A377694 -id:A377694 - cp's OEIS frontend

A378973 Decimal expansion of the surface area of a triakis icosahedron with unit shorter edge length.

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

Offset: 2

Paolo Xausa, Dec 13 2024

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

			26.228595976743751681458195104356801731865266699519...

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

Cf. A378974 (volume), A378975 (inradius), A378976 (midradius), A378977 (dihedral angle).
Cf. A377694 (surface area of a truncated dodecahedron with unit edge length).
Cf. A002163.

First[RealDigits[3*Sqrt[(173 - 9*Sqrt[5])/2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisIcosahedron", "SurfaceArea"], 10, 100]]

Equals 3*sqrt((173 - 9*sqrt(5))/2) = 3*sqrt((173 - 9*A002163)/2).

A377695 Decimal expansion of the volume of a truncated dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 04 2024

			85.039664559370881554679651012654159610712109542...

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

Cf. A377694 (surface area), A377696 (circumradius), A377697 (midradius), A377698 (Dehn invariant, negated).
Cf. A102769 (analogous for a regular dodecahedron).
Cf. A002163.

First[RealDigits[5/12*(99 + 47*Sqrt[5]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedDodecahedron", "Volume"], 10, 100]]

Equals (5/12)*(99 + 47*sqrt(5)) = (5/12)*(99 + 47*A002163).

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.

A386465 Decimal expansion of the surface area of an augmented truncated dodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 3

Paolo Xausa, Jul 25 2025

The augmented truncated dodecahedron is Johnson solid J_68.

			102.18209222021391857798854245281533205298421595361...

Paolo Xausa, Table of n, a(n) for n = 3..10000
Wikipedia, Augmented truncated dodecahedron.

Cf. A386464 (volume).

Cf. A002194, A010476, A377694, A386412, A386460, A386543, A386545.

First[RealDigits[(20 + 25*Sqrt[3] + 110*Sqrt[#] + Sqrt[5*#])/4 & [5 + Sqrt[20]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J68", "SurfaceArea"], 10, 100]]

Equals (20 + 25*sqrt(3) + 110*sqrt(5 + 2*sqrt(5)) + sqrt(5*(5 + 2*sqrt(5))))/4 = (20 + 25*A002194 + 110*sqrt(5 + A010476) + sqrt(5*(5 + A010476)))/4.

Equals the largest root of 256*x^8 - 10240*x^7 - 3955200*x^6 + 122240000*x^5 + 16152924000*x^4 - 343551280000*x^3 - 11461251137500*x^2 + 131995515375000*x + 634637481578125.

A386543 Decimal expansion of the surface area of a parabiaugmented truncated dodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 3

Paolo Xausa, Jul 28 2025

The parabiaugmented truncated dodecahedron is Johnson solid J_69.

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

			103.37342428732584861123113591699400755105334133204...

Paolo Xausa, Table of n, a(n) for n = 3..10000
Wikipedia, Metabiaugmented truncated dodecahedron.
Wikipedia, Parabiaugmented truncated dodecahedron.

Cf. A386466 (volume).

Cf. A002194, A010476, A377694, A385803, A386465, A386545.

First[RealDigits[(20 + 15*Sqrt[3] + 50*Sqrt[#] + Sqrt[5*#])/2 & [5 + Sqrt[20]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J69", "SurfaceArea"], 10, 100]]

Equals (20 + 15*sqrt(3) + 50*sqrt(5 + 2*sqrt(5)) + sqrt(5*(5 + 2*sqrt(5))))/2 = (20 + 15*A002194 + 50*sqrt(5 + A010476) + sqrt(5*(5 + A010476)))/2.

Equals the largest root of x^8 - 80*x^7 - 11400*x^6 + 796000*x^5 + 31475250*x^4 - 1804610000*x^3 - 8296459375*x^2 + 548931187500*x - 2544044046875.

A386545 Decimal expansion of the surface area of a triaugmented truncated dodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 3

Paolo Xausa, Jul 28 2025

The triaugmented truncated dodecahedron is Johnson solid J_71.

			104.56475635443777864447372938117268304912246671047...

Paolo Xausa, Table of n, a(n) for n = 3..10000
Wikipedia, Triaugmented truncated dodecahedron.

Cf. A386544 (volume).

Cf. A002194, A010476, A377694, A385805, A386465, A386543.

First[RealDigits[(60 + 35*Sqrt[3] + 90*Sqrt[#] + 3*Sqrt[5*#])/4 & [5 + Sqrt[20]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J71", "SurfaceArea"], 10, 100]]

Equals (60 + 35*sqrt(3) + 90*sqrt(5 + 2*sqrt(5)) + 3*sqrt(5*(5 + 2*sqrt(5))))/4 = (60 + 35*A002194 + 90*sqrt(5 + A010476) + 3*sqrt(5*(5 + A010476)))/4.

Equals the largest root of 256*x^8 - 30720*x^7 - 1574400*x^6 + 238464000*x^5 + 68364000*x^4 - 390828240000*x^3 + 4437895162500*x^2 + 78660973125000*x - 1021409416546875.

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

A378973 Decimal expansion of the surface area of a triakis icosahedron with unit shorter edge length.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A377695 Decimal expansion of the volume of a truncated dodecahedron with unit edge length.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A386465 Decimal expansion of the surface area of an augmented truncated dodecahedron with unit edges.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A386543 Decimal expansion of the surface area of a parabiaugmented truncated dodecahedron with unit edges.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A386545 Decimal expansion of the surface area of a triaugmented truncated dodecahedron with unit edges.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments