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Cube Tetrahedron Picking

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CubeTetrahedronPicking

Given four points chosen at random inside a unit cube, the average volume of the tetrahedron determined by these points is given by

V^_=(int_0^1...int_0^1_()_(12)|V(x_i)|dx_1...dx_4dy_1...dy_4dz_1...dz_4)/(int_0^1...int_0^1_()_(12)dx_1...dx_4dy_1...dy_4dz_1...dz_4),
(1)

where the polyhedron vertices are located at (x_i,y_i,z_i) where i=1, ..., 4, and the (signed) volume is given by the determinant

V=1/(3!)|x_1 y_1 z_1 1; x_2 y_2 z_2 1; x_3 y_3 z_3 1; x_4 y_4 z_4 1|.
(2)

The integral is extremely difficult to compute, but the analytic result for the mean tetrahedron volume is

V^_=(3977)/(216000)-(pi^2)/(2160)=0.01384277...
(3)

(OEIS A093524; Zinani 2003). Note that the result quoted in the reply to Seidov (2000) actually refers to the average volume for tetrahedron tetrahedron picking.

See also

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

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References

Do, K.-A. and Solomon, H. "A Simulation Study of Sylvester's Problem in Three Dimensions." J. Appl. Prob. 23, 509-513, 1986.Seidov, Z. F. "Letters: Random Triangle." Mathematica J. 7, 414, 2000.Sloane, N. J. A. Sequence A093524 in "The On-Line Encyclopedia of Integer Sequences."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.

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Cube Tetrahedron Picking

Cite this as:

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

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