Tetrahedron Inscribing

Pick four points at random on the surface of a unit Sphere. Find the distribution of possible volumes of(nonregular) Tetrahedra. Without loss of generality, the first point can be chosen as (1, 0, 0).Designate the other points , , and . Then the distances from the first Vertex are

			(1)
			(2)
			(3)

(4)

(5)

	(6)

⁷ computer Trials give

			(7)
			(8)
			(9)

See also Point-Point Distance--1-D, Triangle Inscribing in a Circle,Triangle Inscribing in an Ellipse

References

Buchta, C. ``A Note on the Volume of a Random Polytope in a Tetrahedron.'' Ill. J. Math. 30, 653-659, 1986.

• Mathematics Prizes
• Mathieu Differential Equation
• Mathieu Function
• Mathieu Groups
• Matrix
• Matrix Addition
• Matrix Decomposition Theorem
• Matrix Diagonalization
• Matrix Direct Product
• Matrix Equality
• Matrix Equation
• Matrix Exponential
• Matrix Group
• Matrix Inverse
• Matrix Multiplication
• Matrix Norm
• Matrix Polynomial Identity
• Matrix Product
• Matrix Transpose
• Matroid
• Maurer Rose
• Maximal Ideal
• Maximally Linearly Independent
• Maximal Sum-Free Set
• Maximal Zero-Sum-Free Set