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

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

Author

Christof Weber, Feb 04 2008

Keywords

• cons
• easy
• nonn

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

• Bernd Kawohl and Christof Weber, Meissner's Mysterious Bodies, Mathematical Intelligencer, Volume 33, Number 3, 2011, pp. 94-101.
• SwissEduc: Teaching and Learning Mathematics, Gleichdick - Koerper konstanter Breite (in German and English)

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.