A377695 -id:A377695 - cp's OEIS frontend

A378974 Decimal expansion of the volume of a triakis icosahedron with unit shorter edge length.

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

Offset: 2

Paolo Xausa, Dec 14 2024

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

			12.017220926874316513329814423376647765182009668737...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Triakis Icosahedron.
Wikipedia, Triakis icosahedron.

Cf. A378973 (surface area), A378975 (inradius), A378976 (midradius), A378977 (dihedral angle).
Cf. A377695 (volume of a truncated dodecahedron with unit edge length).
Cf. A002163.

First[RealDigits[(19 + 13*Sqrt[5])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisIcosahedron", "Volume"], 10, 100]]

Equals (19 + 13*sqrt(5))/4 = (19 + 13*A002163)/4.

A377694 Decimal expansion of the surface area of a truncated dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 3

Paolo Xausa, Nov 04 2024

			100.990760153101988544745948988636656554915090575...

Eric Weisstein's World of Mathematics, Truncated Dodecahedron.
Wikipedia, Truncated dodecahedron.

Cf. A377695 (volume), A377696 (circumradius), A377697 (midradius), A377698 (Dehn invariant, negated).
Cf. A131595 (analogous for a regular dodecahedron).
Cf. A002194, A010476.

First[RealDigits[5*(Sqrt[3] + 6*Sqrt[5 + Sqrt[20]]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedDodecahedron", "SurfaceArea"], 10, 100]]

Equals 5*(sqrt(3) + 6*sqrt(5 + 2*sqrt(5))) = 5*(A002194 + 6*sqrt(5 + A010476)).

A377697 Decimal expansion of the midradius of a truncated dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 05 2024

			2.9270509831248422723068802515484571765804637697...

Eric Weisstein's World of Mathematics, Truncated Dodecahedron.
Wikipedia, Truncated dodecahedron.

Cf. A377694 (surface area), A377695 (volume), A377696 (circumradius), A377698 (Dehn invariant, negated).
Cf. A239798 (analogous for a regular dodecahedron).
Cf. A010499, A205769.

First[RealDigits[(5 + Sqrt[45])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedDodecahedron", "Midradius"], 10, 100]]

Equals (5 + 3*sqrt(5))/4 = (5 + A010499)/4.

Equals A205769 - 1/2.

A386464 Decimal expansion of the volume of an augmented truncated dodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 25 2025

The augmented truncated dodecahedron is Johnson solid J_68.

			87.3637098777040746856191001251416771010058551...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Augmented truncated dodecahedron.

Cf. A386465 (surface area).

Cf. A179590, A204188, A377695, A386411, A386459, A386466, A386544.

First[RealDigits[505/12 + 81/4*Sqrt[5], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J68", "Volume"], 10, 100]]

Equals 505/12 + 81*sqrt(5)/4 = 505/12 + 81*A204188.
Equals A377695 + A179590.
Equals the largest root of 36*x^2 - 3030*x - 10055.

A386466 Decimal expansion of the volume of a parabiaugmented truncated dodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 25 2025

The parabiaugmented truncated dodecahedron is Johnson solid J_69.

Also the volume of a metabiaugmented truncated dodecahedron (Johnson solid J_70) with unit edges.

			89.68775519603726781655854923762919459129960068854...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Metabiaugmented truncated dodecahedron.
Wikipedia, Parabiaugmented truncated dodecahedron.

Cf. A386543 (surface area).

Cf. A002163, A179590, A377695, A385578, A385802, A386464, A386544.

First[RealDigits[(515 + 251*Sqrt[5])/12, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J69", "Volume"], 10, 100]]

Equals (515 + 251*sqrt(5))/12 = (515 + 251*A002163)/12.
Equals A377695 + 2*A179590.
Equals the largest root of 36*x^2 - 3090*x - 12445.

A386544 Decimal expansion of the volume of a triaugmented truncated dodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 28 2025

The triaugmented truncated dodecahedron is Johnson solid J_71.

			92.01180051437046094749799835011671208159334626...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Triaugmented truncated dodecahedron.

Cf. A386545 (surface area).

Cf. A002163, A179590, A377695, A385506, A385804, A386464, A386466.

First[RealDigits[7/12*(75 + 37*Sqrt[5]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J71", "Volume"], 10, 100]]

Equals (7/12)*(75 + 37*sqrt(5)) = (7/12)*(75 + 37*A002163).
Equals A377695 + 3*A179590.
Equals the largest root of 36*x^2 - 3150*x - 14945.

A377696 Decimal expansion of the circumradius of a truncated dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 04 2024

			2.9694490158633984670421666956925979636007477003...

Eric Weisstein's World of Mathematics, Truncated Dodecahedron.
Wikipedia, Truncated dodecahedron.

Cf. A377694 (surface area), A377695 (volume), A377697 (midradius), A377698 (Dehn invariant, negated).
Cf. A179296 (analogous for a regular dodecahedron).
Cf. A002163.

First[RealDigits[Sqrt[74 + 30*Sqrt[5]]/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedDodecahedron", "Circumradius"], 10, 100]]

Equals sqrt(74 + 30*sqrt(5))/4 = sqrt(74 + 30*A002163)/4.

A381692 Decimal expansion of the isoperimetric quotient of a truncated dodecahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 07 2025

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

			0.7940548943037979813018428272822581808271192993785...

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

Cf. A377694 (surface area), A377695 (volume).

Cf. A000796, A002163, A002194, A010476, A381684.

First[RealDigits[Pi/20*(99 + 47*Sqrt[5])^2/(Sqrt[3] + 6*Sqrt[5 + Sqrt[20]])^3, 10, 100]]

Equals 36*Pi*A377695^2/(A377694^3).

Equals (Pi/20)*(99 + 47*sqrt(5))^2/((sqrt(3) + 6*sqrt(5 + 2*sqrt(5)))^3) = (A000796/20)*(99 + 47*A002163)^2/((A002194 + 6*sqrt(5 + A010476))^3).

cp's OEIS Frontend

A378974 Decimal expansion of the volume of a triakis icosahedron with unit shorter edge length.

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A377694 Decimal expansion of the surface area of a truncated dodecahedron with unit edge length.

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A377697 Decimal expansion of the midradius of a truncated dodecahedron with unit edge length.

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A386464 Decimal expansion of the volume of an augmented truncated dodecahedron with unit edges.

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A386466 Decimal expansion of the volume of a parabiaugmented truncated dodecahedron with unit edges.

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A386544 Decimal expansion of the volume of a triaugmented truncated dodecahedron with unit edges.

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A377696 Decimal expansion of the circumradius of a truncated dodecahedron with unit edge length.

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A381692 Decimal expansion of the isoperimetric quotient of a truncated dodecahedron.

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