A093577 -id:A093577 - cp's OEIS frontend

A378393 Decimal expansion of the midradius of a deltoidal icositetrahedron with unit shorter edge length.

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

Offset: 1

Paolo Xausa, Nov 30 2024

The deltoidal icositetrahedron is the dual polyhedron of the (small) rhombicuboctahedron.

			1.5606601717798212866012665431572735589272539065327...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2.

Cf. A378390 (surface area), A378391 (volume), A378392 (inradius), A378394 (dihedral angle).
Cf. A285871 (midradius of a (small) rhombicuboctahedron with unit edge).
Cf. A010474, A093577.

First[RealDigits[(2 + Sqrt[18])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["DeltoidalIcositetrahedron", "Midradius"], 10, 100]]

Equals (2 + 3*sqrt(2))/4 = (2 + A010474)/4.

A378207 Decimal expansion of the midradius of a triakis tetrahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Nov 21 2024

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

			0.589255650988789603667370301754040866070696614740...

Index entries for algebraic numbers, degree 2.

Cf. A378204 (surface area), A378205 (volume), A378206 (inradius), A378208 (dihedral angle).
Cf. A093577 (midradius of a truncated tetrahedron with unit edge).
Cf. A010524.

First[RealDigits[5/Sqrt[72], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisTetrahedron", "Midradius"], 10, 100]]

5/sqrt(72) \\ Charles R Greathouse IV, Feb 11 2025

Equals 5/(6*sqrt(2)) = 5/A010524.

A377275 Decimal expansion of the volume of a truncated tetrahedron with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Oct 23 2024

			2.7105759945484321768699033880685879839252044278...

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

Cf. A377274 (surface area), A377276 (circumradius), A093577 (midradius), A377277 (Dehn invariant).
Cf. A020829 (analogous for a regular tetrahedron).
Cf. A002193.

First[RealDigits[23/12*Sqrt[2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedTetrahedron", "Volume"], 10, 100]]

Equals (23/12)*sqrt(2) = (23/12)*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.

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.

A243309 Decimal expansion of DeVicci's tesseract constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 03 2014

This "tesseract" constant is the edge length of the largest 3-dimensional cube that can be inscribed within a unit 4-dimensional cube.
From Amiram Eldar, May 29 2021: (Start)
Named by Finch (2003) after Kay R. Pechenick DeVicci Shultz.
The problem was apparently first posed by Gardner (1966). According to Gardner (2001), he had received the correct answers to the problem from Eugen I. Bosch (1966), G. de Josselin de Jong (1971), Hermann Baer (1974) and Kay R. Pechenick (1983). (End)

			1.00743475688427937609825359523109914192569...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.14 DeVicci's tesseract constant, p. 524.
Martin Gardner, Is It Possible to Visualize a Four-Dimensional Figure?, Mathematical Games, Sci. Amer., Vol. 215, No. 5, (Nov. 1966), pp. 138-143.
Martin Gardner, Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American, New York: Vintage Books, 1977, Chapter 4, "Hypercubes", pp. 41-54.
Martin Gardner, The Colossal Book of Mathematics, New York, London: W. W. Norton & Co., 2001, Chapter 13, "Hypercubes", pp. 162-174.

Hallard T. Croft, Kenneth Falconer and Richard K. Guy, Unsolved Problems in Geometry, Springer-Verlag New York, 1991, Section B4, p. 53.
Richard K. Guy and Richard J. Nowakowski, Monthly Unsolved Problems, 1969-1997, The American Mathematical Monthly, Vol. 104, No. 10 (1997), pp. 967-973.
Greg Huber, Kay Pechenick Shultz and John E. Wetzel, The n-cube is Rupert, The American Mathematical Monthly, Vol. 125, No. 6 (2018), pp. 505-512.
Kay R. Pechenick DeVicci Shultz, Largest m-Cube in an n-Cube: Partial Solution, Notes written in 1996 and assembled in 2013 with a preface by Greg Huber, KITP preprint NSF-ITP-13-142.
Eric Weisstein's World of Mathematics, Prince Rupert's Cube.
Wikipedia, Prince Rupert's cube.
Index entries for algebraic numbers, degree 8

Cf. A093577.

Root[4*x^8 - 28*x^6 - 7*x^4 + 16*x^2 + 16, x, 3] // RealDigits[#, 10, 103]& // First

polrootsreal(4*x^8-28*x^6-7*x^4+16*x^2+16)[3] \\ Charles R Greathouse IV, Apr 07 2016

sqrt(polrootsreal(Pol([4,-28,-7,16,16]))[1]) \\ Charles R Greathouse IV, Apr 07 2016

Positive root of the polynomial 4*x^8 - 28*x^6 - 7*x^4 + 16*x^2 + 16.

A243313 Decimal expansion of a 5-dimensional analog of DeVicci's tesseract constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 03 2014

This constant is the edge length of the largest 3-dimensional cube that can be inscribed within a unit 5-dimensional cube.

Also, the smallest positive root in x^4 - 22*x^2 + 25 = 0.

			1.0963763171773128040759311069135237901965384969435155182975524965...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.14 DeVicci's tesseract constant, p. 525.

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Eric Weisstein's MathWorld, Prince Rupert's Cube.

Cf. A093577, A243309.

RealDigits[Sqrt[11-4*Sqrt[6]], 10, 104] // First

2*sqrt(2)-sqrt(3) \\ Stefano Spezia, Dec 24 2024

Equals sqrt(11-4*sqrt(6)) = 2*sqrt(2)-sqrt(3).

cp's OEIS Frontend

A378393 Decimal expansion of the midradius of a deltoidal icositetrahedron with unit shorter edge length.

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A378207 Decimal expansion of the midradius of a triakis tetrahedron with unit shorter edge length.

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A377275 Decimal expansion of the volume of a truncated tetrahedron with unit 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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A377276 Decimal expansion of the circumradius of a truncated tetrahedron with unit edge length.

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A243309 Decimal expansion of DeVicci's tesseract constant.

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A243313 Decimal expansion of a 5-dimensional analog of DeVicci's tesseract constant.

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