Meissner Tetrahedra

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

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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 .

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


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References

[1] B. Kawohl and C. Weber. "Meissner's Mysterious Bodies." (Jun 19, 2011) www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/pub100.pdf.

[2] E. Meissner, "Über Punktmengen konstanter Breite," Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 56(42–50), 1911. www.archive.org/stream/vierteljahrsschr56natu# 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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