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Robbins Constant

The Robbins constant is the mean line segment length, i.e., the expected distance between two points chosen at random in cube line picking, namely

Delta(3)=1/(105)[4+17sqrt(2)-6sqrt(3)+21ln(1+sqrt(2))+42ln(2+sqrt(3))-7pi]
(1)
=1/(105)[4+17sqrt(2)-6sqrt(3)+21sinh^(-1)1+42ln(2+sqrt(3))-7pi]
(2)
=0.66170...
(3)

(OEIS A073012; Robbins 1978, Le Lionnais 1983, Beck 2023).

This value is implemented in the Wolfram Language as PolyhedronData["Cube", "MeanInteriorLineSegmentLength"].

See also

Cube Line Picking, Geometric Probability, Hypercube Line Picking, Mean Line Segment Length, Robin's Constant

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References

Beck, D. "Mean Distance in Polyhedra." 22 Sep 2023. https://arxiv.org/abs/2309.13177.Finch, S. R. "Geometric Probability Constants." §8.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 479-484, 2003.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 30, 1983.Robbins, D. "Average Distance between Two Points in a Box." Amer. Math. Monthly 85, 278, 1978.Sloane, N. J. A. Sequences A073012, A160693, and A160694 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Robbins Constant

Cite this as:

Weisstein, Eric W. "Robbins Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RobbinsConstant.html

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