3,8,9,4,4,0,4,1,5,3,7,0,1,
%T A361602 4,9,3,3,9,2,8,6,2,1,0,3,9,4,7,6,0,5,6,3,0,7,4,1,2,3,7,4,8,0,4,2,3,8,
%U A361602 0,0,7,2,4,4,1,5,8,7,6,7,8,7,9,1,0,5,1,3,3,2,0,4,4,7,2,6,8,6,0,6,7,2,7,1,2
%N A361602 Decimal expansion of the mean of the distribution of disorientation angles between two identical cubes (in radians).
%C A361602 The probability distribution function of disorientation angles was calculated for random rotations uniformly distributed with respect to Haar measure (see, e.g., Rummler, 2002).
%C A361602 See A361601 for more details.
%C A361602 The angle in degrees is 40.7358443613...
%H A361602 Amiram Eldar, <a href="/A361602/a361602.txt">Mathematica code for A361602, A361603 and A361604</a>.
%H A361602 D. C. Handscomb, <a href="https://doi.org/10.4153/CJM-1958-010-0">On the random disorientation of two cubes</a>, Canadian Journal of Mathematics, Vol. 10 (1958), pp. 85-88.
%H A361602 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 A361602 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 A361602 Hansklaus Rummler, <a href="https://doi.org/10.1007/BF03025318">On the distribution of rotation angles how great is the mean rotation angle of a random rotation?</a>, The Mathematical Intelligencer, Vol. 24, No. 4 (2002), pp. 6-11; <a href="https://core.ac.uk/download/pdf/159153606.pdf">alternative link</a>.
%H A361602 Wikipedia, <a href="https://en.wikipedia.org/wiki/Misorientation">Misorientation</a>.
%F A361602 Equals Integral_{t=0..tmax} t * P(t) dt, where tmax = A361601 and P(t) is
%F A361602 1) (24/Pi) * (1-cos(t)) for 0 <= t <= Pi/4.
%F A361602 2) (24/Pi) * (3*(sqrt(2)-1)*sin(t) - 2*(1-cos(t))) for Pi/4 <= t <= Pi/3.
%F A361602 3) (24/Pi) * ((3*(sqrt(2)-1) + 4/sqrt(3)) * sin(t) - 6*(1-cos(t))) for Pi/3 <= t <= 2 * arctan(sqrt(2) * (sqrt(2)-1)).
%F A361602 4) (24/Pi) * ((3*(sqrt(2)-1) + 4/sqrt(3)) * sin(t) - 6*(1-cos(t))) - (288*sin(t)/Pi^2) * (2*(sqrt(2)-1) * arccos(f(t) * cot(t/2)) + (1/sqrt(3)) * arccos(g(t) * cot(t/2))) + (288*(1-cos(t))/Pi^2) * (2*arccos(f(t) * (sqrt(2)+1)/sqrt(2)) + arccos(g(t) * (sqrt(2)+1)/sqrt(2))) for 2 * arctan(sqrt(2) * (sqrt(2)-1)) <= t <= tmax, where f(t) = (sqrt(2)-1)/sqrt(1-(sqrt(2)-1)^2 * cot(t/2)^2) and g(t) = (sqrt(2) - 1)^2/sqrt(3 - cot(t/2)^2).
%e A361602 0.71097460768605911916438944041537014933928621039476...
%t A361602 (* See the program in the links section. *)
%Y A361602 Cf. A361601, A361603, A361604.
%K A361602 nonn,cons
%O A361602 0,1
%A A361602 _Amiram Eldar_, Mar 17 2023