A020829 -id:A020829 - cp's OEIS frontend

A102208 Decimal expansion of the volume of an icosahedron with unit edge length.

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

Offset: 1

Bryan Jacobs (bryanjj(AT)gmail.com), Feb 17 2005

			2.181694990624912373503822...

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 = 1..1000
Eric Weisstein's World of Mathematics, Icosahedron.
Wikipedia, Icosahedron - Area and volume.
Index entries for algebraic numbers, degree 2.

Cf. A001622 (phi), A020829 (regular tetrahedron volume), A131594 (regular octahedron volume), A102769 (regular dodecahedron volume).

Cf. A071402.

RealDigits[5/12*(3+Sqrt[5]),10, 200][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

5*(3 + sqrt(5))/12 \\ G. C. Greubel, Jul 06 2017

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

Equals 5*phi^2/6, phi being the golden ratio. - Stanislav Sykora, Nov 23 2013

A102769 Decimal expansion of the volume of a dodecahedron with each edge of unit length.

Original entry on oeis.org

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

Offset: 1

Bryan Jacobs (bryanjj(AT)gmail.com), Feb 10 2005

Equals 5*phi^3/(2*xi^2), phi being the golden ratio (A001622) and xi its associate (A182007). - Stanislav Sykora, Nov 23 2013

			7.663118960624631968716053920...

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 = 1..1000
Eric Weisstein's World of Mathematics, Dodecahedron.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A001622 (phi), A182007 (phi associate), A020829 (regular tetrahedron volume), A131594 (regular octahedron volume), A102208 (regular icosahedron volume).

evalf((15+7*sqrt(5))/4,100); # Wesley Ivan Hurt, Jan 29 2017

RealDigits[(Sqrt[5]/2)*(GoldenRatio)^4, 10, 50][[1]] (* G. C. Greubel, Jul 06 2017 *)

(7*sqrt(5)+15)/4 \\ Charles R Greathouse IV, Apr 25 2016

Equals (15 + 7 sqrt(5)) / 4.

Equals (sqrt(5)/2)*(phi)^4, where phi is the golden ratio. - G. C. Greubel, Jul 06 2017

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

A232812 Decimal expansion of the surface index of a regular tetrahedron.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Dec 01 2013

Equivalently, the surface area of a regular tetrahedron with unit volume. Among Platonic solids, surface indices decrease with increasing number of faces: this one, 6.0 (cube = hexahedron), A232811 (octahedron), A232810 (dodecahedron), and A232809 (icosahedron).

			7.20562173105601636005279232409725707779044450935589335...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 6

Cf. A002194, A020829, A232808 (surface index of a sphere), A232809, A232810, A232811.

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

sqrtn(139968,6) \\ Charles R Greathouse IV, Apr 25 2016

Equals 2*sqrt(3)*3^(2/3).

Equals A002194/A020829^(2/3).

A377275 Decimal expansion of the volume of a truncated tetrahedron with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Oct 23 2024

			2.7105759945484321768699033880685879839252044278...

Eric Weisstein's World of Mathematics, Truncated Tetrahedron.
Wikipedia, Truncated tetrahedron.

Cf. A377274 (surface area), A377276 (circumradius), A093577 (midradius), A377277 (Dehn invariant).
Cf. A020829 (analogous for a regular tetrahedron).
Cf. A002193.

First[RealDigits[23/12*Sqrt[2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedTetrahedron", "Volume"], 10, 100]]

Equals (23/12)*sqrt(2) = (23/12)*A002193.

A380738 Decimal expansion of the largest vertex angle, in radians, in a disdyakis dodecahedron face.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Feb 01 2025

A disdyakis dodecahedron face is a scalene triangle with three acute angles.

			1.5219613819557816631607772718833843821006183306...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Disdyakis dodecahedron.

Cf. A380736 (face smallest angle), A380737 (face medium angle).

Cf. A020829, A378393, A378712, A378713, A378714, A378715, A380734, A380735.

