A377996 -id:A377996 - cp's OEIS frontend

A379711 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a disdyakis triacontahedron.

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

Offset: 1

Paolo Xausa, Dec 31 2024

The disdyakis triacontahedron is the dual polyhedron of the truncated icosidodecahedron (great rhombicosidodecahedron).

			2.8778366104612242809434504548179917754749428665406...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Disdyakis Triacontahedron.
Wikipedia, Disdyakis triacontahedron.

Cf. A379708 (surface area), A379709 (volume), A379710 (inradius), A379388 (midradius).
Cf. A344075, A377995 and A377996 (dihedral angles of a truncated icosidodecahedron (great rhombicosidodecahedron)).
Cf. A002163.

First[RealDigits[ArcCos[(-179 - 24*Sqrt[5])/241], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["DisdyakisTriacontahedron", "DihedralAngles"]], 10, 100]]

Equals arccos((-179 - 24*sqrt(5))/241) = arccos((-179 - 24*A002163)/241).

A379389 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a deltoidal hexecontahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 23 2024

The deltoidal hexecontahedron is the dual polyhedron of the (small) rhombicosidodecahedron.

			2.6899252342065763400728815146316168300353303724921...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Deltoidal Hexecontahedron.
Wikipedia, Deltoidal hexecontahedron.

Cf. A379385 (surface area), A379386 (volume), A379387 (inradius), A379388 (midradius).
Cf. A377995 and A377996 (dihedral angles of a (small) rhombicosidodecahedron).
Cf. A002163.

First[RealDigits[ArcCos[-(19 + 8*Sqrt[5])/41], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["DeltoidalHexecontahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-(19 + 8*sqrt(5))/41) = arccos(-(19 + 8*A002163)/41).

A377995 Decimal expansion of the dihedral angle, in radians, between square and pentagonal faces in a (small) rhombicosidodecahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 15 2024

Also the dihedral angle, in radians, between square and 10-gonal faces in a truncated icosidodecahedron (great rhombicosidodecahedron).

			2.588018294692747986954110653190234364162145576674...

Eric Weisstein's World of Mathematics, Great Rhombicosidodecahedron.
Eric Weisstein's World of Mathematics, Small Rhombicosidodecahedron.

Cf. A242671, A377996.

First[RealDigits[ArcCos[-Sqrt[(5 + Sqrt[5])/10]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["Rhombicosidodecahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-sqrt((5 + sqrt(5))/10)) = arccos(-sqrt(A242671)).

A387189 Decimal expansion of the smallest dihedral angle, in radians, in a pentagonal bipyramid (Johnson solid J_13).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 21 2025

This is the dihedral angle between triangular faces at the edge where the two pyramidal parts of the solid meet.

Also the dihedral angle between triangular faces in a pentagonal orthobicupola (Johnson solid J_30).

			1.3047162795687363719907812632876451487306158399...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Pentagonal bipyramid.
Wikipedia, Pentagonal orthobicupola.

Cf. A236367 (J_13 smallest dihedral angle).
Cf. other J_30 dihedral angles: A105199, A377995, A377996.
Cf. A179641 (J_13 volume), A120011 (J_13 surface area, divided by 10).
Cf. A384624 (J_30 volume), A384625 (J_30 surface area).
Cf. A010532, A386852.

First[RealDigits[ArcCos[(Sqrt[80] - 5)/15], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J13", "DihedralAngles"]], 10, 100]]

Equals arccos((4*sqrt(5) - 5)/15) = arccos((A010532 - 5)/15).

Equals 2*A386852.

A387190 Decimal expansion of the second smallest dihedral angle, in radians, in an elongated pentagonal cupola (Johnson solid J_20).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 22 2025

This is the dihedral angle between adjacent square faces at the edge where the prism and cupola parts of the solid meet.
Also the analogous dihedral angle in Johnson solids J_38-J_41.
Also the dihedral angle between a square face and a decagonal face in Johnson solids J_76-J_83.

			2.124370685691941870739854421729019962133608522388...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Elongated pentagonal cupola.

Cf. other J_20 dihedral angles: A019669, A228824, A377995, A377996, A387147.
Cf. A384144 (J_20 volume), A179591 (J_20 surface area - 10).
Cf. A002163.

