A179292 -id:A179292 - cp's OEIS frontend

A179296 Decimal expansion of circumradius of a regular dodecahedron with edge length 1.

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 09 2010

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

Appears as a coordinate in a degree-7 quadrature formula on 12 points over the unit circle [Stroud & Secrest]. - R. J. Mathar, Oct 12 2011

			1.40125853844407354467667793532206799444393173977549286366084518639135...

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

Chai Wah Wu, Table of n, a(n) for n = 1..10001
A. H. Stroud and Don Secrest, Approximate integration formulas for certain spherically symmetric regions, Math. Comp. 17 (82) (1963) 105.
Wikipedia, Dodecahedron.
Wolfram Alpha, Dodecahedron.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A102208, A102769, A131595, A179290, A179292, A179294.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A019881 (icosahedron), A187110 (tetrahedron). - Stanislav Sykora, Feb 10 2014

RealDigits[(Sqrt[3]+Sqrt[15])/4, 10, 175][[1]]

(1+sqrt(5))*sqrt(3)/4 \\ Stefano Spezia, Jan 27 2025

Equals (sqrt(3) + sqrt(15))/4 = sqrt((9 + 3*sqrt(5))/8).
The minimal polynomial is 16*x^4 - 36*x^2 + 9. - Joerg Arndt, Feb 05 2014
Equals (sqrt(3)/2) * phi = A010527 * A001622. - Amiram Eldar, Jun 02 2023

A179591 Decimal expansion of the surface area of pentagonal cupola with edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal cupola: 15 vertices, 25 edges, and 12 faces.

			16.5797497529881970460940463443632246181026360961176551818747440...

Wolfram Alpha, Johnson solid 5
Index entries for algebraic numbers, degree 8.

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587, A179588, A179589, A179590.

RealDigits[N[(20+Sqrt[10*(80+31*Sqrt[5]+Sqrt[2175+930*Sqrt[5]])])/4,200]]

Digits of (20+sqrt(10*(80+31*sqrt(5)+sqrt(2175+930*sqrt(5)))))/4.

A179588 Decimal expansion of the surface area of square cupola with edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Square cupola: 12 vertices, 20 edges, and 10 faces.

			11.56047793231506739113082378992526852408214900456427677440...

Wolfram Alpha, Johnson solid 4
Index entries for algebraic numbers, degree 4

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587.

Mathematica
```
RealDigits[N[7+2*Sqrt[2]+Sqrt[3],200]]
```

Digits of 7 + 2*sqrt(2) + sqrt(3).

A179637 Decimal expansion of the surface area of pentagonal rotunda with edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Pentagonal rotunda: 20 vertices, 35 edges, and 17 faces.

			22.3472002653941282767984141581886130738180135134316226129799763167102...

Wolfram Alpha, Johnson solid 6

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449-A179452, A179552-A179554, A179587-A179593.

RealDigits[N[Sqrt[5*(145+58*Sqrt[5]+2*Sqrt[30*(65+29*Sqrt[5])])]/2,200]]

Digits of sqrt(5*(145+58*sqrt(5)+2*sqrt(30*(65+29*sqrt(5)))))/2.

Offset corrected by R. J. Mathar, Aug 15 2010

A185093 Decimal expansion of the volume of small rhombicosidodecahedron with edge = 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Feb 04 2012

Small Rhombicosidodecahedron: 62 faces, 60 vertices, and 120 edges.
Surface Area = 30+sqrt(30*(10+3*sqrt(5)+sqrt(75+30*sqrt(5)))) = 59.30598284491...
Circumradius = sqrt(11+4*sqrt(5))/2 = 2.23295050941569004950041538324968277293...
Midradius = sqrt(10+4*sqrt(5))/2 = 2.17625089948282151110005286599776788019807...
Quadratic number with denominator 3 and minimal polynomial 9x^2 - 360x - 605. - Charles R Greathouse IV, Apr 25 2016

			41.6153237824979670652886787977356702759259774762447486679520...

