A394105 - OEIS

%I #7 Mar 10 2026 14:03:33

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

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

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

%N Decimal expansion of the mean distance between two points uniformly and independently selected at random in a regular dodecahedron of unit volume.

%H Dominik Beck, <a href="https://arxiv.org/abs/2309.13177">Mean distance in polyhedra</a>, arXiv:2309.13177 [math.PR], 2023.

%H Dominik Beck, <a href="https://www2.karlin.mff.cuni.cz/~beckd/lectures/BONNMeanDistancePresentation_Final.pdf">Mean distance in polyhedra</a>, lecture, 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.

%F Equals (1/(30 + 14*sqrt(5))^(1/3)) * (1516/1575 + 2*sqrt(2/5)/45 - 124*sqrt(3/5)/175 - 71*sqrt(2)/1575 - 12*sqrt(3)/35 + 342/(175*sqrt(5)) + 493*Pi/23625 + 67*Pi/(945*sqrt(5)) + (397 - 244*sqrt(5))*arccot(2)/18900 + (24023 + 11788*sqrt(5))*(arccos(2/3) - arccos(1/3))/94500 - (461 + 212*sqrt(5))*(arccos(23/41) + acos(39/41))/1000 - (1031 + 521*sqrt(5))*arccosh(13/3)/75600 + (367 + 163*sqrt(5))*arccosh(9)/16800 + (22197 + 8149*sqrt(5))*(arccosh(121/41) - arccosh(57/41))/84000 + (15763 + 7063*sqrt(5))*(arccosh(7/3) - arccosh(3))/21000 + (288889 + 129739*sqrt(5))*log(3)/378000 + 2*(423 + 187*sqrt(5))*(arccosh(4) - arccosh(2))/875 + (109 - 3143*sqrt(5))*log(5)/151200).

%e 0.642520677094876011622551913553681055654417848315244...

%t RealDigits[1/(30 + 14*Sqrt[5])^(1/3)*(1516/1575 + (2*Sqrt[2/5])/45 - (124*Sqrt[3/5])/175 - (71*Sqrt[2])/1575 - (12*Sqrt[3])/35 + 342/(175*Sqrt[5]) + (493*Pi)/23625 + (67*Pi)/(945*Sqrt[5]) + ((397 - 244*Sqrt[5])*ArcCot[2])/18900 + ((24023 + 11788*Sqrt[5])*(ArcCos[2/3] - ArcCos[1/3]))/94500 - ((461 + 212*Sqrt[5])*(ArcCos[23/41] + ArcCos[39/41]))/1000 - ((1031 + 521*Sqrt[5])*ArcCosh[13/3])/75600 + ((367 + 163*Sqrt[5])*ArcCosh[9])/16800 + ((22197 + 8149*Sqrt[5])*(ArcCosh[121/41] - ArcCosh[57/41]))/84000 + ((15763 + 7063*Sqrt[5])*(ArcCosh[7/3] - ArcCosh[3]))/21000 + ((288889 + 129739*Sqrt[5])*Log[3])/378000 + (2*(423 + 187*Sqrt[5])*(ArcCosh[4] - ArcCosh[2]))/875 + ((109 - 3143*Sqrt[5])*Log[5])/151200), 10, 120][[1]]

%o (PARI) (1/(30 + 14*sqrt(5))^(1/3)) * (1516/1575 + 2*sqrt(2/5)/45 - 124*sqrt(3/5)/175 - 71*sqrt(2)/1575 - 12*sqrt(3)/35 + 342/(175*sqrt(5)) + 493*Pi/23625 + 67*Pi/(945*sqrt(5)) + (397 - 244*sqrt(5))*atan(1/2)/18900 + (24023 + 11788*sqrt(5))*(acos(2/3) - acos(1/3))/94500 - (461 + 212*sqrt(5))*(acos(23/41) + acos(39/41))/1000 - (1031 + 521*sqrt(5))*acosh(13/3)/75600 + (367 + 163*sqrt(5))*acosh(9)/16800 + (22197 + 8149*sqrt(5))*(acosh(121/41) - acosh(57/41))/84000 + (15763 + 7063*sqrt(5))*(acosh(7/3) - acosh(3))/21000 + (288889 + 129739*sqrt(5))*log(3)/378000 + 2*(423 + 187*sqrt(5))*(acosh(4) - acosh(2))/875 + (109 - 3143*sqrt(5))*log(5)/151200)

%Y Analogous constants: A073012 (cube), A366019 (regular tetrahedron), A394103 (regular octahedron), A394104 (regular icosahedron), this constant (regular dodecahedron).

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Mar 10 2026