Reuleaux tetrahedron

The volume of a Reuleaux tetrahedron is :\frac{s^3}{12}\big(3\sqrt2 - 49\pi + 162\tan^{-1}\sqrt{2}\big) = \frac{s^3}{12}\left(32\pi - 81\cos^{-1}\left(\tfrac 1 3\right) + 3\sqrt{2}\right) \approx 0.422\,s^3. The surface area is : \left[8\pi - 18\cos^{-1}\left(\tfrac 1 3\right)\right] s^2 \approx 2.975\,s^2. == Meissner bodies ==

Ernst Meissner and Friedrich Schilling showed how to modify the Reuleaux tetrahedron to form a surface of constant width, by replacing three of its edge arcs by curved patches formed as the surfaces of rotation of a circular arc. According to which three edge arcs are replaced (three that have a common vertex or three that form a triangle) there result two noncongruent shapes that are sometimes called Meissner bodies or Meissner tetrahedra. Bonnesen and Fenchel conjectured that Meissner tetrahedra are the minimum-volume three-dimensional shapes of constant width, a conjecture which is still open. In 2011 Anciaux and Guilfoyle proved that the minimizer must consist of pieces of spheres and tubes over curves, which, being true for the Meissner tetrahedra, supports the conjecture. In connection with this problem, Campi, Colesanti and Gronchi showed that the minimum-volume surface of revolution with constant width is the surface of revolution of a Reuleaux triangle through one of its symmetry axes. Man Ray's painting Hamlet was based on a photograph he took of a Meissner tetrahedron, which he thought of as resembling both Yorick's skull and Ophelia's breast from Shakespeare's Hamlet. == References ==