A179590 -id:A179590 - cp's OEIS frontend

A386691 Decimal expansion of the volume of a parabidiminished rhombicosidodecahedron with unit edges.

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

Offset: 2

Paolo Xausa, Jul 30 2025

The parabidiminished rhombicosidodecahedron is Johnson solid J_80.

Also the volume of a metabidiminished rhombicosidodecahedron and a gyrate bidiminished rhombicosidodecahedron (Johnson solids J_81 and J_82, respectively) with unit edges.

			36.967233145831580803409780572760635295338486330...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Gyrate bidiminished rhombicosidodecahedron.
Wikipedia, Metabidiminished rhombicosidodecahedron.
Wikipedia, Parabidiminished rhombicosidodecahedron.

Cf. A386692 (surface area).

Cf. A001622, A002163, A134946, A179590, A185093, A386689, A386693.

First[RealDigits[5/3*(11 + 5*Sqrt[5]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J80", "Volume"], 10, 100]]

Equals (5/3)*(11 + 5*sqrt(5)) = (5/3)*(11 + 5*A002163).
Equals A185093 - 2*A179590.
Equals (50/3)*A001622 + 10 = A134946*100 + 10.
Equals the largest root of 9*x^2 - 330*x - 100.

A386693 Decimal expansion of the volume of a tridiminished rhombicosidodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 31 2025

The tridiminished rhombicosidodecahedron is Johnson solid J_83.

			34.64318782749838767247033146027311780504474075702...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Tridiminished rhombicosidodecahedron.

Cf. A386694 (surface area).

Cf. A002163, A179590, A185093, A386689, A386691.

First[RealDigits[35/2 + 23/3*Sqrt[5], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J83", "Volume"], 10, 100]]

Equals 35/2 + (23/3)*sqrt(5) = 35/2 + (23/3)*A002163.
Equals A185093 - 3*A179590.
Equals the largest root of 36*x^2 - 1260*x + 445.

A179639 Decimal expansion of the volume of gyroelongated pentagonal pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Gyroelongated pentagonal pyramid: 11 vertices,25 edges,and 16 faces.

			1.88019215822908780282010679244089525495689855152098881326825313369561...

Wolfram Alpha, Johnson solid 11

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A179554, A179587, A179588, A179589, A179590, A179591, A179592, A179593, A179637, A179638.

Mathematica
```
RealDigits[N[(25+9*Sqrt[5])/24,200]]
```

Digits of (25+9*sqrt(5))/24.

A179640 Decimal expansion of the surface area of gyroelongated pentagonal pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Gyroelongated pentagonal pyramid: 11 vertices, 25 edges, and 16 faces.

			8.21566792897225677348693575803563097544289387179912568441637087996861...

Wolfram Alpha, Johnson solid 11

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A179554, A179587, A179588, A179589, A179590, A179591, A179592, A179593, A179637, A179638, A179639.

RealDigits[N[Sqrt[5/2*(70+Sqrt[5]+3*Sqrt[75+30*Sqrt[5]])]/2,200]]

Digits of sqrt(5/2*(70+sqrt(5)+3*sqrt(75+30*sqrt(5))))/2.

A386852 Decimal expansion of the dihedral angle, in radians, between the pentagonal face and a triangular face in a pentagonal pyramid with equal edges (Johnson solid J_2).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Aug 05 2025

Also the dihedral angle, in radians, between the 10-gonal face and a triangular face in a pentagonal cupola (Johnson solid J_5)

			0.65235813978436818599539063164382257436530791996...

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

Cf. A179552 (J_2 volume), A179553 (J_2 surface area).
Cf. A179590 (J_5 volume), A179591 (J_5 surface area).
Cf. A002163, A010476.

First[RealDigits[ArcSec[Sqrt[15 - 6*Sqrt[5]]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J2", "DihedralAngles"]], 10, 100]]

acos(sqrt((5+2*sqrt(5))/15)) \\ Charles R Greathouse IV, Aug 19 2025

Equals arcsec(sqrt(15 - 6*sqrt(5))) = arcsec(sqrt(15 - 6*A002163)).

Equals arccos(sqrt((5 + 2*sqrt(5))/15)) = arccos(sqrt((5 + A010476)/15)).

