Well, here's the MathWorld description of the Meissner tetrahedron: ----- 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 or Meissner bodies (Lachand-Robert and Oudet 2007). ----- which seems to have less than full tetrahedral symmetry. --Dan On 2013-01-10, at 6:48 AM, Fred lunnon wrote:
On 1/10/13, Dan Asimov <dasimov@earthlink.net> wrote:
Scratch that idea!
Okay, second try: Build an equilateral triangle on the longest side of the scalene triangle, so that it contains the vertex opposite that side.
Then the Reuleaux triangle in which this equilateral one is inscribed seems to be the natural shape of constant width containing the original scalene triangle and having the same diameter.
Agreed --- once I had realised this, I felt much better about the theorem for arbitrary subsets of |E^n --- at any rate, for n = 2 ! Furthermore, in this case at least, the superset is fairly easily seen to be unique.
...
I think the Meissner tetrahedron has constant width, but without full tetrahedral symmetry.
Huh? It must have tetrahedral symmetry, by construction: 4 congruent spherical segments atop faces, 6 congruent cyclidal segments atop edges.
A trivial computation --- can't understand why I balked at this before --- shows that, along a "bi-altitude" through the mid-points of the edges, the Meissner undershoots unit width by exactly the same amount that the "Reuleaux" overshoots it: sqrt(3) - 1/sqrt(2) - 1 = 0.0249 ; each attempt proves equally feeble (the same applies to my generalisation).
Presumably a better approximation would be the mean of the two surfaces. Intuitively, I think it's pretty clear that the constant minimum width surface does exist in this case too (and is also unique). Simply place a hard Meissner inside a soft Reuleaux between parallel abrasive planes at unit distance, and randomly grind away ...
Fred Lunnon
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