A361603 Decimal expansion of the standard deviation of the distribution of disorientation angles between two identical cubes (in radians).

1, 9, 7, 4, 8, 3, 0, 2, 6, 7, 7, 9, 4, 9, 4, 1, 6, 4, 0, 2, 6, 0, 7, 9, 9, 2, 7, 7, 5, 3, 7, 8, 4, 2, 5, 4, 9, 8, 5, 3, 8, 6, 4, 7, 6, 3, 0, 2, 9, 8, 4, 5, 3, 7, 0, 8, 4, 9, 7, 9, 7, 4, 2, 3, 0, 3, 4, 2, 9, 1, 5, 2, 8, 1, 2, 1, 9, 1, 2, 7, 1, 8, 5, 7, 6, 0, 5, 5, 8, 0, 2, 5, 2, 6, 0, 6, 8, 1, 6, 1, 7, 7, 6, 9, 2

Offset

0,2

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 11.3149439599...

Links

Table of n, a(n) for n=0..104.

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.

Formula

Equals sqrt(<t^2> - <t>^2), where <t^k> = Integral_{t=0..tmax} t^k * P(t) dt, tmax = A361601, and P(t) is given in the Formula section of A361602.

Example

0.19748302677949416402607992775378425498538647630298...

Mathematica

See the program in the links section.

Crossrefs

Cf. A361601, A361602, A361604.

Sequence in context: A154482 A155958 A010546 * A232735 A191760 A237841

Adjacent sequences: A361600 A361601 A361602 * A361604 A361605 A361606

Keyword

nonn,cons

Author

Amiram Eldar, Mar 17 2023

Status

approved