Meissner Tetrahedra

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).

See also

Meissner Polyhedron, Reuleaux Tetrahedron, Solid of Constant Width

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Antunes, P. R. S. and Bogosel, B. "Parametric Shape Optimization Using the Support Function." Comput. Optim. Appl. 82, 107-138, 2022.Bogosel, B. "Volume Computation for Meissner Polyhedra and Applications." 25 Oct 2023., H. T.; Falconer, K. J.; and Guy, R. K. "Minimal Bodies of Constant Width." §A22 in Unsolved Problems in Geometry. New York: Springer-Verlag, p. 34, 1991.Hynd, R. "The Perimeter and Volume of a Reuleaux Polyhedron." 12 Oct 2023., I. M. and Boltjanskiĭ, V. G. Ch. 7 in Convex Figures. New York: Holt, Rinehart and Winston, 1960.Kawohl, B. and Weber, C. "MeissnerÕs Mysterious Bodies." Math. Intelligencer 33, 94-101, 2011.Lachand-Robert, R. and Oudet, É. "Spheroforms.", T. and Oudet, É. "Bodies of Constant Width in Arbitrary Dimension." Math. Nachr. 280, 740-750, 2007.Martini, H.; Montejano, L.; and Oliveros, D. §8.3 in Bodies of Constant Width. Cham, Switzerland: Birkhäuser/Springer, 2019.Meissner, E. "Über Punktmengen konstanter Breite." Vierteljahresschr. naturforsch. Ges. Zürich 56, 42-50, 1911.Meissner, E. and. Schilling, F. "Drei Gipsmodelle von Flächen konstanter Breite." Z. Math. Phys. 60, 92-94, 1912.

Cite this as:

Weisstein, Eric W. "Meissner Tetrahedra." From MathWorld--A Wolfram Web Resource.

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