OFFSET
0,3
COMMENTS
Average volume of a tetrahedron picked at random in a unit cube.
From Amiram Eldar, Aug 25 2020: (Start)
The exact value of this constant was first calculated by Zinani (2003).
Equals (1/5) times the probability that 5 points independently and uniformly chosen in a cube are the vertices of a re-entrant (concave) polyhedron, i.e., one of the points falls within the tetrahedron formed by the other 4 points. Do and Solomon (1986) evaluated this probability using simulations, and their result is equivalent to an estimate of 0.0139 of this constant, with a 95% confidence interval of [0.01358, 0.01420] (Zinani, 2003). (End)
LINKS
Kim-Anh Do and Herbert Solomon, A simulation study of Sylvester's problem in three dimensions, Journal of applied probability, Vol. 23, No. 2 (1986), pp. 509-513, alternative link.
Johan Philip, The Expected Volume of a Random Tetrahedron in a Cube, 2007.
Eric Weisstein's World of Mathematics, Cube Tetrahedron Picking.
Alessandro Zinani, The expected volume of a tetrahedron whose vertices are chosen at random in the interior of a cube, Monatshefte für Mathematik, Vol. 139, No. 4 (2003), pp. 341-348.
EXAMPLE
0.0138427757...
MATHEMATICA
RealDigits[3977/216000-Pi^2/2160, 10, 120][[1]] (* Harvey P. Dale, Oct 31 2013 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Mar 30 2004
EXTENSIONS
Added initial 0 to match offset. - N. J. A. Sloane, Feb 08 2015
STATUS
approved