A394105 Decimal expansion of the mean distance between two points uniformly and independently selected at random in a regular dodecahedron of unit volume.

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, 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, 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

Offset

0,1

Links

Table of n, a(n) for n=0..104.

Dominik Beck, Mean distance in polyhedra, arXiv:2309.13177 [math.PR], 2023.

Dominik Beck, Mean distance in polyhedra, lecture, 2024.

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

Formula

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).

Example

0.642520677094876011622551913553681055654417848315244...

Mathematica

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]]

Prog

(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)

Crossrefs

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

Sequence in context: A145011 A173625 A086036 * A345644 A019849 A244685

Adjacent sequences: A394102 A394103 A394104 * A394106 A394107 A394108

Keyword

nonn,cons

Author

Amiram Eldar, Mar 10 2026

Status

approved