A361601 -id:A361601 - cp's OEIS frontend

A361618 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).

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

Offset: 0

Amiram Eldar, Mar 18 2023

The angle in degrees is 23.1637293985...

			0.404283345044893585...

Amiram Eldar, Mathematica code for A361618 and A361619.
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.

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

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

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
P1(t) = (288/Pi^2) * sin(t) * (Pi^2/32 - Integral_{x=0..Pi/4} arcsin(tan(t/2)*cos(x)) dx),
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),
g(t) = arcsin(sqrt(((sqrt(2) + 1)^2 * tan(t/2)^2 - 1)/(4 * sqrt(2) * tan(t/2)^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).

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

Original entry on oeis.org

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

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 11.3149439599...

			0.19748302677949416402607992775378425498538647630298...

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, A361604.

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

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

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.

A361619 Decimal expansion of the standard deviation of the distribution of the least of the nine acute angles between pairs of edges of two randomly disoriented cubes (in radians).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 18 2023

The corresponding value in degrees is 10.3116974681...

			0.179973072288580123...

Amiram Eldar, Mathematica code for A361618 and A361619.
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.

Cf. A228496, A361601, A361618, A361620, A361621.

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

Equals sqrt( - ^2), where = Integral_{t=0..arccos(2/3)} t^k * P(t) dt, and P(t) is given in A361618.

A361620 Decimal expansion of the median of the distribution of the least of the nine acute angles between pairs of edges of two randomly disoriented cubes (in radians).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 18 2023

The corresponding value in degrees is 23.1834079659...

			0.40462680083850138475144500357414183647267233632878...

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.
Eric Weisstein's World of Mathematics, Median.
Wikipedia, Misorientation.

Cf. A228496, A361601, A361618, A361619, A361621.

wp = 110; p[a_?NumericQ] := If[a <= 0 || a >= Pi/4, 0, (288/Pi^2) * Sin[a]*(Pi^2/32 - NIntegrate[ArcSin[Tan[a/2]*Cos[x]], {x, 0, Pi/4}, WorkingPrecision -> wp])]; f[y_?NumericQ] := NIntegrate[p[a], {a, 0, y}, WorkingPrecision -> wp]; RealDigits[y /. FindRoot[f[y] == 1/2, {y, 0.5}, WorkingPrecision -> wp], 10, 100][[1]]

Equals c such that Integral_{t=0..c} P(t) dt = 1/2, where P(t) is given in the Formula section of A361618.

A361621 Decimal expansion of the mode of the distribution of the least of the nine acute angles between pairs of edges of two randomly disoriented cubes (in radians).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 18 2023

The value of t where P(t) takes its maximum value, where P(t) is given in A361618.

The corresponding value in degrees is 23.3085980601...

			0.40681178017420877101384268811808080344448609291346...

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.
Eric Weisstein's World of Mathematics, Mode.
Wikipedia, Misorientation.

Cf. A228496, A361601, A361618, A361619, A361620.

wp = 200; f[a_] := If[a <= 0 || a >= Pi/4, 0, Sin[a] * (Pi^2/32 - NIntegrate[ArcSin[Tan[a/2]*Cos[x]], {x, 0, Pi/4}, WorkingPrecision -> wp])]; RealDigits[t /. (NMaximize[f[t], {t}, WorkingPrecision -> wp][[2]]), 10, 100][[1]]

A380702 Decimal expansion of the acute vertex angles, in radians, in a (small) triakis octahedron face.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jan 30 2025

			0.54802840762031274453086328282062867847971236365920...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Triakis octahedron.

Cf. A380703 (face obtuse angle).

Cf. A020765, A361601, A378351, A378352, A378353, A378354.

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

Equals arccos(1/2 + sqrt(2)/4) = arccos(1/2 + A020765).
Equals (Pi - A380703)/2.
Equals A361601/2. - Hugo Pfoertner, Jan 30 2025

cp's OEIS Frontend

A361618 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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A361602 Decimal expansion of the mean of the distribution of disorientation angles between two identical cubes (in radians).

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A361603 Decimal expansion of the standard deviation 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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A361619 Decimal expansion of the standard deviation 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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A361620 Decimal expansion of the median 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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A361621 Decimal expansion of the mode 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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Comments

Examples

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A380702 Decimal expansion of the acute vertex angles, in radians, in a (small) triakis octahedron face.

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