A374772 -id:A374772 - cp's OEIS frontend

A374753 Decimal expansion of the volume of a regular dodecahedron having unit inradius.

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

Offset: 1

Paolo Xausa, Jul 19 2024

The dodecahedral conjecture (proved in 1988 by Thomas C. Hales and Sean McLaughlin, see links) states that, in any packing of unit spheres in the Euclidean 3-space, every Voronoi cell has volume at least equal to this value.

			5.55029102851551026907043211366183924073759821288...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Thomas C. Hales and Sean McLaughlin, A proof of the dodecahedral conjecture, arXiv:math/9811079 [math.MG], 2008.
Thomas C. Hales and Sean McLaughlin, The Dodecaheral Conjecture, Journal of the American Mathematical Society, Vol. 23, No. 2, April 2010, pp. 299-344.
Wikipedia, Dodecahedral conjecture.
Wikipedia, Regular dodecahedron.
Index entries for algebraic numbers, degree 4.

Cf. A019699, A374755 (strong dodecahedral conjecture), A374772 (density).

First[RealDigits[10*Sqrt[130 - 58*Sqrt[5]], 10, 100]]

Equals (4/3)*Pi/A374772 = 10*A019699/A374772.
Equals 10*sqrt(130 - 58*sqrt(5)).
Equals A374755/3.

A381671 Decimal expansion of the isoperimetric quotient of a regular tetrahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 03 2025

Polya (1954) defines the isoperimetric quotient of a solid as 36*Pi*V^2/(S^3), where V and S are the volume and surface area of the solid, respectively.

The isoperimetric quotient of a sphere is 1.

			0.30229989403903630843234637627369262204734437468212...

George Polya, Mathematics and Plausible Reasoning, Vol. 1: Induction and Analogy in Mathematics, Princeton University Press, Princeton, New Jersey, 1954. See pp. 188-189, exercise 43.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A273633 (sphericity).
Cf. isoperimetric quotient of other Platonic solids: A019673 (cube), A073010 (octahedron), A374772 (dodecahedron), A381672 (icosahedron).
Cf. A002194, A019673.

First[RealDigits[Pi/(6*Sqrt[3]), 10, 100]]

Equals Pi/(6*sqrt(3)) = A019673/A002194.

A374755 Decimal expansion of the surface area of a regular dodecahedron having unit inradius.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 20 2024

Bezdek's strong dodecahedral conjecture (proved by Hales, see links) states that, in any packing of unit spheres in the Euclidean 3-space, the surface area of every bounded Voronoi cell is at least this value.

			16.6508730855465308072112963409855177222127946386...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Károly Bezdek, On a stronger form of Rogers' lemma and the minimum surface area of Voronoi cells in unit ball packings, Journal für die reine und angewandte Mathematik, No. 518, 2000, pp. 131-143.
Thomas C. Hales, The Strong Dodecahedral Conjecture and Fejes Toth's Conjecture on Sphere Packings with Kissing Number Twelve, arXiv:1110.0402 [math.MG], 2012.
Wikipedia, Regular dodecahedron.

Cf. A374753 (dodecahedral conjecture), A374772, A374837, A374838.

Cf. A001622, A098317, A131595.

First[RealDigits[30*Sqrt[130 - 58*Sqrt[5]], 10, 100]]

Equals 30*sqrt(130 - 58*sqrt(5)).
Equals 60*sqrt(3 - A001622)/A098317.
Equals 4*Pi/A374772.
Equals 3*A374753.
Minimal polynomial: x^4 - 234000*x^2 + 64800000. - Stefano Spezia, Sep 03 2025

A374837 Decimal expansion of Bezdek and Daróczy-Kiss's upper bound for the surface area density of a unit ball in any face cone of a Voronoi cell in an arbitrary packing of unit balls in the Euclidean 3-space.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 21 2024

See Theorem 1.1 in Bezdek and Daróczy-Kiss (2005).

See A374772 for an improved bound.

			0.7783683851377739227957671666059435201971163186281...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Károly Bezdek and Endre Daróczy-Kiss, Finding the Best Face on a Voronoi Polyhedron--The Strong Dodecahedral Conjecture Revisited, Monatshefte für Mathematik, Vol. 145, No. 3, July 2005, pp. 191-206.

Cf. A374753 (dodecahedral conjecture), A374755 (strong dodecahedral conjecture), A374772, A374838.

First[RealDigits[(30*ArcCos[Sqrt[3]/2*Sin[Pi/5]] - 9*Pi)/(5*Tan[Pi/5]), 10, 100]]

Equals (30*arccos((sqrt(3)/2)*sin(Pi/5)) - 9*Pi)/(5*tan(Pi/5)).

Equals 4*Pi/A374838.

A374771 Decimal expansion of the volume of the sphere inscribed in a regular dodecahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 19 2024

			5.78333595039657417842182321041033675553722324626...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Regular dodecahedron.

Cf. A000796, A001622, A019699, A102769, A237603 (radius), A374772.

