Cube Line Picking -- from Wolfram MathWorld

The average distance between two points chosen at random inside a unit cube (the case of hypercube line picking), sometimes known as the Robbins constant, is

			(1)
			(2)
			(3)

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

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

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 for , 2, ... are 1, 1/2, 11/30, 211/630, 187/525, 3524083/6306300, ... (OEIS A160693 and A160694).

Pick 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.


5	1.1180339887498
6	1.0606601482100
7	1
8	1
9	0.86602540378463
10	0.74999998333331
11	0.70961617562351
12	0.70710678118660
13	0.70710678118660
14	0.70710678118660
15	0.625

Cube Line Picking

See also