A377275 -id:A377275 - cp's OEIS frontend

A381684 Decimal expansion of the isoperimetric quotient of a truncated tetrahedron.

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

Offset: 0

Paolo Xausa, Mar 04 2025

Polya (1954) defines the isoperimetric quotient of a solid as 36*Pi*V^2/S^3, where V and S are the volume and surface area of the solid, respectively.

The isoperimetric quotient of a sphere is 1.

			0.4662292826432950646084875599089894958106273300491...

George Polya, Mathematics and Plausible Reasoning, Vol. 1: Induction and Analogy in Mathematics, Princeton University Press, Princeton, New Jersey, 1954. See pp. 188-189, exercise 43.

Paolo Xausa, Table of n, a(n) for n = 0..10000
P. K. Aravind, How Spherical Are the Archimedean Solids and Their Duals?, The College Mathematics Journal, Vol. 42, Issue 2, 2011, pp. 98-107.
Index entries for transcendental numbers.

Cf. A377274 (surface area), A377275 (volume).

Cf. A000796, A002194.

First[RealDigits[529*Pi/(2058*Sqrt[3]), 10, 100]]

529*Pi/2058/sqrt(3) \\ Charles R Greathouse IV, Aug 19 2025

Equals 36*Pi*A377275^2/(A377274^3).

Equals 529*Pi/(2058*sqrt(3)) = 529*A000796/(2058*A002194).

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.

A377274 Decimal expansion of the surface area of a truncated tetrahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 23 2024

			12.12435565298214105469212439054110656859963677667...

Eric Weisstein's World of Mathematics, Truncated Tetrahedron.
Wikipedia, Truncated tetrahedron.

Cf. A377275 (volume), A377276 (circumradius), A093577 (midradius), A377277 (Dehn invariant).

Cf. A002194 (analogous for a regular tetrahedron).

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

Equals 7*sqrt(3) = 7*A002194.

A386411 Decimal expansion of the volume of an augmented truncated tetrahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 21 2025

The augmented truncated tetrahedron is Johnson solid J_65.

			3.889087296526011384204643991576669716066597657...

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

Cf. A386412 (surface area).

Cf. A010466, A020829, A377275.

First[RealDigits[11/Sqrt[8], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J65", "Volume"], 10, 100]]

Equals 11/(2*sqrt(2)) = 11/A010466.
Equals A377275 + 10*A020829.
Equals the largest root of 8*x^2 - 121.

A377276 Decimal expansion of the circumradius of a truncated tetrahedron with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Oct 23 2024

			1.17260393995585738864140752838611657014705708835...

Eric Weisstein's World of Mathematics, Truncated Tetrahedron.
Wikipedia, Truncated tetrahedron.

Cf. A377274 (surface area), A377275 (volume), A093577 (midradius), A377277 (Dehn invariant).
Cf. A187110 (analogous for a regular tetrahedron).
Cf. A010478.

First[RealDigits[Sqrt[22]/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedTetrahedron", "Circumradius"], 10, 100]]

Equals sqrt(22)/4 = A010478/4.

A379732 Decimal expansion of 207/208.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Dec 31 2024

Conjectured densest packing of truncated tetrahedra.

			0.995192307692307692307692307692307692307692307692...

Pablo F. Damasceno, Michael Engel, and Sharon C. Glotzer, Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces, arXiv:1109.1323 [cond-mat.soft], 2011.
Yang Jiao and Sal Torquato, Analytical Construction of A Dense Packing of Truncated Tetrahedra, arXiv:1107.2300 [cond-mat.soft], 2011.
Wikipedia, Truncated tetrahedron.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A374772, A377274, A377275.

Mathematica
```
First[RealDigits[207/208, 10, 100]]
```

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A381684 Decimal expansion of the isoperimetric quotient of a truncated tetrahedron.

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

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A377274 Decimal expansion of the surface area of a truncated tetrahedron with unit edge length.

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A386411 Decimal expansion of the volume of an augmented truncated tetrahedron with unit edge.

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A377276 Decimal expansion of the circumradius of a truncated tetrahedron with unit edge length.

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A379732 Decimal expansion of 207/208.

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