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A137617
Decimal expansion of volume of the solid of revolution generated by a Reuleaux triangle rotated around one of its symmetry axes.
3
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
0,1
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).
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)
FORMULA
2/3 * Pi - Pi^2 / 6
EXAMPLE
0.44946103...
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}]
CROSSREFS
KEYWORD
cons,easy,nonn
AUTHOR
Christof Weber, Feb 04 2008
EXTENSIONS
Link corrected by Christof Weber, Jan 06 2013
STATUS
approved