A378207 -id:A378207 - cp's OEIS frontend

A020829 Decimal expansion of 1/sqrt(72) = 1/(3*2^(3/2)) = sqrt(2)/12.

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

Offset: 0

N. J. A. Sloane

Volume of regular tetrahedron with unit edge. - Stanislav Sykora, May 31 2012
In the dragon curve fractal, (5/6)*sqrt(2) = 1.1785.... is the maximum distance of any point from curve start.  Such a maximum must be to a vertex of the convex hull.  Hull vertices are shown by Benedek and Panzone (theorem 3, page 85) and their P8 = 7/6 - (1/6)i at distance sqrt((7/6)^2 + (1/6)^2) is the maximum. - Kevin Ryde, Nov 22 2019
With offset 1, volume of a triangular cupola (Johnson solid J_3) with unit edges. - Paolo Xausa, Aug 04 2025

			0.117851130197757920733474...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Agnes I. Benedek and Rafael Panzone, On Some Notable Plane Sets, II: Dragons, Revista de la Unión Matemática Argentina, volume 39, numbers 1-2, 1994, pages 76-90.
Wikipedia, Platonic solid.
Wikipedia, Tetrahedron.
Wikipedia, Triangular cupola.
Index entries for algebraic numbers, degree 2

Cf. A131594 (regular octahedron volume), A102208 (regular icosahedron volume), A102769 (regular dodecahedron volume).

Cf. A010524, A020765, A020775, A378207.

RealDigits[Sqrt[2]/12, 10, 50][[1]] (* G. C. Greubel, Jul 06 2017 *)

sqrt(2)/12 \\ G. C. Greubel, Jul 06 2017

Equals Integral_{x=0..Pi/4} sin(x)^2 * cos(x) dx. - Amiram Eldar, May 31 2021

Equals 1/A010524 = A020765/3 = A020775/2 = A378207/5. - Hugo Pfoertner, Jan 26 2025

A378204 Decimal expansion of the surface area of a triakis tetrahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 20 2024

The triakis tetrahedron is the dual polyhedron of the truncated tetrahedron.

			5.5277079839256664151915545611178111398784809093...

Eric Weisstein's World of Mathematics, Triakis Tetrahedron.
Wikipedia, Triakis tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A378205 (volume), A378206 (inradius), A378207 (midradius), A378208 (dihedral angle).
Cf. A377274 (surface area of a truncated tetrahedron with unit edge).
Cf. A010468.

First[RealDigits[5*Sqrt[11]/3, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisTetrahedron", "SurfaceArea"], 10, 100]]

Equals (5/3)*sqrt(11) = (5/3)*A010468.

A378205 Decimal expansion of the volume of a triakis tetrahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Nov 20 2024

The triakis tetrahedron is the dual polyhedron of the truncated tetrahedron.

			0.9820927516479826727789505029234014434511610245673...

Eric Weisstein's World of Mathematics, Triakis Tetrahedron.
Wikipedia, Triakis tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A378204 (surface area), A378206 (inradius), A378207 (midradius), A378208 (dihedral angle).
Cf. A377275 (volume of a truncated tetrahedron with unit edge).
Cf. A002193.

First[RealDigits[25/36*Sqrt[2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisTetrahedron", "Volume"], 10, 100]]

Equals (25/36)*sqrt(2) = (25/36)*A002193.

A378206 Decimal expansion of the inradius of a triakis tetrahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Nov 21 2024

The triakis tetrahedron is the dual polyhedron of the truncated tetrahedron.

			0.53300179088902608574609433108459844097593504016042...

Index entries for algebraic numbers, degree 2.

Cf. A378204 (surface area), A378205 (volume), A378207 (midradius), A378208 (dihedral angle).

Cf. A010539.

First[RealDigits[5/Sqrt[88], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisTetrahedron", "Inradius"], 10, 100]]

Equals 5/(2*sqrt(22)) = 5/A010539.

A378208 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a triakis tetrahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 21 2024

The triakis tetrahedron is the dual polyhedron of the truncated tetrahedron.

			2.2605713275803962793413578116086559655552841805381...

Wikipedia, Triakis tetrahedron.
Index entries for transcendental numbers.

Cf. A378204 (surface area), A378205 (volume), A378206 (inradius), A378207 (midradius).

Cf. A137914 and A156546 (dihedral angles of a truncated tetrahedron).

First[RealDigits[ArcCos[-7/11], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["TriakisTetrahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-7/11).

A380700 Decimal expansion of the acute vertex angles, in radians, in a triakis tetrahedron face.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jan 30 2025

			0.58568554345715095961775753847751776620036106717164...

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

Cf. A380701 (face obtuse angle).

Cf. A378204, A378205, A378206, A378207, A378208.

Mathematica
```
First[RealDigits[ArcCos[5/6], 10, 100]]
```

Equals arccos(5/6).

Equals (Pi - A380701)/2.

A380701 Decimal expansion of the obtuse vertex angle, in radians, in a triakis tetrahedron face.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jan 30 2025

			1.9702215666754913192271283063244673517964472650318...

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

Cf. A380700 (face acute angles).

Cf. A378204, A378205, A378206, A378207, A378208.

First[RealDigits[ArcCos[-7/18], 10, 100]]

Equals arccos(-7/18).

Equals Pi - 2*A380700.

cp's OEIS Frontend

A020829 Decimal expansion of 1/sqrt(72) = 1/(3*2^(3/2)) = sqrt(2)/12.

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A378204 Decimal expansion of the surface area of a triakis tetrahedron with unit shorter edge length.

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A378205 Decimal expansion of the volume of a triakis tetrahedron with unit shorter edge length.

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A378206 Decimal expansion of the inradius of a triakis tetrahedron with unit shorter edge length.

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A378208 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a triakis tetrahedron.

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A380700 Decimal expansion of the acute vertex angles, in radians, in a triakis tetrahedron face.

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A380701 Decimal expansion of the obtuse vertex angle, in radians, in a triakis tetrahedron face.

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