cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Christof Weber

Christof Weber's wiki page.

Christof Weber has authored 4 sequences.

A137617 Decimal expansion of volume of the solid of revolution generated by a Reuleaux triangle rotated around one of its symmetry axes.

Original entry on oeis.org

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

Views

Author

Christof Weber, Feb 04 2008

Keywords

Comments

The rotated Reuleaux triangle is not only a body of constant width, it is the minimum volume surface of revolution with constant width (Campi et al. 1996).

Examples

			0.44946103...

References

  • St. Campi, A. Colesanti and P. Gronchi, Minimum problems for volumes of convex bodies, Partial Differential Equations and Applications - Collected Papers in Honor of Carlo Pucci, Marcel Dekker (1996), pp. 43-55.

Links

  • Bernd Kawohl and Christof Weber, Meissner's Mysterious Bodies, Mathematical Intelligencer, Volume 33, Number 3, 2011, pp. 94-101.
  • SwissEduc: Teaching and Learning Mathematics, Bodies of Constant Width (with informations on bodies of constant width like the rotated Reuleaux Triangle and others)

Crossrefs

Cf. A102888, A137615, A137616, A137618.

Programs

  • Mathematica
    k1[x_] := Sqrt[1 - (x - Sqrt[3]/2)^2]; k2[x_] := Sqrt[1 - x^2] - 1/2; Pi * Integrate[k1[x]^2, {x, Sqrt[3]/2 - 1, 0}] + Pi * Integrate[k2[x]^2, {x, 0, Sqrt[3]/2}]

Formula

2/3 * Pi - Pi^2 / 6

Extensions

Link corrected by Christof Weber, Jan 06 2013

A137618 Decimal expansion of surface area of the solid of revolution generated by a Reuleaux triangle rotated around one of its symmetry axes.

Original entry on oeis.org

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

Views

Author

Christof Weber, Feb 04 2008

Keywords

Comments

The rotated Reuleaux triangle is not only a surface of constant width, it is the minimum area surface of revolution with constant width (Campi et al. 1996).

Examples

			2.99331717348313360398045643326695538995643899633661...

References

  • St. Campi, A. Colesanti and P. Gronchi, Minimum problems for volumes of convex bodies, Partial Differential Equations and Applications - Collected Papers in Honor of Carlo Pucci, Marcel Dekker (1996), pp. 43-55.

Links

Crossrefs

Cf. A102888, A137615, A137616, A137617.

Programs

  • Mathematica
    k1[x_] := Sqrt[1 - (x - Sqrt[3]/2)^2]; k2[x_] := Sqrt[1 - x^2] - 1/2; 2*Pi*Integrate[k1[x]*Sqrt[1+D[k1[x],x]^2],{x,Sqrt[3]/2-1,0}] + 2*Pi*Integrate[k2[x]*Sqrt[1+D[k2[x],x]^2], {x, 0, Sqrt[3]/2}]
RealDigits[2*Pi - Pi^2/3, 10, 120][[1]] (* Amiram Eldar, May 22 2023 *)

Formula

Equals 2*Pi - Pi^2 /3.

A137616 Decimal expansion of surface area of the Meissner Body.

Original entry on oeis.org

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

Views

Author

Christof Weber, Feb 04 2008

Keywords

Comments

The Meissner body is a three-dimensional generalization of the Reuleaux triangle having constant width 1. Although it is based on the Reuleaux tetrahedron, it is different from that. The Meissner body exists in two different versions.

Examples

			2.93411519432335593899268892412301181446158693642737...

References

  • Johannes Boehm and E. Quaisser, Schoenheit und Harmonie geometrischer Formen - Sphaeroformen und symmetrische Koerper, Berlin: Akademie Verlag (1991), S. 71.
  • G. D. Chakerian and H. Groemer, Convex Bodies of Constant Width, in: P. Gruber and J. Wills (Editors), Convexity and its Applications, Basel / Boston / Stuttgart: Birkhäuser (1983), p. 68.

Links

  • Bernd Kawohl and Christof Weber, Meissner's Mysterious Bodies, Mathematical Intelligencer, Volume 33, Number 3, 2011, pp. 94-101.
  • SwissEduc: Teaching and Learning Mathematics, Bodies of Constant Width (with information, animations and interactive pictures of both Meissner bodies).

Crossrefs

Cf. A102888, A137615, A137617, A137618.

Programs

  • Mathematica
    RealDigits[(2 - Sqrt[3]/2 * ArcCos[1/3])* Pi, 10, 120][[1]] (* Amiram Eldar, May 27 2023 *)

Formula

Equals (2 - sqrt(3)/2 * arccos(1/3)) * Pi.

Extensions

Link corrected by Christof Weber, Jan 06 2013

A137615 Decimal expansion of volume of the Meissner Body.

Original entry on oeis.org

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

Views

Author

Christof Weber, Feb 04 2008

Keywords

Comments

The Meissner body is a three-dimensional generalization of the Reuleaux triangle having constant width 1. Although it is based on the Reuleaux tetrahedron, it is different from that. The Meissner body exists in two different versions.

Examples

			0.41986004596508022334213000096833827916507033508865...

References

  • Johannes Boehm and E. Quaisser, Schoenheit und Harmonie geometrischer Formen - Sphaeroformen und symmetrische Koerper, Berlin: Akademie Verlag (1991), p. 71.
  • G. D. Chakerian and H. Groemer, Convex Bodies of Constant Width, in: P. Gruber and J. Wills (Editors), Convexity and its Applications, Basel / Boston / Stuttgart: Birkhäuser (1983), p. 68.
  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.10 Reuleaux Triangle Constants, p. 513.

Links

Crossrefs

Cf. A102888, A137616, A137617, A137618.

Programs

  • Mathematica
    RealDigits[(2/3 - Sqrt[3]/4 * ArcCos[1/3])* Pi, 10, 120][[1]] (* Amiram Eldar, May 27 2023 *)

Formula

Equals (2/3 - sqrt(3)/4 * arccos(1/3))* Pi.

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