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

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

Author

Christof Weber, Feb 04 2008

Keywords

• cons
• easy
• nonn

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

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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