A179290 -id:A179290 - cp's OEIS frontend

A179590 Decimal expansion of the volume of pentagonal cupola with edge length 1.

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

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

			2.32404531833319313093944911248751749029374557307435048284726483027368...

Wolfram Alpha, Johnson solid 5
Index entries for algebraic numbers, degree 2

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

Mathematica
```
RealDigits[N[(5+4*Sqrt[5])/6,200]]
```

Digits of (5+4*sqrt(5))/6.

A179553 Decimal expansion of the surface area of pentagonal pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal pyramid: 6 faces, 6 vertices, and 10 edges.

			3.885540910050063539668319904270950058085880737273174114276853431338785...

Wolfram Alpha, Johnson solid 2
Index entries for algebraic numbers, degree 8

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

RealDigits[N[Sqrt[5/2*(10+Sqrt[5]+Sqrt[75+30*Sqrt[5]])]/2,200]]

Digits of sqrt(5/2*(10+sqrt(5)+sqrt(75+30sqrt(5))))/2.

A121570 Decimal expansion of cosecant of 36 degrees = csc(Pi/5) = 1/sin(Pi/5).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Aug 08 2006

1 + csc(Pi/5) is the radius of the smallest circle into which 5 unit circles can be packed ("r=2.701+ Proved by Graham in 1968.", according to the Friedman link, which has a diagram).
csc(Pi/5) = 1/A019845 is the distance between the center of the larger circle and the center of each unit circle.
The problem of finding the diameter d of the circumscribing circle of a regular pentagon of side s = 10 (in some length units) appears as an example in Abū Kāmil's treatise on the pentagon and decagon (see the Havil reference) and Abū Kāmil links. The answer is d/s = 1/sin(Pi/5). - Wolfdieter Lang, Mar 01 2018
Longer diagonal of golden rhombus with unit edge length. - Eric W. Weisstein, Dec 11 2018
The length of the longer side of a golden rectangle inscribed in a unit circle. - Michal Paulovic, Sep 01 2022
The radius of a common circle surrounded by 5 tangent unit circles is A121570 - 1. - Thomas Otten, Dec 27 2023

			1.701301616704079864363080994126...

Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 58.

G. C. Greubel, Table of n, a(n) for n = 1..10000
E. Friedman, Erich's Packing Center: "Circles in Circles"
I. C. Karpinski, The Algebra of Abu Kamil, Amer. Math. Month. XXI,2 (1914), 37-48.
MacTutor History of Mathematics, Abu Kamil Shuja.
Eric Weisstein's World of Mathematics, Golden Rhombus
Wikipedia, Abu Kamil.
Index entries for algebraic numbers, degree 4

Cf. A001622, A019845 (inverse), A182007 (2/A121570).

Cf. A179290 (shorter golden rhombus diagonal).

SetDefaultRealField(RealField(100)); R:= RealField(); 1/Sin(Pi(R)/5); // G. C. Greubel, Nov 02 2018

evalf(1/sin(Pi/5),130); # Muniru A Asiru, Nov 02 2018

RealDigits[Csc[Pi/5], 10, 100][[1]] (* G. C. Greubel, Nov 02 2018 *)

PARI
```
1/sin(Pi/5)
```

numerical_approx(1/sin(pi/5), digits=100) # G. C. Greubel, Dec 12 2018

Equals 1/A019845.
Equals 2*(2*phi - 1)*sqrt(2 + phi)/5, with the golden ratio phi = A001622. - Wolfdieter Lang, Mar 01 2018
Equals sqrt(2 + 2 / sqrt(5)). - Michal Paulovic, Sep 01 2022
The minimal polynomial is 5*x^4 - 20*x^2 + 16. - Joerg Arndt, Sep 09 2022

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A179590 Decimal expansion of the volume of pentagonal cupola with edge length 1.

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A179553 Decimal expansion of the surface area of pentagonal pyramid with edge length 1.

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A121570 Decimal expansion of cosecant of 36 degrees = csc(Pi/5) = 1/sin(Pi/5).

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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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