A361603 -id:A361603 - cp's OEIS frontend

A361601 Decimal expansion of the maximum possible disorientation angle between two identical cubes (in radians).

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

Offset: 1

Amiram Eldar, Mar 17 2023

Mackenzie and Thomson (1957) attributed the idea of finding this angle to the British theoretical physicist Frederick Charles Frank (1911-1988), who proposed this problem in 1949.
The disorientation angle between two identical bodies is the least angle of rotation about an axis through the center of mass of one of the bodies that is needed to bring it into the same orientation as the other body. For two cubes with indistinguishable faces, there are 24 rotations angles that will bring the first cube into coincidence with the second, and the disorientation angle is the least of them.
The rotation which achieves this maximum disorientation can be described as a rotation by 90 degrees about any axis parallel to a face diagonal of the cube.
The angle in degrees is 62.7994296198...
The solution to the analogous two-dimensional problem with two squares is the trivial value Pi/4 (A003881).

			1.09605681524062548906172656564125735695942472731840...

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.
Wikipedia, Misorientation.
Index entries for transcendental numbers.

Cf. A003881, A201488, A361602, A361603, A361604.

RealDigits[ArcCos[Sqrt[2]/2 - 1/4], 10, 100][[1]]

PARI
```
acos(sqrt(2)/2 - 1/4)
```

Equals arccos(sqrt(2)/2 - 1/4).
Equals 2 * arccos(1/2 + sqrt(2)/4).
Equals 2 * arctan((sqrt(2)-1) * sqrt(5-2*sqrt(2))).

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

Original entry on oeis.org

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

Amiram Eldar, Mar 17 2023

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...

			0.71097460768605911916438944041537014933928621039476...

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.

Cf. A361601, A361603, A361604.

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

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).

A361604 Decimal expansion of the median of the distribution of disorientation angles between two identical cubes (in radians).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 17 2023

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 42.3413510913...

			0.73899598628760510179634113561583582476481591764706...

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.

Cf. A361601, A361602, A361603.

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

Equals c such that Integral_{t=0..c} P(t) dt = 1/2, where P(t) is given in the Formula section of A361602.

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A361601 Decimal expansion of the maximum possible disorientation angle between two identical cubes (in radians).

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

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A361604 Decimal expansion of the median of the distribution of disorientation angles between two identical cubes (in radians).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula