| Five Hundred Fifth in the Famous People (FP) Series - Ernst Meissner |
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The Reuleaux tetrahedron is the intersection of four spheres of radius s centered at the vertices of a regular tetrahedron with side length s. The sphere through each vertex passes through the other three vertices, which also form vertices of the Reuleaux tetrahedron. The Reuleaux tetrahedron has the same face structure as a regular tetrahedron, but with curved faces: four vertices, and four curved faces, connected by six circular-arc edges.
Meißner & Schiller (1912) 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 (pictures and films in Weber 2009). Bonnesen & Fenchel (1934) conjectured that Meissner tetrahedra are the minimum-volume three-dimensional shapes of constant width, a conjecture which is still open. In connection with this problem, Campi, Colesanti & Gronchi (1996) 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.
Now for those who truly understand all of that, you're in the wrong hobby. For the rest of us, St. Johns Lutheran Cemetery has an unexploded soda tube very near a Meissner marker.
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