OFFSET
1,1
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).
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)
FORMULA
Equals 2*Pi - Pi^2 /3.
EXAMPLE
2.99331717348313360398045643326695538995643899633661...
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 *)
CROSSREFS
KEYWORD
AUTHOR
Christof Weber, Feb 04 2008
STATUS
approved