A195698 -id:A195698 - cp's OEIS frontend

A195695 Decimal expansion of arcsin(sqrt(1/3)) and of arccos(sqrt(2/3)).

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

Offset: 0

Clark Kimberling, Sep 23 2011

The complementary magic angle, that is, Pi/2 - A195696. The angle between the body-diagonal and a congruent face-diagonal of a cube. And also the polar angle of the cone circumscribed to a regular tetrahedron from one of its vertices. - Stanislav Sykora, Nov 21 2013

This is the value of the angle of the circular cone to the axis, that maximizes the volume of the cone enclosed by a given area. See the +plus link. - Michel Marcus, Aug 27 2017

			arcsin(sqrt(1/3)) = 0.61547970867038734106746458912399...

G. C. Greubel, Table of n, a(n) for n = 0..5000
John D. Barrow, Outer space: Archimedean ice cream cones, +plus magazine.
Wikipedia, Polyhedron, and further links therein.
Index entries for transcendental numbers

Cf. A195696 (magic angle), A195698, A020760, A157697, A243445.

[Arcsin(Sqrt(1/3))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[1/3];
N[ArcSin[r], 100]
RealDigits[%]  (* A195695 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A195696 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A019673 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195698 *)

atan(1/sqrt(2)) \\ Michel Marcus, Aug 27 2017

Also equals arctan(1/sqrt(2)). - Michel Marcus, Aug 27 2017

A378715 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a disdyakis dodecahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 07 2024

The disdyakis dodecahedron is the dual polyhedron of the truncated cuboctahedron (great rhombicuboctahedron).

			2.7066946454792287856258644383068280456984454555717...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Disdyakis dodecahedron.

Cf. A378712 (surface area), A378713 (volume), A378714 (inradius), A378393 (midradius).
Cf. A177870, A195698 and A195702 (dihedral angles of a truncated cuboctahedron (great rhombicuboctahedron)).
Cf. A002193.

First[RealDigits[ArcCos[-(71 + 12*Sqrt[2])/97], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["DisdyakisDodecahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-(71 + 12*sqrt(2))/97) = arccos(-(71 + 12*A002193)/97).

A378354 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a (small) triakis octahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 24 2024

The (small) triakis octahedron is the dual polyhedron of the truncated cube.

			2.57174440034566846791285405092806379355115694111...

Wikipedia, Triakis octahedron.

Cf. A378351 (surface area), A378352 (volume), A378353 (inradius), A201488 (midradius).
Cf. A019669 and A195698 (dihedral angles of a truncated cube).
Cf. A377342.

First[RealDigits[ArcCos[-(3 + 8*Sqrt[2])/17], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["TriakisOctahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-(3 + 8*sqrt(2))/17) = arccos(-(3 + A377342)/17).

A387294 Decimal expansion of the largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 25 2025

This is the dihedral angle between a triangular face in the antiprism part of the solid and a triangular face in the cupola part of the solid.

Also the analogous dihedral angle in a gyroelongated triangular bicupola (Johnson solid J_44).

			2.9570800796354481515618725813450376530518087004...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated triangular bicupola.
Wikipedia, Gyroelongated triangular cupola.

Cf. other J_22 dihedral angles: A195698, A387295, A387296, A387297.
Cf. A344076 (J_22 volume), A344077 (J_22 surface area).
Cf. A385256 (J_44 volume), A385257 (J_44 surface area).
Cf. A137914, A246724.

First[RealDigits[ArcSec[3] + ArcCos[1 - Sqrt[12]/3], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["J22", "DihedralAngles"]], 10, 100]]

Equals arccos(1/3) + arccos(1 - 2*sqrt(3)/3) = A137914 + arccos(-A246724).

A387295 Decimal expansion of the second largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 25 2025

This is the dihedral angle between a triangular face in the antiprism part of the solid and a square face in the cupola part of the solid.

Also the analogous dihedral angle in a gyroelongated triangular bicupola (Johnson solid J_44).

			2.6814372804191827475908005056128080315848833860639...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated triangular bicupola.
Wikipedia, Gyroelongated triangular cupola.

Cf. other J_22 dihedral angles: A195698, A387294, A387296, A387297.
Cf. A344076 (J_22 volume), A344077 (J_22 surface area).
Cf. A385256 (J_44 volume), A385257 (J_44 surface area).
Cf. A195696, A246724.

First[RealDigits[ArcTan[Sqrt[2]] + ArcCos[1 - Sqrt[12]/3], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J22", "DihedralAngles"]], 2], 10, 100]]

Equals arccos(sqrt(3)/3) + arccos(1 - 2*sqrt(3)/3) = A195696 + arccos(-A246724).

A387296 Decimal expansion of the third largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 26 2025

This is the dihedral angle between triangular faces in the antiprism part of the solid.

Also the analogous dihedral angle in a gyroelongated triangular bicupola (Johnson solid J_44).

			2.5346001497151261930915028102102107021498303291935...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated triangular bicupola.
Wikipedia, Gyroelongated triangular cupola.

Cf. other J_22 dihedral angles: A195698, A387294, A387295, A387297.
Cf. A344076 (J_22 volume), A344077 (J_22 surface area).
Cf. A385256 (J_44 volume), A385257 (J_44 surface area).
Cf. A010469.

First[RealDigits[ArcCos[(1 - Sqrt[12])/3], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J22", "DihedralAngles"]], 3], 10, 100]]

Equals arccos((1 - 2*sqrt(3))/3) = arccos((1 - A010469)/3).

A387297 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 26 2025

This is the dihedral angle between a triangular face and the hexagonal face.

			1.7261206622946734694269434030970502773414686910539...

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

Cf. other J_22 dihedral angles: A195698, A387294, A387295, A387296.
Cf. A344076 (J_22 volume), A344077 (J_22 surface area).
Cf. A010469.

First[RealDigits[ArcCos[1 - 2/Sqrt[3]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J22", "DihedralAngles"]], 10, 100]]

Equals arccos(1 - 2*sqrt(3)/3) = arccos(1 - A010469/3).

A378389 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a tetrakis hexahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 27 2024

The tetrakis hexahedron is the dual polyhedron of the truncated octahedron.

			2.498091544796508851659834154562180246155658808...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Tetrakis hexahedron.

Cf. A378388 (surface area), A374359 (volume - 1), A010532 (inradius*10), A179587 (midradius + 1).

Cf. A156546 and A195698 (dihedral angles of a truncated octahedron), A195729.

First[RealDigits[ArcCos[-4/5], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["TetrakisHexahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-4/5).

Equals 2*A195729. - Amiram Eldar, Nov 27 2024

cp's OEIS Frontend

A195695 Decimal expansion of arcsin(sqrt(1/3)) and of arccos(sqrt(2/3)).

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A378715 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a disdyakis dodecahedron.

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A378354 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a (small) triakis octahedron.

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A387294 Decimal expansion of the largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

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A387295 Decimal expansion of the second largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

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A387296 Decimal expansion of the third largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

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A387297 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

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