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Cube Line Picking

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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

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"].

CubeLinePickingDistribution

The probability function as a function of line length, illustrated above, was found in (nearly) closed form by Mathai et al. (1999). After simplifying, correcting typos, and completing the integrals, gives the closed form

P(l)={-l^2[(l-8)l^2+pi(6l-4)] for 0<=l<=1; 2l[(l^2-8sqrt(l^2-1)+3)l^2-4sqrt(l^2-1)+12l^2sec^(-1)l+pi(3-4l)-1/2] for 1<l<=sqrt(2); l[(1+l^2)(6pi+8sqrt(l^2-2)-5-l^2)-16lcsc^(-1)(sqrt(2-2l^(-2)))+16ltan^(-1)(lsqrt(l^2-2))-24(l^2+1)tan^(-1)(sqrt(l^2-2))] for sqrt(2)<l<=sqrt(3).
(4)

The first even raw moments mu_n^' for n=0, 2, ... are 1, 1/2, 11/30, 211/630, 187/525, 3524083/6306300, ... (OEIS A160693 and A160694).

Pick n points on a cube, and space them as far apart as possible. The best value known for the minimum straight line distance between any two points is given in the following table.

nd(n)
51.1180339887498
61.0606601482100
71
81
90.86602540378463
100.74999998333331
110.70961617562351
120.70710678118660
130.70710678118660
140.70710678118660
150.625

See also

Cube Line Picking--Face and Face, Cube Line Picking--Face and Interior, Cube Point Picking, Cube Triangle Picking, Discrepancy Theorem, Hypercube Line Picking, Line Line Picking, Point Picking, Robbins Constant, Square Line Picking

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References

Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006b.Beck, D. "Mean Distance in Polyhedra." 22 Sep 2023. https://arxiv.org/abs/2309.13177.Bolis, T. S. Solution to Problem E2629. "Average Distance between Two Points in a Box." Amer. Math. Monthly 85, 277-278, 1978.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Finch, S. R. "Geometric Probability Constants." §8.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 479-484, 2003.Ghosh, B. "Random Distances within a Rectangle and between Two Rectangles." Bull. Calcutta Math. Soc. 43, 17-24, 1951.Holshouser, A. L.; King, L. R.; and Klein, B. G. Solution to Problem E3217, "Minimum Average Distance between Points in a Rectangle." Amer. Math. Monthly 96, 64-65, 1989.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 30, 1983.Mathai, A. M.; Moschopoulos, P.; and Pederzoli, G. "Distance between Random Points in a Cube." J. Statistica 59, 61-81, 1999.Robbins, D. "Average Distance between Two Points in a Box." Amer. Math. Monthly 85, 278, 1978.Santaló, L. A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976.Schroeppel, R. (results due to R. H. Hardin and N. J. A. Sloane) "points in a cube." math-fun@cs.arizona.edu posting, May 30, 1996.Sloane, N. J. A. Sequences A073012, A160693, and A160694 in "The On-Line Encyclopedia of Integer Sequences."

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Cube Line Picking

Cite this as:

Weisstein, Eric W. "Cube Line Picking." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CubeLinePicking.html

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