A361602 Decimal expansion of the mean of the distribution of disorientation angles between two identical cubes (in radians).

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, 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, 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

Offset: 0

Author

Amiram Eldar, Mar 17 2023

Keywords

• nonn
• cons

Comments

The probability distribution function of disorientation angles was calculated for random rotations uniformly distributed with respect to Haar measure (see, e.g., Rummler, 2002).

See A361601 for more details.

The angle in degrees is 40.7358443613...

Examples

			0.71097460768605911916438944041537014933928621039476...
		

Links

• Amiram Eldar, Mathematica code for A361602, A361603 and A361604.
• D. C. Handscomb, On the random disorientation of two cubes, Canadian Journal of Mathematics, Vol. 10 (1958), pp. 85-88.
• 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.
• Hansklaus Rummler, On the distribution of rotation angles how great is the mean rotation angle of a random rotation?, The Mathematical Intelligencer, Vol. 24, No. 4 (2002), pp. 6-11; alternative link.
• Wikipedia, Misorientation.

Crossrefs

Cf. A361601, A361603, A361604.

Programs

Mathematica

(* See the program in the links section. *)

Formula

Equals Integral_{t=0..tmax} t * P(t) dt, where tmax = A361601 and P(t) is

1) (24/Pi) * (1-cos(t)) for 0 <= t <= Pi/4.

2) (24/Pi) * (3*(sqrt(2)-1)*sin(t) - 2*(1-cos(t))) for Pi/4 <= t <= Pi/3.

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)).

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).