First[RealDigits[Pi*Sqrt[1525 + 682*Sqrt[5]]/30, 10, 100]]

Equals (4/3)*Pi*A237603^3 = 10*A019699*A237603^3.
Equals (1/30)*Pi*sqrt(1525 + 682*sqrt(5)).
Equals (Pi/6)*A001622^6/((3 - A001622)^(3/2)).
Equals A102769*A374772.

A374838 Decimal expansion of Bezdek and Daróczy-Kiss's lower bound for the surface area of any Voronoi cell in an arbitrary packing of unit balls in the Euclidean 3-space.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 21 2024

See Theorem 1.1 in Bezdek and Daróczy-Kiss (2005).

See A374755 for an improved bound (the strong dodecahedral conjecture).

			16.144502852765379806937602328092933543868922009780...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Károly Bezdek and Endre Daróczy-Kiss, Finding the Best Face on a Voronoi Polyhedron--The Strong Dodecahedral Conjecture Revisited, Monatshefte für Mathematik, Vol. 145, No. 3, July 2005, pp. 191-206.

Cf. A374753 (dodecahedral conjecture), A374755 (strong dodecahedral conjecture), A374772, A374837.

First[RealDigits[20*Pi*Tan[Pi/5]/(30*ArcCos[Sqrt[3]/2*Sin[Pi/5]] - 9*Pi), 10, 100]]

Equals 20*Pi*tan(Pi/5)/(30*arccos(sqrt(3)/2*sin(Pi/5)) - 9*Pi).

Equals 4*Pi/A374837.

A381672 Decimal expansion of the isoperimetric quotient of a regular icosahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 03 2025

For the definition of isoperimetric quotient of a solid, see A381671.

			0.8287977192520120215005810038129635758617830308723...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A273637 (sphericity).
Cf. isoperimetric quotient of other Platonic solids: A019673 (cube), A073010 (octahedron), A374772 (dodecahedron), A381671 (tetrahedron).
Cf. A000796, A002194, A374883.

First[RealDigits[Pi*GoldenRatio^4/(15*Sqrt[3]), 10, 100]]

Equals Pi*phi^4/(15*sqrt(3)) = A000796*A374883/(15*A002194).

A374956 Decimal expansion of Muder's 1993 upper bound for the density of packing of unit spheres in the Euclidean 3-space.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 25 2024

See A374772 for an improved bound.

			0.77305589657690889055021755701529047308262451752162...

Douglas J. Muder, A New Bound on the Local Density of Sphere Packings, Discrete & Computational Geometry, Vol. 10, 1993, pp. 351-375.

Cf. A374772, A374837, A374955 (volume).

Module[{beta, r, s},
  s[p_] := Pi - 5*ArcTan[Sqrt[(1 - 2*r^2)/(p*r^2)]];
  beta = 5*r*Sqrt[1 - 2*r^2]/(3*Sqrt[2]) + s[2]/6;
  r = SolveValues[4/13*Pi == 2*s[3] - Sqrt[8/3]*s[2] && r > 0, r, Reals];
  RealDigits[4*Pi/(39*beta), 10, 100][[1,1]]]

Equals 4*Pi/(39*beta), where beta = 5*r*sqrt(1-2*r^2)/(3*sqrt(2)) + (1/6)*(Pi - 5*arctan(sqrt((1 - 2*r^2)/(2*r^2)))) and r is the positive solution to (4/13)*Pi = 2*(Pi - 5*arctan(sqrt((1 - 2*r^2)/(3*r^2)))) - sqrt(8/3)*(Pi - 5*arctan(sqrt((1 - 2*r^2)/(2*r^2)))). See Corollary in Muder (1993), p. 352.

Equals (4/3)*Pi/A374955.

A379732 Decimal expansion of 207/208.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Dec 31 2024

Conjectured densest packing of truncated tetrahedra.

			0.995192307692307692307692307692307692307692307692...

Pablo F. Damasceno, Michael Engel, and Sharon C. Glotzer, Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces, arXiv:1109.1323 [cond-mat.soft], 2011.
Yang Jiao and Sal Torquato, Analytical Construction of A Dense Packing of Truncated Tetrahedra, arXiv:1107.2300 [cond-mat.soft], 2011.
Wikipedia, Truncated tetrahedron.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A374772, A377274, A377275.

Mathematica
```
First[RealDigits[207/208, 10, 100]]
```

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A374753 Decimal expansion of the volume of a regular dodecahedron having unit inradius.

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A381671 Decimal expansion of the isoperimetric quotient of a regular tetrahedron.

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A374755 Decimal expansion of the surface area of a regular dodecahedron having unit inradius.

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A374837 Decimal expansion of Bezdek and Daróczy-Kiss's upper bound for the surface area density of a unit ball in any face cone of a Voronoi cell in an arbitrary packing of unit balls in the Euclidean 3-space.

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A374771 Decimal expansion of the volume of the sphere inscribed in a regular dodecahedron with unit edge.

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A374838 Decimal expansion of Bezdek and Daróczy-Kiss's lower bound for the surface area of any Voronoi cell in an arbitrary packing of unit balls in the Euclidean 3-space.

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A381672 Decimal expansion of the isoperimetric quotient of a regular icosahedron.

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A374956 Decimal expansion of Muder's 1993 upper bound for the density of packing of unit spheres in the Euclidean 3-space.

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