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%I #7 Mar 11 2026 15:06:34
%S 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,
%T 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,
%U 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
%N 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.
%H Dominik Beck, <a href="https://arxiv.org/abs/2412.07952">On Random Simplex Picking Beyond the Blashke Problem</a>, arXiv:2412.07952 [math.MG], 2024.
%H Dominik Beck, <a href="https://www2.karlin.mff.cuni.cz/~beckd/lectures/DISSERTATION.pdf">Random polytopes</a>, doctoral thesis, Mathematical Institute of Charles University, Prague, 2025.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RhombicDodecahedron.html">Rhombic Dodecahedron</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Rhombic_dodecahedron">Rhombic dodecahedron</a>.
%F 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.
%e 0.0129384816563456179539685972153458735614826032417689...
%t 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]]
%o (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
%Y 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).
%K nonn,cons
%O 0,3
%A _Amiram Eldar_, Mar 11 2026