Bodies of Constant Width

Written by Christof Weber (University of Teacher Education FHNW)


ContentsSpheres are not the only three-dimensional bodies of constant width
Target audienceTeachers of mathematics

What's it about?

If a flat plate is placed on three spheres (of equal size) and pushed back and forth, it moves perfectly parallel to the surface of the table. One might suppose that this property of constant width is not only a necessary, but also a sufficient condition for spheres. If you are familiar with the Reuleaux triangle, however, you will rightly suspect that this supposition is incorrect. As in the two-dimensional situation, there exist three-dimensional bodies which are not spherical although they are of constant width. However, the three-dimensional generalisation of the Reuleaux triangle - the Reuleaux tetrahedron (see illustration above left) - is only of almost constant width.
The information (see document below) describes how bodies of exactly constant width can be constructed. In addition to Meissner bodies (see illustration above right), it includes some illustrations of rotationally symmetric bodies of constant width. The two Meissner bodies can be viewed as animations (see animations below) or explored interactively from all sides (see link below).

Downloads

Information  Information "What does this solid have to do with a ball?" PDF [272 KB]
 
Animation  Animation "first Meissner body" mov [874 KB]
Animation  Animation "second Meissner body" mov [928 KB]

Links

Site "Meissner Bodies - interactive"Rotate, discover ... the two Meissner bodies