A236556 -id:A236556 - cp's OEIS frontend

A236555 Decimal expansion of the steradian solid angle subtended by one triangular facet of the cuboctahedron.

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

Offset: 0

Stanislav Sykora, Jan 28 2014

Also the vertex solid angle of a regular tetrahedron.

			0.55128559843253080794214415146445924293385062869017869619170040868416...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Cuboctahedron
Eric Weisstein's World of Mathematics, Spherical Polygon
Wikipedia, Platonic solid
Index entries for transcendental numbers.

Cf. A236556. Icosidodecahedron: A319881, A319883. Vertex Angles: A019669, A236557, A236558.

RealDigits[N[ArcCos[23/27], 107]][[1]] (* T. D. Noe, Jan 29 2014 *)

acos(23/27) \\ Charles R Greathouse IV, Feb 10 2025

arccos(23/27).
3*arcsin(2*sqrt(2)/3) - Pi. - Bradley Klee, Oct 04 2018
8*A236555 + 6*A236556 = 4*Pi. - Bradley Klee, Oct 04 2018
Equals 2*arctan(sqrt(2)/5). - R. J. Mathar, Mar 11 2025

Definition corrected by Bradley Klee, Oct 04 2018

A236557 Vertex solid angle (steradians) in a regular icosahedron.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jan 28 2014

			2.6345470260447546596513034674924022914118092285116431884051417273...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Platonic solid.

Cf. Analogs: A236555 (tetrahedron), A236556 (octahedron), A019669 (cube), A236558 (dodecahedron).

RealDigits[2*Pi - 5*ArcSin[2/3], 10, 120][[1]] (* Amiram Eldar, May 22 2023 *)

PARI
```
2*Pi-5*asin(2/3)
```

Equals 2*Pi - 5*arcsin(2/3).

A236558 Vertex solid angle (steradians) in a regular dodecahedron.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jan 28 2014

			2.9617391537973149678740903860233946481841958625459137019202715623145...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Platonic solid.

Cf. Analogs: A236555 (tetrahedron), A236556 (octahedron), A019669 (cube), A236557 (icosahedron).

RealDigits[Pi - ArcTan[2/11], 10, 120][[1]] (* Amiram Eldar, May 22 2023 *)

PARI
```
Pi-atan(2/11)
```

Equals Pi - arctan(2/11).

A319881 Decimal expansion of the steradian solid angle subtended by one triangular facet of the icosidodecahedron.

Original entry on oeis.org

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

Offset: 0

Bradley Klee, Sep 30 2018

			0.17985349979247827058855299725610823601297353682919...

Eric Weisstein's World of Mathematics, Icosidodecahedron
Eric Weisstein's World of Mathematics, Spherical Polygon

Cf. A319883. Cuboctahedron: A236555, A236556.

RealDigits[3*ArcTan[2] - Pi, 10, 80][[1]]

3*atan(2)-Pi \\ Altug Alkan, Oct 03 2018

Equals 3*arctan(2) - Pi.

Equals (Pi - 3*A319883)/5.

A319883 Decimal expansion of the steradian solid angle subtended by one pentagonal facet of the icosidodecahedron.

Original entry on oeis.org

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

Offset: 0

Bradley Klee, Sep 30 2018

			0.74744171820913396183995946566632056804410057174304...

Eric Weisstein's World of Mathematics, Icosidodecahedron
Eric Weisstein's World of Mathematics, Spherical Polygon

Cf. A319881. Cuboctahedron: A236555, A236556.

RealDigits[2*Pi - 5*ArcTan[2], 10, 80][[1]]

Equals 2*Pi - 5*arctan(2).

Equals (Pi - 5*A319881)/3.

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A236555 Decimal expansion of the steradian solid angle subtended by one triangular facet of the cuboctahedron.

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A236557 Vertex solid angle (steradians) in a regular icosahedron.

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A236558 Vertex solid angle (steradians) in a regular dodecahedron.

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A319881 Decimal expansion of the steradian solid angle subtended by one triangular facet of the icosidodecahedron.

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A319883 Decimal expansion of the steradian solid angle subtended by one pentagonal facet of the icosidodecahedron.

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