Given a regular tetrahedron of unit volume, consider the lengths of line segments connecting pairs of points picked at random inside the tetrahedron. The distribution of lengths is illustrated above and the mean line segment length can be given in closed form as
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(1)
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(2)
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(OEIS A366019; Beck 2023).
This beautiful result supplants the approximate value 
estimated using quasi-Monte Carlo
numerical integration by E. Weisstein in Feb. 2005. (In fact, numerical
integration using a global adaptive method with 
maximum error increases gives a much more accurate estimate
of 0.729462.)
To obtain the mean line segment length for a regular tetrahedron with unit edge lengths (instead of unit volume), solve 
(where 
is the volume of a tetrahedron in terms of its edge length)
for 
to obtain 
and take 
to give
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(3)
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(4)
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This value is implemented in the Wolfram Language as PolyhedronData["Tetrahedron", "MeanInteriorLineSegmentLength"].
