A387322 -id:A387322 - cp's OEIS frontend

A387320 Decimal expansion of the largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

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

Offset: 1

Paolo Xausa, Aug 27 2025

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

Also the analogous dihedral angle in a gyroelongated square bicupola (Johnson solid J_45).

			2.687150505637070622058237671034217872408094243788...

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

Cf. other J_23 dihedral angles: A177870, A195702, A387321, A387322, A387323.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A385258 (J_45 volume), A385259 (J_45 surface area).
Cf. A002193.

First[RealDigits[ArcCos[(1 - Sqrt[8 + Sqrt[32]])/3], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["J23", "DihedralAngles"]], 10, 100]]

Equals arccos((1 - 2*sqrt(2 + sqrt(2)))/3) = arccos((1 - 2*sqrt(2 + A002193))/3).

A387321 Decimal expansion of the second largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 27 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 square bicupola (Johnson solid J_45).

			2.6412090003740395440214510528511358326798716782548...

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

Cf. other J_23 dihedral angles: A177870, A195702, A387320, A387322, A387323.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A385258 (J_45 volume), A385259 (J_45 surface area).
Cf. A002193, A010487, A195696.

First[RealDigits[ArcSec[Sqrt[3]] + ArcCos[-Sqrt[(7 + Sqrt[32] - 2*Sqrt[20 + 14*Sqrt[2]])/3]], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J23", "DihedralAngles"]],2], 10, 100]]

Equals arcsec(sqrt(3)) + arccos(-sqrt((7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)))/3)) = A195696 + arccos(-sqrt((7 + A010487 - 2*sqrt(20 + 14*A002193))/3)).

Equals A195696 + A387323.

A387323 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 29 2025

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

			1.6858923822495302658575939503353780784364569832448...

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

Cf. other J_23 dihedral angles: A177870, A195702, A387320, A387321, A387322.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A002193, A003881, A010487, A195696.

First[RealDigits[ArcCos[-Sqrt[(7 + Sqrt[32] - 2*Sqrt[20 + 14*Sqrt[2]])/3]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J23", "DihedralAngles"]], 10, 100]]

Equals arccos(-sqrt((7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)))/3)) = arccos(-sqrt((7 + A010487 - 2*sqrt(20 + 14*A002193))/3)).

Equals A387321 - A195696 = A387322 - A003881.

cp's OEIS Frontend

A387320 Decimal expansion of the largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

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A387321 Decimal expansion of the second largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

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A387323 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

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