Reuleaux tetrahedron

The Reuleaux tetrahedron is the intersection of four spheres with radius s whose four center points s lie at the corners of a regular tetrahedron with side length. The four corners of the generating tetrahedron form the four corners of the Reuleaux tetrahedron. The Reuleaux tetrahedron has the same structure as its generating tetrahedral: four corners, four faces and six edges. The areas consist of spherical segments, however, and the edges of circle segments.

The Reuleaux tetrahedron is defined and named after its two -dimensional analogue, the Reuleaux triangle, which is named after Franz Reuleaux. But in contrast to this the Reuleaux tetrahedron is not a body of constant width, because the midpoints of two opposite edges have a greater distance

The volume is the Reuleaux tetrahedron

( Weisstein ).

Meissner body

Ernst Meissner and Frederick Schilling (1911, 1912) showed, however, as the Reuleaux tetrahedron can be modified to form a body of constant width. To do the three ( consisting of segments of a circle ) edges are replaced by surfaces that are part of a body of revolution. This rotation axis body than the edge of the associated generating tetrahedron as generating curve is a circle segment that arises when one cuts the Reuleaux tetrahedron with the continuing pages of the generating tetrahedron. Depending on which three edges are replaced ( three sharing a common corner or three that form a triangle), results in two topologically different bodies, which are also called Meissner bodies ( for movies and interactive images see Weber). Tommy BONNESEN and Werner Fenchel (1934 ) suggested that the Meissner bodies are the bodies of constant width with minimal volume, but the evidence is still open ( Kawohl and Weber, 2011). Campi et al. (1996) have shown that the body of rotation having a constant width with a minimum volume is a Reuleaux triangle, which is rotated about one of its axes of symmetry.