A394170 Decimal expansion of the mean volume of a tetrahedron whose vertices are uniformly and independently selected at random in the interior of a regular octahedron of unit volume.

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

Offset

0,3

Comments

The problem of calculating this constant was mentioned by Zinani (2003). It was evaluated by Weisstein as 0.01368, and calculated by Beck (2022).

Links

Table of n, a(n) for n=0..105.

Dominik Beck, Octahedron tetrahedron picking: Expected volume of a random tetrahedron in a regular octahedron, 2022.

Dominik Beck, On Random Simplex Picking Beyond the Blashke Problem, arXiv:2412.07952 [math.MG], 2024.

Dominik Beck, Random polytopes, doctoral thesis, Mathematical Institute of Charles University, Prague, 2025.

Eric Weisstein's World of Mathematics, Octahedron 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.

Index entries for transcendental numbers.

Formula

Equals 19297*Pi^2/3843840 - 6619/184320.

Example

0.0136374112765241754602123153299677982932384778749528...

Mathematica

RealDigits[19297*Pi^2/3843840 - 6619/184320, 10, 120, -1][[1]]

Prog

(PARI) 19297*Pi^2/3843840 - 6619/184320

Crossrefs

Tetrahedron volume in: A093524 (cube), A093525 (tetrahedron), A093591 (ball), A394169 (surface of sphere), this constant (octahedron), A394171 (rhombic dodecahedron), A394172 (cuboctahedron), A394173 (truncated tetrahedron), A394174 (triangular bipyramid), A394175 (triangular prism), A394176 (square pyramid).

Sequence in context: A170859 A123146 A396028 * A016661 A376827 A378935

Adjacent sequences: A394167 A394168 A394169 * A394171 A394172 A394173

Keyword

nonn,cons

Author

Amiram Eldar, Mar 11 2026

Status

approved