A102208 -id:A102208 - cp's OEIS frontend

A179451 Decimal expansion of the surface area of an icosidodecahedron with side length 1.

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.

			29.3059828449119895407453754361926770276025163091742830907641713815460...

Index entries for algebraic numbers, degree 8

Cf. A010527, A102208, A179290, A179292, A179294, A179449, A179450.

RealDigits[N[Sqrt[30*(10+3*Sqrt[5]+Sqrt[75+30*Sqrt[5]])],200]]

polrootsreal(x^8 - 1200*x^6 + 324000*x^4 - 27000000*x^2 + 324000000)[8] \\ Charles R Greathouse IV, Oct 30 2023

Sqrt(30*(10+3*sqrt(5)+sqrt(75+30*sqrt(5))))

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

A179450 Decimal expansion of the volume of an icosidodecahedron with edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.

			13.83552593624940413982599206140528266708175201889932288543420886199647...

Index entries for algebraic numbers, degree 2

Cf. A010527, A102208, A179290, A179292, A179294, A179449.

Mathematica
```
RealDigits[N[(45+17*Sqrt[5])/6,200]]
```

(45 + 17*sqrt(5))/6 \\ Charles R Greathouse IV, Oct 30 2023

(45 + 17*sqrt(5))/6.

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

A020829 Decimal expansion of 1/sqrt(72) = 1/(3*2^(3/2)) = sqrt(2)/12.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Volume of regular tetrahedron with unit edge. - Stanislav Sykora, May 31 2012
In the dragon curve fractal, (5/6)*sqrt(2) = 1.1785.... is the maximum distance of any point from curve start.  Such a maximum must be to a vertex of the convex hull.  Hull vertices are shown by Benedek and Panzone (theorem 3, page 85) and their P8 = 7/6 - (1/6)i at distance sqrt((7/6)^2 + (1/6)^2) is the maximum. - Kevin Ryde, Nov 22 2019
With offset 1, volume of a triangular cupola (Johnson solid J_3) with unit edges. - Paolo Xausa, Aug 04 2025

			0.117851130197757920733474...

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
Agnes I. Benedek and Rafael Panzone, On Some Notable Plane Sets, II: Dragons, Revista de la Unión Matemática Argentina, volume 39, numbers 1-2, 1994, pages 76-90.
Wikipedia, Platonic solid.
Wikipedia, Tetrahedron.
Wikipedia, Triangular cupola.
Index entries for algebraic numbers, degree 2

Cf. A131594 (regular octahedron volume), A102208 (regular icosahedron volume), A102769 (regular dodecahedron volume).

Cf. A010524, A020765, A020775, A378207.

RealDigits[Sqrt[2]/12, 10, 50][[1]] (* G. C. Greubel, Jul 06 2017 *)

sqrt(2)/12 \\ G. C. Greubel, Jul 06 2017

Equals Integral_{x=0..Pi/4} sin(x)^2 * cos(x) dx. - Amiram Eldar, May 31 2021

Equals 1/A010524 = A020765/3 = A020775/2 = A378207/5. - Hugo Pfoertner, Jan 26 2025

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

Original entry on oeis.org

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.

A131594 Decimal expansion of sqrt(2)/3, the volume of a regular octahedron with edge length 1.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Aug 30 2007

Volume of a regular octahedron: V = ((sqrt(2))/3)* a^3, where 'a' is the edge.

			0.471404520791031682933896...

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. A020829 (regular tetrahedron volume), A102208 (regular icosahedron volume), A102769 (regular dodecahedron volume).

Cf. A179587.

RealDigits[Sqrt[2]/3,10,120][[1]] (* Harvey P. Dale, May 27 2012 *)

PARI
```
sqrt(2)/3 \\ G. C. Greubel, Jul 06 2017
```

Equals A002193/3 = A010464/A010482. - R. J. Mathar, Dec 11 2009

More digits from R. J. Mathar, Dec 11 2009

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.

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).

A232809 Decimal expansion of the surface index of a regular icosahedron.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Dec 01 2013

Equivalently, surface area of a regular icosahedron with unit volume. Among Platonic solids, surface indices decrease with increasing number of faces: A232812 (tetrahedron), 6.0 (cube = hexahedron), A232811 (octahedron), A232810 (dodecahedron), and this one.

An algebraic integer of degree 12 with minimal polynomial x^12 - 41115600x^6 + 765275040000. - Charles R Greathouse IV, Apr 25 2016

			5.14834855619951564633081294611601906410086411638672414845...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 12

Cf. A010527, A102208 (solid index of a sphere), A232808, A232810, A232811, A232812.

RealDigits[5*Sqrt[3]/(5*(3+Sqrt[5])/12)^(2/3), 10, 120][[1]] (* Amiram Eldar, May 25 2023 *)

5*sqrt(3)/(5*(3+sqrt(5))/12)^(2/3) \\ Charles R Greathouse IV, Apr 25 2016

Equals 5*sqrt(3)/(5*(3+sqrt(5))/12)^(2/3).

Equals 10*A010527/A102208^(2/3).

cp's OEIS Frontend

A179451 Decimal expansion of the surface area of an icosidodecahedron with side length 1.

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A179450 Decimal expansion of the volume of an icosidodecahedron with edge length 1.

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A020829 Decimal expansion of 1/sqrt(72) = 1/(3*2^(3/2)) = sqrt(2)/12.

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

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A131594 Decimal expansion of sqrt(2)/3, the volume of a regular octahedron 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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A179296 Decimal expansion of circumradius of a regular dodecahedron with edge length 1.

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