A394171 Decimal expansion of the mean volume of a tetrahedron whose vertices are uniformly and independently selected at random in the interior of a rhombic dodecahedron of unit volume.

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

Offset

0,3

Links

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

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, Rhombic Dodecahedron.

Wikipedia, Rhombic dodecahedron.

Formula

Equals 2421179003623/17933819904000 + 37061863*Pi^2/29889699840 - 9406373047*log(2)/9340531200 - 1757220593*log(2)^2/2490808320 + 282589831*log(3)/283852800 - 6078271*PolyLog(2, 1/4)/8515584.

Example

0.0129384816563456179539685972153458735614826032417689...

Mathematica

RealDigits[2421179003623/17933819904000 + 37061863*Pi^2/29889699840 - 9406373047*Log[2]/9340531200 - 1757220593*Log[2]^2/2490808320 + 282589831*Log[3]/283852800 - 6078271*PolyLog[2, 1/4]/8515584, 10, 120, -1][[1]]

Prog

(PARI) 2421179003623/17933819904000 + 37061863*Pi^2/29889699840 - 9406373047*log(2)/9340531200 - 1757220593*log(2)^2/2490808320 + 282589831*log(3)/283852800 - 6078271*polylog(2, 1/4)/8515584

Crossrefs

Tetrahedron volume in: A093524 (cube), A093525 (tetrahedron), A093591 (ball), A394169 (surface of sphere), A394170 (octahedron), this constant (rhombic dodecahedron), A394172 (cuboctahedron), A394173 (truncated tetrahedron), A394174 (triangular bipyramid), A394175 (triangular prism), A394176 (square pyramid).

Sequence in context: A260525 A366249 A381236 * A302973 A258403 A270202

Adjacent sequences: A394168 A394169 A394170 * A394172 A394173 A394174

Keyword

nonn,cons

Author

Amiram Eldar, Mar 11 2026

Status

approved