First[RealDigits[ArcCos[1/6 - Sqrt[2]/12], 10, 100]]

Equals arccos(1/6 - sqrt(2)/12) = arccos(1/6 - A020829).

Equals Pi - A380736 - A380737.

A363437 Decimal expansion of the volume of the regular tetrahedron inscribed in the unit-radius sphere.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 02 2023

			0.51320023927966734623035447155729551613120155668455...

Eric Weisstein's World of Mathematics, Regular Tetrahedron.
Wikipedia, Tetrahedron: Regular tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A118273 (cube), A122553 (regular octahedron), A339259 (regular icosahedron), A363438 (regular dodecahedron).

Other constants related to the regular tetrahedron: A020781, A020829, A137914, A156546, A187110, A210974, A232812, A236555.

Mathematica
```
RealDigits[8/(9*Sqrt[3]), 10, 120][[1]]
```

8/9/sqrt(3) \\ Charles R Greathouse IV, Feb 07 2025

Equals 8/(9*sqrt(3)).
Equals A118273 / 3.
Equals A020829 / A187110 ^ 3.

A364895 Decimal expansion of the 4-volume of the unit regular pentachoron (5-cell).

Original entry on oeis.org

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

Offset: 0

Jianing Song, Aug 12 2023

Decimal expansion of sqrt(5)/96.

In general, the n-volume of the unit regular n-simplex is sqrt(n+1)/(n!*2^(n/2)).

			Equals 0.02329237476562280933...

Wikipedia, 5-cell
Polytope Wiki, Pentachoron

Decimal expansion of 4-volumes: this sequence (5-cell), A000007 = 1 (8-cell or tesseract), A020793 = 1/6 (16-cell), A000038 = 2 (24-cell), A364896 (120-cell), A364897 (600-cell).

Decimal expansion of the n-volume of the unit regular n-simplex: A120011 (n=2), A020829 (n=3), this sequence (n=4).

First[RealDigits[Sqrt[5]/96, 10, 100, -1]] (* Paolo Xausa, Jun 12 2024 *)

PARI
```
sqrt(5)/96
```

A386000 Decimal expansion of the volume of a tridiminished icosahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 14 2025

The tridiminished icosahedron is Johnson solid J_63.

			1.277186493437438661452675653379955568670180...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Tridiminished icosahedron.

Cf. A386001 (surfacea area).

Cf. A002163, A020829, A102208, A179552, A385857, A386002.

First[RealDigits[5/8 + 7*Sqrt[5]/24, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J63", "Volume"], 10, 100]]

Equals 5/8 + 7*sqrt(5)/24 = 5/8 + 7*A002163/24.
Equals A102208 - 3*A179552 = A386002 - A020829.
Equals the largest root of 144*x^2 - 180*x - 5.

A386002 Decimal expansion of the volume of an augmented tridiminished icosahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 18 2025

The augmented tridiminished icosahedron is Johnson solid J_64.

			1.3950376236351965821861497137307637418843196778...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Augmented tridiminished icosahedron.

Cf. A386003 (surface area).

Cf. A002163, A010466, A020829, A102208, A179552, A385857, A386000.

First[RealDigits[(15 + Sqrt[8] + 7*Sqrt[5])/24, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J64", "Volume"], 10, 100]]

Equals (15 + 2*sqrt(2) + 7*sqrt(5))/24 = (15 + A010466 + 7*A002163)/24.
Equals A102208 - 3*A179552 + A020829 = A386000 + A020829.
Equals the largest root of 2304*x^4 - 5760*x^3 + 3376*x^2 + 280*x - 49.

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A102208 Decimal expansion of the volume of an icosahedron with unit edge length.

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A102769 Decimal expansion of the volume of a dodecahedron with each edge of unit length.

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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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A232812 Decimal expansion of the surface index of a regular tetrahedron.

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A377275 Decimal expansion of the volume of a truncated tetrahedron with unit edge length.

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A380738 Decimal expansion of the largest vertex angle, in radians, in a disdyakis dodecahedron face.

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A363437 Decimal expansion of the volume of the regular tetrahedron inscribed in the unit-radius sphere.

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