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.
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
Examples
0.5977557435927337398151960798274735969724820222495278516...
Links
- 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.
Programs
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Mathematica
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]]
Formula
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.