Meissner Tetrahedra

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faces
1
2
3
4
Reuleaux wedges
1
2
3
4
5
6
Meissner wedges
1
2
3
4
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6
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The two Meissner bodies are solids of constant width. Others are spheres and certain solids of revolution.
The Reuleaux tetrahedron
T
is the intersection of four balls of radius 1, each centered at a vertex of a regular tetrahedron
R
with side length 1. Each of the six curved edges of
T
is the intersection of two spheres; three edges meet at each vertex and three surround each face.
For a curved edge
E
, let
F
be the corresponding straight edge of
R
and let
A
and
B
be the faces of
R
that meet at
F
. The planes containing
A
and
B
cut a wedge
V
out of
T
with edges that are circular arcs
C
and
D
. The wedge
W
is formed by rotating
C
into
D
around
F
. Rounding
E
means to replace
V
with
W
.
The first kind of Meissner body is obtained by rounding the three edges at a vertex of
T
and the second by rounding the three edges around a face of
T
.

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.

External Links

Meissner Tetrahedron (Wolfram MathWorld)
Reuleaux Tetrahedron (Wolfram MathWorld)
Curves and Surfaces of Constant Width

Permanent Citation

Izidor Hafner
​
​"Meissner Tetrahedra"​
​http://demonstrations.wolfram.com/MeissnerTetrahedra/​
​Wolfram Demonstrations Project​
​Published: January 7, 2014