Meißner (1911) showed how to modify the Reuleaux tetrahedron (which is not a solid
of constant width) 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. Depending on which three edge arcs are replaced (three that have a common vertex
or three that form a triangle), one of two noncongruent shapes can be produced that
are called Meissner tetrahedra (Lachand-Robert and Oudet 2007).
The figure above (Bogosel 2023) illustrates the two types of Meissner tetrahedra, with the top row showing the case where three edges having a common vertex are smoothed and the bottom row showing the case where three edges adjacent to a common face are smoothed. In the figure, spherical parts are represented in sky blue, wedge surfaces are shown in mauve, and spindle surfaces in hunter green.
It is conjectured that the Meissner tetrahedra minimize the volume among all three dimensional bodies with fixed constant width, but proof or refutation remains open (Antunes and Bogosel 2022, Bogosel 2023).
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