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# Tetrahedron Line Picking

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

 (1) (2)

(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

 (3) (4)

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

Mean Line Segment Length, Regular Tetrahedron, Tetrahedron Tetrahedron Picking, Tetrahedron Triangle Picking

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## References

Beck, D. "Mean Distance in Polyhedra." 22 Sep 2023. https://arxiv.org/abs/2309.13177.Sloane, N. J. A. Sequence A366019 in "The On-Line Encyclopedia of Integer Sequences."

## Referenced on Wolfram|Alpha

Tetrahedron Line Picking

## Cite this as:

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