The average distance between two points chosen at random inside a unit cube (the n = 3 case of hypercube line picking), sometimes known as the Robbins constant, is Δ(3) | = | 1/105[4 + 17sqrt(2) - 6sqrt(3) + 21ln(1 + sqrt(2)) + 42ln(2 + sqrt(3)) - 7π] | = | 1/105[4 + 17sqrt(2) - 6sqrt(3) + 21sinh^(-1) 1 + 42ln(2 + sqrt(3)) - 7π] | = | 0.6617... (OEIS A073012; Robbins 1978, Le Lionnais 1983). The probability function as a function of line length, illustrated above, was found in (nearly) closed form by Mathai et al. (1999).
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