A348681 -id:A348681 - cp's OEIS frontend

A348680 Decimal expansion of the average length of a chord in a unit square defined by a point on the perimeter and a direction, both uniformly and independently chosen at random.

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

Offset: 0

Amiram Eldar, Oct 29 2021

			0.70980150661400782746374731464451797194994085344545...

A. M. Mathai, An introduction to geometrical probability: distributional aspects with applications, Amsterdam: Gordon and Breach, 1999, p. 221, ex. 2.3.7.

Rodney Coleman, Random paths through convex bodies, Journal of Applied Probability, Vol. 6, No. 2 (1969), pp. 430-441; alternative link; author's link.
Maurice Horowitz, Probability of random paths across elementary geometrical shapes, Journal of Applied Probability, Vol. 2, No. 1 (1965), pp. 169-177; Correction, ibid., Vol. 3, No. 1 (1966), p. 285.
Philip W. Kuchel and Rodney J. Vaughan, Average Lengths of Chords in a Square, Mathematics Magazine, Vol. 54, No. 5 (1981), pp. 261-269.

Cf. A091505, A247674, A348681, A348682, A348683.

RealDigits[(3 * Log[1 + Sqrt[2]] + 1 - Sqrt[2])/Pi, 10, 100][[1]]

(3*log(1 + sqrt(2)) + 1 - sqrt(2))/Pi \\ Michel Marcus, Oct 29 2021

Equals (3*log(1 + sqrt(2)) + 1 - sqrt(2))/Pi.

A348682 Decimal expansion of the average length of a chord in a unit cube defined by a point on the surface and a direction, both uniformly and independently chosen at random.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Oct 29 2021

			0.5977557435927337398151960798274735969724820222495278516...

Rodney Coleman, Random paths through convex bodies, Journal of Applied Probability, Vol. 6, No. 2 (1969), pp. 430-441; alternative link; author's link.
Maurice Horowitz, Probability of random paths across elementary geometrical shapes, Journal of Applied Probability, Vol. 2, No. 1 (1965), pp. 169-177; Correction, ibid., Vol. 3, No. 1 (1966), p. 285.

Cf. A073012, A093066, A348680, A348681, A348683.

RealDigits[N[(1/(3*Pi)) * (2*Pi - 6 + 2*Log[2] + 7*Log[3]/2 + 4*Sqrt[2]*ArcCot[Sqrt[2]]) - (4/Pi) * Integrate[Sqrt[x^2-1] * (x * ArcCot[x] + Log[1 + x^2]/2) / x, {x, 1, Sqrt[2]}], 110], 10, 100][[1]]

Equals (1/(3*Pi)) * (2*Pi - 6 + 2*log(2) + 7*log(3)/2 + 4*sqrt(2)*arccot(sqrt(2))) - (4/Pi) * Integral_{x=1..sqrt(2)} (sqrt(x^2-1) * (x * arccot(x) + log(1 + x^2)/2) / x) dx.

A348683 Decimal expansion of the average length of a chord in a unit cube defined by the intersection of the surface with a straight line passing through 2 points uniformly and independently chosen at random in the interior of the cube.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 29 2021

			1.10284530377862705859305188874735596957733355965158...

Rodney Coleman, Random paths through convex bodies, Journal of Applied Probability, Vol. 6, No. 2 (1969), pp. 430-441; alternative link; author's link.

Cf. A073012, A093066, A348680, A348681, A348682.

RealDigits[4/63 - Pi/9 + 17*Sqrt[2]/63 - 2*Sqrt[3]/21 + Log[1+Sqrt[2]]/3 + 2*Log[2+Sqrt[3]]/3, 10, 100][[1]]

4/63 - Pi/9 + 17*sqrt(2)/63 - 2*sqrt(3)/21 + log(1+sqrt(2))/3 + 2*log(2+sqrt(3))/3 \\ Michel Marcus, Oct 29 2021

Equals 4/63 - Pi/9 + 17*sqrt(2)/63 - 2*sqrt(3)/21 + log(1+sqrt(2))/3 + 2*log(2+sqrt(3))/3.

A355415 Decimal expansion of the average distance between the center of a unit cube to a point on its surface uniformly chosen by a random direction from the center.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 01 2022

If the point is uniformly chosen at random on the surface, then the average is A097047.

			0.61068740195158385653466722967371662846911552581907...

Mathematics Stack Exchange, Average Width of the Cube, or Average Radius of a Regular Polyhedron, 2021.
Mathematics Stack Exchange, What is the average radius of the hypercube?, 2019.

Cf. A006752, A073012, A093066, A097047, A130590, A135691, A348680, A348681, A348682, A348683, A355186 (2D analog).

RealDigits[N[(3/Pi)*Integrate[ArcCot[Sqrt[1 + x^2]]/Sqrt[1 + x^2], {x, 0, 1}], 101], 10, 100][[1]]
(* or *)
RealDigits[3 * ((Im[PolyLog[2, (3 - 2*Sqrt[2])*I]] - Catalan)/Pi - Log[17 - 12*Sqrt[2]]/8), 10, 100][[1]]

Equals (1/2) * Integral_{x=-1..1, y=-1..1} (1 + x^2 + y^2)^(-1) dx dy / Integral_{x=-1..1, y=-1..1} (1 + x^2 + y^2)^(-3/2) dx dy.
Equals (3/Pi) * Integral_{x=0..1} arccot(sqrt(1+x^2))/sqrt(1+x^2) dx.
Equals (6/Pi) * Integral_{x=0..Pi/4} log(sqrt(1+cos(x)^2)/cos(x)) dx.
Equals 3 * ((Im(Li_2((3-2*sqrt(2))*i)) - G)/Pi - log(17-12*sqrt(2))/8), where Li_2 is the dilogarithm function, i is the imaginary unit, and G is Catalan's constant (A006752).

cp's OEIS Frontend

A348680 Decimal expansion of the average length of a chord in a unit square defined by a point on the perimeter and a direction, both uniformly and independently chosen at random.

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A348682 Decimal expansion of the average length of a chord in a unit cube defined by a point on the surface and a direction, both uniformly and independently chosen at random.

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A348683 Decimal expansion of the average length of a chord in a unit cube defined by the intersection of the surface with a straight line passing through 2 points uniformly and independently chosen at random in the interior of the cube.

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A355415 Decimal expansion of the average distance between the center of a unit cube to a point on its surface uniformly chosen by a random direction from the center.

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