First[RealDigits[ArcCos[-Sqrt[(5 - Sqrt[5])/10]], 10, 100]] (* or *)
First[RealDigits[RankedMin[Union[PolyhedronData["J20", "DihedralAngles"]], 2], 10, 100]]

Equals arccos(-sqrt((5 - sqrt(5))/10)) = arccos(-sqrt((5 - A002163)/10)).

A387607 Decimal expansion of the largest dihedral angle, in radians, in a gyroelongated pentagonal cupola (Johnson solid J_24).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Sep 04 2025

This is the dihedral angle between triangular faces in the antiprism part of the solid.
Also the analogous dihedral angle in a gyroelongated pentagonal rotunda, gyroelongated pentagonal bicupola, gyroelongated pentagonal cupolarotunda and gyroelongated pentagonal birotunda (Johnson solids J_25, J_46, J_47 and J_48, respectively).
Also the analogous dihedral angle in a decagonal antiprism.

			2.7783286661990213551074289190050780042508333364090...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Polytope Wiki, Decagonal antiprism.
Polytope Wiki, Gyroelongated pentagonal bicupola.
Polytope Wiki, Gyroelongated pentagonal birotunda.
Polytope Wiki, Gyroelongated pentagonal cupola.
Polytope Wiki, Gyroelongated pentagonal cupolarotunda.
Polytope Wiki, Gyroelongated pentagonal rotunda.
Wikipedia, Gyroelongated pentagonal bicupola.
Wikipedia, Gyroelongated pentagonal birotunda.
Wikipedia, Gyroelongated pentagonal cupola.
Wikipedia, Gyroelongated pentagonal cupolarotunda.
Wikipedia, Gyroelongated pentagonal rotunda.

Cf. other J_24 dihedral angles: A377995, A377996, A387608, A387609, A387610.
Cf. A384283 (J_24 volume), A384284 (J_24 surface area).
Cf. A384285 (J_25 volume), A384286 (J_25 surface area).
Cf. A385260 (J_46 volume), A385261 (J_46 surface area).
Cf. A385262 (J_47 volume), A385263 (J_47 surface area).
Cf. A385264 (J_48 volume), A385488 (J_48 surface area).
Cf. A010476.

First[RealDigits[ArcCos[(1 - Sqrt[10 + Sqrt[20]])/3], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["J24", "DihedralAngles"]], 10, 100]]

Equals arccos((1 - sqrt(10 + 2*sqrt(5)))/3) = arccos((1 - sqrt(10 + A010476))/3).

A387608 Decimal expansion of the fourth largest dihedral angle, in radians, in a gyroelongated pentagonal cupola (Johnson solid J_24).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Sep 04 2025

This is the dihedral angle between adjacent triangular faces at the edge where the antiprism and cupola parts of the solid meet.

Also the analogous dihedral angle in a gyroelongated pentagonal bicupola and gyroelongated pentagonal cupolarotunda (Johnson solids J_46 and J_47, respectively).

			2.314725687375130081473793791474182971134043297238...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Polytope Wiki, Gyroelongated pentagonal bicupola.
Polytope Wiki, Gyroelongated pentagonal cupola.
Polytope Wiki, Gyroelongated pentagonal cupolarotunda.
Wikipedia, Gyroelongated pentagonal bicupola.
Wikipedia, Gyroelongated pentagonal cupola.
Wikipedia, Gyroelongated pentagonal cupolarotunda.

Cf. other J_24 dihedral angles: A377995, A377996, A387607, A387609, A387610.
Cf. A384283 (J_24 volume), A384284 (J_24 surface area).
Cf. A385260 (J_46 volume), A385261 (J_46 surface area).
Cf. A385262 (J_47 volume), A385263 (J_47 surface area).
Cf. A002163, A002194, A010476, A386852.

First[RealDigits[ArcSec[Sqrt[15 - 6*#]] + ArcCos[(Sqrt[5 + 2*#] - # - 1)/Sqrt[3]] & [Sqrt[5]], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J24", "DihedralAngles"]],4], 10, 100]]

Equals arccos(sqrt((5 + 2*sqrt(5))/15)) + arccos((sqrt(5 + 2*sqrt(5)) - sqrt(5) - 1)/sqrt(3)) = arccos(sqrt((5 + A010476)/15)) + arccos((sqrt(5 + A010476) - A002163 - 1)/A002194).

