A131595 -id:A131595 - cp's OEIS frontend

A377804 Decimal expansion of the surface area of a snub dodecahedron with unit edge length.

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

Offset: 2

Paolo Xausa, Nov 08 2024

			55.2867449584451489436570705587807625317445951163...

Eric Weisstein's World of Mathematics, Snub Dodecahedron.
Wikipedia, Snub dodecahedron.
Index entries for algebraic numbers, degree 8.

Cf. A377805 (volume), A377806 (circumradius), A377807 (midradius).
Cf. A131595 (analogous for a regular dodecahedron).
Cf. A002194.

First[RealDigits[20*Sqrt[3] + 3*Sqrt[25 + 10*Sqrt[5]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["SnubDodecahedron", "SurfaceArea"], 10, 100]]

Equals 20*sqrt(3) + 3*sqrt(25 + 10*sqrt(5)) = 20*A002194 + A131595.

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

A363438 Decimal expansion of the volume of the regular dodecahedron inscribed in the unit-radius sphere.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 02 2023

			2.78516386312262296729255491273594698789932177207633...

Eric Weisstein's World of Mathematics, Regular Dodecahedron.
Wikipedia, Regular dodecahedron.
Index entries for algebraic numbers, degree 4.

Cf. A118273 (cube), A122553 (regular octahedron), A339259 (regular icosahedron), A363437 (regular tetrahedron).
Cf. A001622.
Other constants related to the regular dodecahedron: A102769, A131595, A179296, A232810, A237603, A239798, A341906.

RealDigits[(2*(5 + Sqrt[5]))/(3*Sqrt[3]), 10, 120][[1]]

2*sqrt(5+sqrt(5))/sqrt(27) \\ Charles R Greathouse IV, Feb 07 2025

Equals 2*sqrt(5+sqrt(5))/(3*sqrt(3)).
Equals 4*(phi+2)/(3*sqrt(3)), where phi is the golden ratio (A001622).
Equals A102769 / A179296 ^ 3.

A341906 Decimal expansion of the moment of inertia of a solid regular dodecahedron with a unit mass and a unit edge length.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 04 2021

The moments of inertia of the five Platonic solids were apparently first calculated by the Canadian physicist John Satterly (1879-1963) in 1957.
The moment of inertia of a solid regular dodecahedron with a uniform mass density distribution, mass M, and edge length L is I = c*M*L^2, where c is this constant.
The corresponding values of c for the other Platonic solids are:
Tetrahedron: 1/20 (= A020761/10).
Octahedron: 1/10 (= A000007).
Cube: 1/6 (= A020793).
Icosahedron: (3 + sqrt(5))/20 (= A104457/10).

			0.60735550374163932719985924360173257727394705341616...

P. K. Aravind, Gravitational collapse and moment of inertia of regular polyhedral configurations, American Journal of Physics, Vol. 59, No. 7 (1991), pp. 647-652.
Frédéric Perrier, Moments of inertia of Archimedean solids, 2015.
John Satterly, Moments of Inertia about Selected Axes of Regular Polygons, Right Pyramids on Regular Polygonal Bases, and of the Platonic and Some Archimedian Polyhedra, American Journal of Physics, Vol. 25, No. 7 (1957), pp. 489-490.
John Satterly, The Moments of Inertia of Some Polyhedra, The Mathematical Gazette, Vol. 42, No. 339 (1958), pp. 11-13.
Wikipedia, List of moments of inertia.
Wikipedia, Moment of inertia.
Wikipedia, Regular dodecahedron.
Index entries for sequences related to moment of inertia.

Cf. A000007, A001622, A020761, A020793, A104457.

Other constants related to the regular dodecahedron: A102769, A131595, A179296, A232810, A237603, A239798.

RealDigits[(95 + 39*Sqrt[5])/300, 10, 100][[1]]

Equals (95 + 39*sqrt(5))/300.

Equals (28 + 39*phi)/150, where phi is the golden ratio (A001622).

cp's OEIS Frontend

A377804 Decimal expansion of the surface area of a snub dodecahedron with unit edge length.

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

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A363438 Decimal expansion of the volume of the regular dodecahedron inscribed in the unit-radius sphere.

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A341906 Decimal expansion of the moment of inertia of a solid regular dodecahedron with a unit mass and a unit edge length.

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