A378354 -id:A378354 - cp's OEIS frontend

A378351 Decimal expansion of the surface area of a (small) triakis octahedron with unit shorter edge length.

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

Offset: 2

Paolo Xausa, Nov 23 2024

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

			10.672941873983546705150008922490160564590104237715...

Eric Weisstein's World of Mathematics, Small Triakis Octahedron.
Wikipedia, Triakis octahedron.

Cf. A378352 (volume), A378353 (inradius), A201488 (midradius), A378354 (dihedral angle).
Cf. A377298 (surface area of a truncated cube with unit edge).
Cf. A010487.

First[RealDigits[3*Sqrt[7 + Sqrt[32]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisOctahedron", "SurfaceArea"], 10, 100]]

Equals 3*sqrt(7 + 4*sqrt(2)) = 3*sqrt(7 + A010487).

A378352 Decimal expansion of the volume of a (small) triakis octahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 23 2024

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

			2.9142135623730950488016887242096980785696718753769...

Eric Weisstein's World of Mathematics, Small Triakis Octahedron.
Wikipedia, Triakis octahedron.

Cf. A378351 (surface area), A378353 (inradius), A201488 (midradius), A378354 (dihedral angle).
Cf. A377299 (volume of a truncated cube with unit edge).
Cf. A156035.
Essentially the same as A002193 and A188582.

First[RealDigits[Sqrt[2] + 3/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisOctahedron", "Volume"], 10, 100]]

Equals sqrt(2) + 3/2 = A002193 + 3/2.

Equals A156035/2. - Hugo Pfoertner, Nov 24 2024

A378353 Decimal expansion of the inradius of a (small) triakis octahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Nov 23 2024

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

			0.81914066340325716171549134573565316624155520306132...

Cf. A378351 (surface area), A378352 (volume), A201488 (midradius), A378354 (dihedral angle).

Cf. A010487.

First[RealDigits[Sqrt[23/68 + Sqrt[32]/17], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisOctahedron", "Inradius"], 10, 100]]

Equals sqrt(23/68 + 4*sqrt(2)/17) = sqrt(23/68 + A010487/17).

A380702 Decimal expansion of the acute vertex angles, in radians, in a (small) triakis octahedron face.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jan 30 2025

			0.54802840762031274453086328282062867847971236365920...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Triakis octahedron.

Cf. A380703 (face obtuse angle).

Cf. A020765, A361601, A378351, A378352, A378353, A378354.

First[RealDigits[ArcCos[1/2 + Sqrt[2]/4], 10, 100]]

Equals arccos(1/2 + sqrt(2)/4) = arccos(1/2 + A020765).
Equals (Pi - A380703)/2.
Equals A361601/2. - Hugo Pfoertner, Jan 30 2025

A380703 Decimal expansion of the obtuse vertex angle, in radians, in a (small) triakis octahedron face.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jan 30 2025

			2.045535838349167749400916817638245527237744672...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Triakis octahedron.

Cf. A380702 (face obtuse angles).

Cf. A010503, A378351, A378352, A378353, A378354.

First[RealDigits[ArcCos[1/4 - Sqrt[2]/2], 10, 100]]

Equals arccos(1/4 - sqrt(2)/2) = arccos(1/2 + A010503).

Equals Pi - 2*A380702.

cp's OEIS Frontend

A378351 Decimal expansion of the surface area of a (small) triakis octahedron with unit shorter edge length.

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A378352 Decimal expansion of the volume of a (small) triakis octahedron with unit shorter edge length.

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A378353 Decimal expansion of the inradius of a (small) triakis octahedron with unit shorter edge length.

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A380702 Decimal expansion of the acute vertex angles, in radians, in a (small) triakis octahedron face.

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A380703 Decimal expansion of the obtuse vertex angle, in radians, in a (small) triakis octahedron face.

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