Cube Line Picking--Face and Face

Instead of picking two points from the interior of the cube, instead pick two points on different faces of the unit cube. In this case, the average distance between the points is

Delta_f(3)=4/5int_0^1int_0^1int_0^1int_0^1sqrt(x^2+y^2+(z-w)^2)dwdxdydz+1/5int_0^1int_0^1int_0^1int_0^1sqrt(1+(y-u)^2+(z-w)^2)dudwdydz
=1/(75)[4+17sqrt(2)-6sqrt(3)+21ln(1+sqrt(2))+42ln(2+sqrt(3))-7pi]
=0.92639...

(1)

(OEIS A093066; Borwein and Bailey 2003, p. 26; Borwein et al. 2004, pp. 66-67). Interestingly,

(2)

as apparently first noted by M. Trott (pers. comm., Mar. 21, 2008).

The two integrals above can be written in terms of sums as

			(3)
			(4)

(Borwein et al. 2004, p. 67), where however appears to be classically divergent and perhaps must be interpreted in some regularized sense.

Consider a line whose endpoints are picked at random on opposite sides of the unit cube. The probability density function for the length of this line is given by

$P(l)={2l[l^2-4sqrt(l^2-1)+pi-1] for 1<=<sqrt(2); 2l[-l^2+4sqrt(l^2-2)-4sec^(-1)(sqrt(l^2-1))+pi-1] for sqrt(2)<=l<=sqrt(3)$

(5)

(Mathai 1999; after simplification). The mean length is

			(6)
			(7)

The first even raw moments for , 2, 4, ... are 1, 4/3, 167/90, 284/105, 931/225, 9868/1485, ....

Consider a line whose endpoints are picked at random on adjacent sides of the unit cube. The probability density function for the length of this line is given by

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