G. C. Greubel, Table of n, a(n) for n = 2..5001 [Offset adapted by _Georg Fischer_, Oct 18 2021]

Cf. A102208, A179290, A179292.

RealDigits[N[20 + (29*Sqrt[5])/3, 200]][[1]]
RealDigits[PolyhedronData["Rhombicosidodecahedron","Volume"],10,100][[1]] (* Harvey P. Dale, Aug 04 2025 *)

29*sqrt(5)/3+20 \\ Charles R Greathouse IV, Oct 01 2012

Offset changed by Georg Fischer, Jul 29 2021

A179589 Decimal expansion of the circumradius of square cupola with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Square cupola: 12 vertices, 20 edges, and 10 faces.

			1.398966325965906702031540539431998764673522563866238879930...

Wolfram Alpha, Johnson solid 4
Index entries for algebraic numbers, degree 4

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587, A179588.

Mathematica
```
RealDigits[N[Sqrt[5+2*Sqrt[2]]/2,200]]
```

Digits of sqrt(5+2*sqrt(2))/2.

A179638 Decimal expansion of the volume of gyroelongated square pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Gyroelongated square pyramid: 9 vertices, 20 edges, and 13 faces.

			1.19270224223223255906019842837725158155262551828862015707793142188822...

Wolfram Alpha, Johnson solid 10

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A179554, A179587, A179588, A179589, A179590, A179591, A179592, A179593, A179637.

RealDigits[N[(Sqrt[2]+2*Sqrt[4+3*Sqrt[2]])/6,200]]

Digits of (sqrt(2)+2 sqrt(4+3 sqrt(2)))/6.

A179593 Decimal expansion of the volume of pentagonal rotunda with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal rotunda: 20 vertices, 35 edges, and 17 faces.

			6.91776296812470206991299603070264133354087600944966144271710443099823...

Wolfram Alpha, Johnson solid 6
Index entries for algebraic numbers, degree 2.

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587, A179588, A179589, A179590, A179591, A179592.

Mathematica
```
RealDigits[N[(45+17*Sqrt[5])/12,200]]
```

Digits of (45+17*sqrt(5))/12.

A179592 Decimal expansion of the circumradius of pentagonal cupola with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal cupola: 15 vertices, 25 edges, and 12 faces.

			2.232950509415690049500415383249682772934080730579181647457441260...

Wolfram Alpha, Johnson solid 5
Index entries for algebraic numbers, degree 4.

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587, A179588, A179589, A179590, A179591.

Mathematica
```
RealDigits[N[Sqrt[11+4*Sqrt[5]]/2,200]]
```

Digits of sqrt(11+4*sqrt(5))/2.

A179639 Decimal expansion of the volume of gyroelongated pentagonal pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Gyroelongated pentagonal pyramid: 11 vertices,25 edges,and 16 faces.

			1.88019215822908780282010679244089525495689855152098881326825313369561...

Wolfram Alpha, Johnson solid 11

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A179554, A179587, A179588, A179589, A179590, A179591, A179592, A179593, A179637, A179638.

Mathematica
```
RealDigits[N[(25+9*Sqrt[5])/24,200]]
```

Digits of (25+9*sqrt(5))/24.

cp's OEIS Frontend

A179296 Decimal expansion of circumradius of a regular dodecahedron with edge length 1.

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A179591 Decimal expansion of the surface area of pentagonal cupola with edge length 1.

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A179588 Decimal expansion of the surface area of square cupola with edge length 1.

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A179637 Decimal expansion of the surface area of pentagonal rotunda with edge length 1.

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A185093 Decimal expansion of the volume of small rhombicosidodecahedron with edge = 1.

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A179589 Decimal expansion of the circumradius of square cupola with edge length 1.

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A179638 Decimal expansion of the volume of gyroelongated square pyramid with edge length 1.

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