A377750 -id:A377750 - 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).

A377751 Decimal expansion of the volume of a truncated icosahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 07 2024

			55.28773075812273923639861693886121953098664736582...

Eric Weisstein's World of Mathematics, Truncated Icosahedron.
Wikipedia, Truncated icosahedron.
Index entries for algebraic numbers, degree 2.

Cf. A377750 (surface area), A377752 (circumradius), A205769 (midradius + 1), A377787 (Dehn invariant).
Cf. A102208 (analogous for a regular icosahedron).
Cf. A002163.

First[RealDigits[(125 + 43*Sqrt[5])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedIcosahedron", "Volume"], 10, 100]]

(125 + 43*sqrt(5))/4 \\ Charles R Greathouse IV, Feb 05 2025

Equals (125 + 43*sqrt(5))/4 = (125 + 43*A002163)/4.

A377752 Decimal expansion of the circumradius of a truncated icosahedron with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 07 2024

			2.47801865906761553756640791226630780693649473...

Eric Weisstein's World of Mathematics, Truncated Icosahedron.
Wikipedia, Truncated icosahedron.
Index entries for algebraic numbers, degree 4.

Cf. A377750 (surface area), A377751 (volume), A205769 (midradius + 1), A377787 (Dehn invariant).
Cf. A019881 (analogous for a regular icosahedron).
Cf. A002163.

First[RealDigits[Sqrt[58 + 18*Sqrt[5]]/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedIcosahedron", "Circumradius"], 10, 100]]

sqrt(58 + 18*sqrt(5))/4 \\ Charles R Greathouse IV, Feb 05 2025

Equals sqrt(58 + 18*sqrt(5))/4 = sqrt(58 + 18*A002163)/4.

A381693 Decimal expansion of the isoperimetric quotient of a truncated icosahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 08 2025

For the definition of isoperimetric quotient of a solid, references and links, see A381684.

			0.90317079253486933111536917369134449479629558352448...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A377750 (surface area), A377751 (volume).

Cf. A019679, A002163, A002194, A010476, A381684.

First[RealDigits[Pi/12*(125 + 43*Sqrt[5])^2/(Sqrt[300] + Sqrt[25 + Sqrt[500]])^3, 10, 100]]

Equals 36*Pi*A377751^2/(A377750^3).

Equals (Pi/12)*(125 + 43*sqrt(5))^2/((10*sqrt(3) + sqrt(5*(5 + 2*sqrt(5))))^3) = A019679*(125 + 43*A002163)^2/((10*A002194 + sqrt(5*(5 + A010476)))^3).

cp's OEIS Frontend

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

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A377751 Decimal expansion of the volume of a truncated icosahedron with unit edge length.

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A377752 Decimal expansion of the circumradius of a truncated icosahedron with unit edge length.

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A381693 Decimal expansion of the isoperimetric quotient of a truncated icosahedron.

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