A093525 Decimal expansion of 13/720 - Pi^2/15015.

0, 1, 7, 3, 9, 8, 2, 3, 9, 2, 4, 5, 1, 2, 6, 6, 9, 3, 8, 4, 2, 6, 7, 9, 4, 6, 5, 5, 7, 8, 8, 7, 3, 8, 2, 7, 2, 0, 8, 2, 1, 5, 3, 1, 5, 8, 3, 4, 7, 9, 3, 5, 9, 8, 1, 3, 7, 8, 7, 9, 0, 0, 8, 8, 1, 0, 5, 5, 2, 5, 5, 5, 5, 6, 7, 3, 8, 2, 4, 4, 8, 3, 1, 3, 2, 6, 4, 3, 9, 5, 1, 5, 5, 7, 0, 7, 5, 0, 0, 1, 2, 8, 8

Offset: 0

Author

Eric W. Weisstein, Mar 30 2004

Keywords

• nonn
• cons

Comments

Average volume of a tetrahedron picked at random in a tetrahedron with unit volume.

Buchta & Reitzner announced this result in 1992, and Mannion (independently) proved it in 1994. Buchta & Reitzner proved a more general result in 2001. - Charles R Greathouse IV, Sep 04 2015

Klee (1969) conjectured that the average volume is 1/60 and stated that according to Monte Carlo experiments 1/57 is the integer-reciprocal closest to this value. - Amiram Eldar, Apr 09 2022

Examples

			0.0173982392...
		

Links

• Christian Buchta and Matthias Reitzner, What is the expected volume of a tetrahedron whose vertices are chosen at random from a given tetrahedron, Österreichische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse, Vol. 129 (1992), pp. 63-68.
• Christian Buchta and Matthias Reitzner, The convex hull of random points in a tetrahedron: Solution of Blaschke's problem and more general results, J. reine angew. Math., Vol. 536 (2001), pp. 1-29.
• Victor Klee, What is the Expected Volume of a Simplex Whose Vertices are Chosen at Random from a Given Convex Body?, The American Mathematical Monthly, Vol. 76, No. 3 (1969), pp. 286-288.
• David Mannion, The volume of a tetrahedron whose vertices are chosen at random in the interior of a parent tetrahedron, Advances in Applied Probability, Vol. 26, No. 3 (Sep 1994), pp. 577-596.
• Eric Weisstein's World of Mathematics, Tetrahedron Tetrahedron Picking.
• Index entries for transcendental numbers

Crossrefs

Cf. A020829, A093524, A093591.

Programs

Mathematica

Join[{0}, RealDigits[13/720 - Pi^2/15015, 10, 100][[1]]] (* Amiram Eldar, Apr 09 2022 *)

PARI

13/720 - Pi^2/15015 \\ Charles R Greathouse IV, Sep 04 2015

Extensions

Added initial 0 to make offset correct. - N. J. A. Sloane, Feb 08 2015