A334114 Decimal expansion of volume of a sphenomegacorona (J88) with each edge of unit length.

Original entry on oeis.org

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

Offset: 1

Violeta Hernández Palacios, Apr 14 2020

A sphenomegacorona is one of the 92 regular-faced non-isogonal convex polyhedra first enumerated by Norman W. Johnson. It's built out of 2 squares and 12 equilateral triangles.
This number is algebraic, of unknown degree.
It appears that the minimal polynomial is 521578814501447328359509917696*x^32 - 985204427391622731345740955648*x^30 - 16645447351681991898880656015360*x^28 + 79710816694053483249372512649216*x^26 - 152195045391070538203422101864448*x^24 + 156280253448056209478031589244928*x^22 - 96188116617075838858708654227456*x^20 + 30636368373570166303441645731840*x^18 + 5828527077458909552923002273792*x^16 - 8060049780765551057159394951168*x^14 + 1018074792115156107372011716608*x^12 + 35220131544370794950945931264*x^10 + 327511698517355918956755959808*x^8 - 116978732884218191486738706432*x^6 + 10231563774949176791703149568*x^4 - 366323949299263261553952192*x^2 + 3071435678740442112675625. - Joerg Arndt, Apr 16 2020

			1.94810822885947280327067639...

Violeta Hernández Palacios, Table of n, a(n) for n = 1..20000
Norman W. Johnson, Convex Polyhedra with Regular Faces, Canadian Journal of Mathematics, 18 (1966), 169-200.
H. S. Teoh, The Sphenomegacorona
A. V. Timofeenko, The non-platonic and non-Archimedean noncomposite polyhedra, Journal of Mathematical Sciences, 162(2009), 720-722.
Eric Weisstein's World of Mathematics, Sphenomegacorona.

Volumes of other Johnson solids: A179552, A179587, A179590.

k := Root[-23 - 56 x + 200 x^2 + 304 x^3 - 776 x^4 + 240 x^5 +
   2000 x^6 - 5584 x^7 - 3384 x^8 + 17248 x^9 + 2464 x^10 -
   24576 x^11 + 1568 x^12 + 17216 x^13 - 3712 x^14 - 4800 x^15 +
   1680 x^16, 2];
{{0, 1/2, Sqrt[1 - k^2]}, {k, 1/2, 0}, {0, Sqrt[(3/4 - k^2)/(1 - k^2)] + 1/2, (1/2 - k^2)/Sqrt[1 - k^2]}, {1/2, 0, -Sqrt[1/2 + k - k^2]}, {0, (Sqrt[3/4 - k^2] (2 k^2 - 1))/((k^2 - 1) Sqrt[1 - k^2]) + 1/2, (k^4 - 1/2)/(1 - k^2)^(3/2)}};
v = Union[%, {1, -1, 1}*# & /@ %, {-1, 1, 1}*# & /@ %, {-1, -1,
  1}*# & /@ %];
f := {{2, 3, 12, 11}, {2, 3, 10, 9}, {3, 12, 5}, {3, 10, 5}, {12, 5,
  7}, {10, 5, 7}, {7, 12, 8}, {7, 10, 1}, {12, 8, 11}, {10, 1,
  9}, {8, 1, 7}, {8, 1, 6}, {8, 11, 6}, {1, 9, 6}, {11, 6, 4}, {9,
  6, 4}, {4, 11, 2}, {4, 9, 2}};
RealDigits[N[Volume[Polyhedron[v, f]], 20000]][[1]]

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A386691 Decimal expansion of the volume of a parabidiminished rhombicosidodecahedron with unit edges.

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A386693 Decimal expansion of the volume of a tridiminished rhombicosidodecahedron with unit edges.

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A179639 Decimal expansion of the volume of gyroelongated pentagonal pyramid with edge length 1.

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A179640 Decimal expansion of the surface area of gyroelongated pentagonal pyramid with edge length 1.

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A386852 Decimal expansion of the dihedral angle, in radians, between the pentagonal face and a triangular face in a pentagonal pyramid with equal edges (Johnson solid J_2).

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A334114 Decimal expansion of volume of a sphenomegacorona (J88) with each edge of unit length.

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