Cube line picking on different faces of a unit cube chooses two distinct faces uniformly and then chooses one point uniformly on each face, independently of the other point. The mean 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...](/images/equations/CubeLinePickingFaceandFace/NumberedEquation1.svg) |
(1)
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(OEIS A093066; Borwein and Bailey 2003, p. 26; Borwein et al. 2004, pp. 66-67). Interestingly,
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(2)
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as apparently first noted by M. Trott (pers. comm., Mar. 21, 2008).
The two integrals above can be written in terms of sums as
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(3)
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(4)
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(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 segment whose endpoints are chosen independently and uniformly
on opposite faces of the unit cube.
The probability density function for
the length 
of this line segment is given
by
![P(l)={2l[l^2-4sqrt(l^2-1)+pi-1] for 1<=l<sqrt(2); 2l[-l^2+4sqrt(l^2-2)-4sec^(-1)(sqrt(l^2-1))+pi-1] for sqrt(2)<=l<=sqrt(3)](/images/equations/CubeLinePickingFaceandFace/NumberedEquation3.svg) |
(5)
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(Mathai et al. 1999; after simplification). The mean length is
 |  | ![1/(15)[3sqrt(2)-4sqrt(3)+4-10ln2+20ln(1+sqrt(3))]+sinh^(-1)1-2/9pi](/images/equations/CubeLinePickingFaceandFace/Inline11.svg) |
(6)
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(7)
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The first even raw moments 
for 
,
2, 4, ... are 1, 4/3, 167/90, 284/105, 931/225, 9868/1485, ....
Consider a line segment whose endpoints are chosen independently and uniformly
on adjacent faces of the unit cube.
The probability density function for
the length 
of this line segment is given
by
![P(l)={1/2pil^2(2-l) for 0<=l<1; 2l[(5/4-l)pi+l^2sec^(-1)l-sqrt(l^2-1)] for 1<=l<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)](/images/equations/CubeLinePickingFaceandFace/NumberedEquation4.svg) |
(8)
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(Mathai et al. 1999; after simplification). The mean length is
 |  | ![1/(180)[42sqrt(2)-6sqrt(3)-11pi+18ln(1+sqrt(2))+96ln(2+sqrt(3))]](/images/equations/CubeLinePickingFaceandFace/Inline20.svg) |
(9)
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(10)
|
The first even raw moments 
for 
,
2, 4, ... are 1, 5/6, 41/45, 1469/1260, 5/3, 53947/20790, ....