Equals A386852 + A387610.

A387609 Decimal expansion of the fifth largest (second smallest) dihedral angle, in radians, in a gyroelongated pentagonal cupola (Johnson solid J_24).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Sep 05 2025

This is the dihedral angle between a triangular face and a square face at the edge where the antiprism and cupola parts of the solid meet.

Also the analogous dihedral angle in a gyroelongated pentagonal bicupola and gyroelongated pentagonal cupolarotunda (Johnson solids J_46 and J_47, respectively).

			2.2159419064878071469869358899196289168037591999759...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Polytope Wiki, Gyroelongated pentagonal bicupola.
Polytope Wiki, Gyroelongated pentagonal cupola.
Polytope Wiki, Gyroelongated pentagonal cupolarotunda.
Wikipedia, Gyroelongated pentagonal bicupola.
Wikipedia, Gyroelongated pentagonal cupola.
Wikipedia, Gyroelongated pentagonal cupolarotunda.

Cf. other J_24 dihedral angles: A377995, A377996, A387607, A387608, A387610.
Cf. A384283 (J_24 volume), A384284 (J_24 surface area).
Cf. A385260 (J_46 volume), A385261 (J_46 surface area).
Cf. A385262 (J_47 volume), A385263 (J_47 surface area).
Cf. A002163, A002194, A010476, A195693.

First[RealDigits[ArcTan[2]/2 + ArcCos[(Sqrt[5 + Sqrt[20]] - Sqrt[5] - 1)/Sqrt[3]], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J24", "DihedralAngles"]], 5], 10, 100]]

Equals arctan(2)/2 + arccos((sqrt(5 + 2*sqrt(5)) - sqrt(5) - 1)/sqrt(3)) = A195693 + arccos((sqrt(5 + A010476) - A002163 - 1)/A002194).

Equals A195693 + A387610.

A387610 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated pentagonal cupola (Johnson solid J_24).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Sep 05 2025

This is the dihedral angle between a triangular face and the decagonal face.
Also the analogous dihedral angle in a gyroelongated pentagonal rotunda (Johnson solid J_25).
Also the analogous dihedral angle in a decagonal antiprism.

			1.662367547590761895478403159830360396768735377275...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Polytope Wiki, Decagonal antiprism.
Polytope Wiki, Gyroelongated pentagonal cupola.
Polytope Wiki, Gyroelongated pentagonal rotunda.
Wikipedia, Gyroelongated pentagonal cupola.
Wikipedia, Gyroelongated pentagonal rotunda.

Cf. other J_24 dihedral angles: A377995, A377996, A387607, A387608, A387609.
Cf. A384283 (J_24 volume), A384284 (J_24 surface area).
Cf. A384285 (J_25 volume), A384286 (J_25 surface area).
Cf. A002163, A002194, A010476, A195693, A386852.

First[RealDigits[ArcCos[(Sqrt[5 + Sqrt[20]] - Sqrt[5] - 1)/Sqrt[3]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J24", "DihedralAngles"]], 10, 100]]

Equals arccos((sqrt(5 + 2*sqrt(5)) - sqrt(5) - 1)/sqrt(3)) = arccos((sqrt(5 + A010476) - A002163 - 1)/A002194).
Equals A387608 - A386852.
Equals A387609 - A195693.

cp's OEIS Frontend

A379711 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a disdyakis triacontahedron.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A379389 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a deltoidal hexecontahedron.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A377995 Decimal expansion of the dihedral angle, in radians, between square and pentagonal faces in a (small) rhombicosidodecahedron.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A387189 Decimal expansion of the smallest dihedral angle, in radians, in a pentagonal bipyramid (Johnson solid J_13).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A387190 Decimal expansion of the second smallest dihedral angle, in radians, in an elongated pentagonal cupola (Johnson solid J_20).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A387607 Decimal expansion of the largest dihedral angle, in radians, in a gyroelongated pentagonal cupola (Johnson solid J_24).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A387608 Decimal expansion of the fourth largest dihedral angle, in radians, in a gyroelongated pentagonal cupola (Johnson solid J_24).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A387609 Decimal expansion of the fifth largest (second smallest) dihedral angle, in radians, in a gyroelongated pentagonal cupola (Johnson solid J_24).

Views