On 1/10/13, Dan Asimov <dasimov@earthlink.net> wrote:
Consider an equilateral triangle.
--Dan
On 2013-01-09, at 1:51 PM, Fred lunnon wrote:
In fact, just at the moment, I am having difficulty imagining a set which is _not_ contained in a sphere of the same diameter ...
Groan --- or indeed a regular tetrahedron, which is where we came in ... Or indeed _any_ triangle or tetrahedron, according to the theorem I quoted earlier. On 1/10/13, Dan Asimov <dasimov@earthlink.net> wrote:
...
OOPS. The Reuleaux tetrahedron, for instance, is *not* of constant width. (It's also spelled with an "a".)
So Fillmore's Corollary is in serious doubt!!! In any case, the argument in his paper includes a crucial mistake.
...
So: I still don't know if there's a surface of constant width in 3-space having the symmetry group of the tetrahedron.
A positive answer is guaranteed by that theorem (to which I can't discover the reference, discourtesy of Googlebooks). Part of the problem is the misnomer "Reuleaux tetrahedron" attached to the figure you described, which is as you observe irrelevant to the discussion. The "Meissner" tetrahedron --- a special case of the construction I posed earlier --- may well have constant width, but my own previous attempts to decide this question have been inconclusive. These fellows are presumably in possession of some high-powered construction. So what does the constant-width superset (with the same diameter) of a scalene triangle (including interior) look like, anyway? And is it unique, given the triangle? Fred Lunnon