A237603 -id:A237603 - cp's OEIS frontend

A179294 Decimal expansion of radius of inscribed sphere about a regular icosahedron with edge = 1.

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 09 2010

Icosahedron: A three-dimensional figure with 20 equilateral triangle faces, 12 vertices, and 30 edges.

			0.75576131407617073048013370202500139263844478889356105922958289203910...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Dr. Math, Icosahedron.
Wikipedia, Icosahedron.
MathWorld, Icosahedron.
Index entries for algebraic numbers, degree 4.

Cf. A010527, A102208, A179290, A179292.

Cf. Platonic solids inradii: A020781 (tetrahedron), A020763 (octahedron), A237603 (dodecahedron).

RealDigits[(Sqrt[42+18Sqrt[5]]/12), 10, 175][[1]]

sqrt((7+3*sqrt(5))/6)/2 \\ Stefano Spezia, Jan 27 2025

Equals sqrt(42 + 18*sqrt(5))/12.

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

A020761 Decimal expansion of 1/2.

Original entry on oeis.org

5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Real part of all nontrivial zeros of the Riemann zeta function (assuming the Riemann hypothesis to be true). - Alonso del Arte, Jul 02 2011
Radius of a sphere with surface area Pi. - Omar E. Pol, Aug 09 2012
Radius of the midsphere (tangent to the edges) in a regular octahedron with unit edges. Also radius of the inscribed sphere (tangent to faces) in a cube with unit edges. - Stanislav Sykora, Mar 27 2014
Construct a rectangle of maximal area inside an arbitrary triangle. The ratio of the rectangle's area to the triangle's area is 1/2. - Rick L. Shepherd, Jul 30 2014

			1/2 = 0.50000000000000...

Michael Penn, A creative approach to a scary looking integral., YouTube video, 2020.
Michael Penn, I really like this sum!, YouTube video, 2021.
Wikipedia, Riemann zeta function
Wikipedia, Platonic solid
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. In platonic solids:
midsphere radii:
A020765 (tetrahedron),
A010503 (cube),
A019863 (icosahedron),
A239798 (dodecahedron);
insphere radii:
A020781 (tetrahedron),
A020763 (octahedron),
A179294 (icosahedron),
A237603 (dodecahedron).

Digits:=100; evalf(1/2); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[1/2, 10, 128][[1]] (* Alonso del Arte, Dec 13 2013 *)
LinearRecurrence[{1},{5,0},99] (* Ray Chandler, Jul 15 2015 *)

{ default(realprecision); x=1/2*10; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Felix Fröhlich, Jul 24 2014

a(n) = 5*(n==0); \\ Michel Marcus, Jul 25 2014

Equals Sum_{k>=1} (1/3^k). Hence 1/2 = 0.1111111111111... in base 3.
Cosine of 60 degrees, i.e., cos(Pi/3).
-zeta(0), zeta being the Riemann function. - Stanislav Sykora, Mar 27 2014
a(0) = 5; a(n) = 0, n > 0. - Wesley Ivan Hurt, Mar 27 2014
a(n) = 5 * floor(1/(n + 1)). - Wesley Ivan Hurt, Mar 27 2014
Equals 2*A019824*A019884. - R. J. Mathar, Jan 17 2021

A020763 Decimal expansion of 1/sqrt(6).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Radius of the inscribed sphere (tangent to all faces) in a regular octahedron with unit edge. - Stanislav Sykora, Nov 21 2013

			0.408248290463863016366214012450981898660991246776111688072115427875...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A010464, A157697.

Cf. Platonic solids in radii: A020781 (tetrahedron), A179294 (icosahedron), A237603 (dodecahedron). - Stanislav Sykora, Feb 25 2014

evalf(1/sqrt(6)); # Michal Paulovic, Dec 09 2022

RealDigits[1/Sqrt[6], 10, 100][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
realDigitsRecip[Sqrt[6]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Jan 10 2025 *)

From Michal Paulovic, Dec 09 2022: (Start)
Equals A157697/2 = A010503 * A020760 = 1/A010464.
Equals [0, 2; 2, 4] (periodic continued fraction expansion). (End)

A020781 Decimal expansion of 1/sqrt(24).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Radius of the inscribed sphere (tangent to the faces) for a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013

			1/sqrt(24) = 0.20412414523193150818310700622549094933... . - _Vladimir Joseph Stephan Orlovsky_, May 30 2010

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. Platonic solids inradii: A020763 (octahedron), A179294 (icosahedron), A237603 (dodecahedron). - Stanislav Sykora, Feb 25 2014

Cf. A010464, A010480, A020763, A020853, A021028, A187110, A244980.

RealDigits[N[1/Sqrt[24],200]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

Equals A010464/12. - Stefano Spezia, Jan 26 2025

Equals 1/A010480 = A020763/2 = 2*A020853 = A187110/3 = A244980/Pi. - Hugo Pfoertner, Jan 26 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.

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.

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

A179294 Decimal expansion of radius of inscribed sphere about a regular icosahedron with edge = 1.

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A020761 Decimal expansion of 1/2.

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A020763 Decimal expansion of 1/sqrt(6).

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A020781 Decimal expansion of 1/sqrt(24).

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

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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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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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