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Octahedron Tetrahedron Picking

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Octahedron tetrahedron picking is the selection of four points independently and uniformly from the interior of a regular octahedron to form the vertices of a tetrahedron. For an octahedron of unit volume, the mean tetrahedron volume is

V^_=(19297pi^2)/(3843840)-(6619)/(184320)=0.013637411....

This exact result (Beck 2026) supersedes the earlier numerical estimate of 0.01368.

See also

Ball Tetrahedron Picking, Cube Tetrahedron Picking, Octahedron, Point Picking, Sphere Tetrahedron Picking, Tetrahedron, Tetrahedron Tetrahedron Picking

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References

Beck, D. "On Random Simplex Picking Beyond the Blaschke Problem." Methodol. Comput. Appl. Probab. 28, 53, 2026. https://doi.org/10.1007/s11009-026-10277-8. https://arxiv.org/abs/2412.07952.Zinani, A. "The Expected Volume of a Tetrahedron Whose Vertices Are Chosen at Random in the Interior of a Cube." Monatshefte Math. 139, 341-348, 2003.

Referenced on Wolfram|Alpha

Octahedron Tetrahedron Picking

Cite this as:

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

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