A379136 -id:A379136 - cp's OEIS frontend

A379132 Decimal expansion of the surface area of a pentakis dodecahedron with unit shorter edge length.

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

Offset: 2

Paolo Xausa, Dec 16 2024

The pentakis dodecahedron is the dual polyhedron of the truncated icosahedron.

			27.93524960070079310581019127996368070525778610907...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Pentakis Dodecahedron.
Wikipedia, Pentakis dodecahedron.
Index entries for algebraic numbers, degree 4.

Cf. A379133 (volume), A379134 (inradius), A379135 (midradius), A379136 (dihedral angle).
Cf. A377750 (surface area of a truncated icosahedron with unit edge length).
Cf. A002163.

First[RealDigits[5/3*Sqrt[(421 + 63*Sqrt[5])/2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["PentakisDodecahedron", "SurfaceArea"], 10, 100]]

sqrt((421 + 63*sqrt(5))/2)*5/3 \\ Charles R Greathouse IV, Feb 05 2025

Equals (5/3)*sqrt((421 + 63*sqrt(5))/2) = (5/3)*sqrt((421 + 63*A002163)/2).

A379133 Decimal expansion of the volume of a pentakis dodecahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Dec 16 2024

The pentakis dodecahedron is the dual polyhedron of the truncated icosahedron.

			13.458569366318714223642964127539153595279924859762...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Pentakis Dodecahedron.
Wikipedia, Pentakis dodecahedron.
Index entries for algebraic numbers, degree 2.

Cf. A379132 (surface area), A379134 (inradius), A379135 (midradius), A379136 (dihedral angle).
Cf. A377751 (volume of a truncated icosahedron with unit edge length).
Cf. A002163.

First[RealDigits[5/36*(41 + 25*Sqrt[5]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["PentakisDodecahedron", "Volume"], 10, 100]]

(41 + 25*sqrt(5))*5/36 \\ Charles R Greathouse IV, Feb 05 2025

Equals (5/36)*(41 + 25*sqrt(5)) = (5/36)*(41 + 25*A002163).

A379135 Decimal expansion of the midradius of a pentakis dodecahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 17 2024

The pentakis dodecahedron is the dual polyhedron of the truncated icosahedron.

			1.4756836610416140907689600838494857255268212565695...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentakis Dodecahedron.
Wikipedia, Pentakis dodecahedron.
Index entries for algebraic numbers, degree 2.

Cf. A379132 (surface area), A379133 (volume), A379134 (inradius), A379136 (dihedral angle).
Cf. A205769 (midradius + 1 of a truncated icosahedron with unit edge length).
Cf. A010499.

First[RealDigits[(11 + Sqrt[45])/12, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["PentakisDodecahedron", "Midradius"], 10, 100]]

(11 + 3*sqrt(5))/12 \\ Charles R Greathouse IV, Feb 05 2025

Equals (11 + 3*sqrt(5))/12 = (11 + A010499)/12.

A379134 Decimal expansion of the inradius of a pentakis dodecahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 17 2024

The pentakis dodecahedron is the dual polyhedron of the truncated icosahedron.

			1.4453319256522148283158512491020811977238711778430...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentakis Dodecahedron.
Wikipedia, Pentakis dodecahedron.
Index entries for algebraic numbers, degree 4.

Cf. A379132 (surface area), A379133 (volume), A379135 (midradius), A379136 (dihedral angle).

Cf. A002163.

First[RealDigits[Sqrt[477/436 + 97*Sqrt[5]/218], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["PentakisDodecahedron", "Inradius"], 10, 100]]

sqrt(477/436 + 97*sqrt(5)/218) \\ Charles R Greathouse IV, Feb 05 2025

Equals sqrt(477/436 + 97*sqrt(5)/218) = sqrt(477/436 + 97*A002163/218).

Equals the largest root of 1744*x^4 - 3816*x^2 + 361.

A380795 Decimal expansion of the smallest acute vertex angles, in radians, in a pentakis dodecahedron face.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Feb 04 2025

A pentakis dodecahedron face is an acute isoscele triangle with the smallest angles being this constant and the largest angle being A380796.

			0.97198502241670672469065747671869944918763790417527...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Pentakis Dodecahedron.
Wikipedia, Pentakis dodecahedron.

Cf. A001622, A379132, A379133, A379135, A379136, A380796.

First[RealDigits[ArcCos[(5 - GoldenRatio)/6], 10, 100]]

Equals arccos((5 - A001622)/6).

Equals (Pi - A380796)/2.

A380796 Decimal expansion of the largest acute vertex angle, in radians, in a pentakis dodecahedron face.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Feb 04 2025

A pentakis dodecahedron face is an acute isoscele triangle with the smallest angles being A380795 and the largest angle being this constant.

			1.1976226087563797890813284298421039858218935910...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentakis Dodecahedron.
Wikipedia, Pentakis dodecahedron.

Cf. A001622, A379132, A379133, A379135, A379136, A380795.

First[RealDigits[ArcCos[(9*GoldenRatio - 8)/18], 10, 100]]

Equals arccos((9*A001622 - 8)/18).

Equals Pi - 2*A380795.

cp's OEIS Frontend

A379132 Decimal expansion of the surface area of a pentakis dodecahedron with unit shorter edge length.

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A379133 Decimal expansion of the volume of a pentakis dodecahedron with unit shorter edge length.

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A379135 Decimal expansion of the midradius of a pentakis dodecahedron with unit shorter edge length.

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A379134 Decimal expansion of the inradius of a pentakis dodecahedron with unit shorter edge length.

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A380795 Decimal expansion of the smallest acute vertex angles, in radians, in a pentakis dodecahedron face.

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A380796 Decimal expansion of the largest acute vertex angle, in radians, in a pentakis dodecahedron face.

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Mathematica

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