A344075 -id:A344075 - 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).

A378977 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a triakis icosahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 14 2024

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

			2.8032178560848059621034493264877253281152659880354...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triakis Icosahedron.
Wikipedia, Triakis icosahedron.

Cf. A378973 (surface area), A378974 (volume), A378975 (inradius), A378976 (midradius).
Cf. A137218 and A344075 (dihedral angles of a truncated dodecahedron).
Cf. A002163.

First[RealDigits[ArcCos[-3*(8 + 5*Sqrt[5])/61], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["TriakisIcosahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-3*(8 + 5*sqrt(5))/61) = arccos(-3*(8 + 5*A002163)/61).

A379136 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a pentakis dodecahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 17 2024

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

			2.7352547614903346619898560183934957927169693...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentakis Dodecahedron.
Wikipedia, Pentakis dodecahedron.

Cf. A379132 (surface area), A379133 (volume), A379134 (inradius), A379135 (midradius).
Cf. A236367 and A344075 (dihedral angles of a truncated icosahedron).
Cf. A002163.

First[RealDigits[ArcCos[-(80 + 9*Sqrt[5])/109], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["PentakisDodecahedron", "DihedralAngles"]], 10, 100]]

acos(-(80 + 9*sqrt(5))/109) \\ Charles R Greathouse IV, Feb 05 2025

Equals arccos(-(80 + 9*sqrt(5))/109) = arccos(-(80 + 9*A002163)/109).

A386530 Decimal expansion of the largest dihedral angle, in radians, in an elongated pentagonal rotunda (Johnson solid J_21).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 22 2025

This is the dihedral angle between a triangular face and a square face (at the edge where the prism and rotunda parts of the solid meet).

Also the analogous dihedral angle in Johnson solids J_40-J_43.

			2.9528821228062311686815089830968947118603985336982...

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

Cf. other J_21 dihedral angles: A019669, A228824, A344075, A387191.
Cf. A384213 (J_21 volume), A179637 (J_21 surface area - 10).
Cf. A002163.

First[RealDigits[ArcCos[-Sqrt[2*(5 + Sqrt[5])/15]], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["J21", "DihedralAngles"]], 10, 100]]

Equals arccos(-sqrt(2*(5 + sqrt(5))/15)) = arccos(-sqrt(2*(5 + A002163)/15)).

A387191 Decimal expansion of the second largest dihedral angle, in radians, in an elongated pentagonal rotunda (Johnson solid J_21).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 22 2025

This is the dihedral angle between a square face and a pentagonal face.

Also one of the dihedral angles in Johnson solids J_40-J_43, J_72-J_75, J_77-J_79 and J_82.

			2.677945044588987122248387151818288482168632345...

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

Cf. other J_21 dihedral angles: A019669, A228824, A344075, A386530.
Cf. A384213 (J_21 volume), A179637 (J_21 surface area - 10).
Cf. A010476, A105199.

First[RealDigits[Pi/2 + ArcTan[2], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J21", "DihedralAngles"]], 2], 10, 100]]

Equals Pi/2 + arctan(2) = A019669 + A105199.

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

A386853 Decimal expansion of the dihedral angle, in radians, between the 10-gonal face and a triangular face in a pentagonal rotunda (Johnson solid J_6).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 06 2025

			1.38208579601133454945018729145714326976181383401...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Pentagonal rotunda.
Index entries for transcendental numbers.

Cf. A179593 (volume), A179637 (surface area).
Cf. other J_6 dihedral angles: A105199, A344075.
Cf. A010476, A386852.

First[RealDigits[ArcCos[Sqrt[(5 - Sqrt[20])/15]], 10, 100]] (* or *)
First[RealDigits[RankedMin[Union[PolyhedronData["J6", "DihedralAngles"]], 2], 10, 100]]

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

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

cp's OEIS Frontend

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

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A378977 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a triakis icosahedron.

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A379136 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a pentakis dodecahedron.

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A386530 Decimal expansion of the largest dihedral angle, in radians, in an elongated pentagonal rotunda (Johnson solid J_21).

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A387191 Decimal expansion of the second largest dihedral angle, in radians, in an elongated pentagonal rotunda (Johnson solid J_21).

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A386853 Decimal expansion of the dihedral angle, in radians, between the 10-gonal face and a triangular face in a pentagonal rotunda (Johnson solid J_6).

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