Meissner Tetrahedra

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The two Meissner bodies are solids of constant width. Others are spheres and certain solids of revolution.


The Reuleaux tetrahedron is the intersection of four balls of radius 1, each centered at a vertex of a regular tetrahedron with side length 1. Each of the six curved edges of is the intersection of two spheres; three edges meet at each vertex and three surround each face.

For a curved edge , let be the corresponding straight edge of and let and be the faces of that meet at . The planes containing and cut a wedge out of with edges that are circular arcs and . The wedge is formed by rotating into around . Rounding means to replace with .

The first kind of Meissner body is obtained by rounding the three edges at a vertex of and the second by rounding the three edges around a face of .


Contributed by: Izidor Hafner (January 2014)
Open content licensed under CC BY-NC-SA




[1] B. Kawohl and C. Weber. "Meissner's Mysterious Bodies." (Jun 19, 2011)

[2] E. Meissner, "Über Punktmengen konstanter Breite," Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 56(42–50), 1911. page/n53/mode/2up.

[3] E. Meissner and F. Schilling, "Drei Gipsmodelle von Flächen konstanter Breite," Zeitschrift für angewandte Mathematik und Physik, 60(92–94), 1912.

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