On 1/11/13, Dan Asimov <dasimov@earthlink.net> wrote:
Right, assuming the proof holds up.
There does seem to be an lot of detail involved in Roberts' presentation. As far as I can tell however, it does at least attempt to cover all the options.
But I'm more inclined to believe the validity of the claim in the Math Intelligencer article that you cite. ... P.S. Fred, what do you mean by "elsewhere" in your posted quoted below?
Kawohl & Weber (Math. Intell., Scott's link) give no justification or reference for the constant-width claim, which I consider on the face of it very surprising and quite possibly erroneous [it takes one to know one ...]; "elsewhere" refers to width along diametral lines away from planes of symmetry. Notice that the relevant portion of the Reuleaux cross-section comprises a pair of unit radius circular arcs meeting at a singular point, whereas the Meisner has a single small-radius circular arc, tangent to the larger pair. The Minkowski sum (ie. mean) of these two cannot possibly be the Roberts, which for part of its length coincides (only) with the Reuleaux: indeed, it can't contain circular arcs at all. So the Minkowski yields a distinct surface. Fred Lunnon
In fact, they write:
----- Incidentally, the Minkowski sum (½)M_V ⊕ (½)M_F, which one obtains half way in the process of morphing M_V into M_F, would render a body with tetrahedal symmetry. It actually has the same constant width as MV and MF .
Its volume, however, is larger than that of the Meissner bodies, due to the Brunn-Minkowski inequality. -----
Here M_V is the Meissner tetrahedron where starting from the Reuleaux tetrahedron, 3 edges sharing a vertex have been modified in the same way, to get a body of constant width. Likewise for M_F, but in this case the 3 edges bound one face of the Reuleaux tetrahedron.
I believe the example in the example Scott Huddleston linked to below is identical to the example (½)M_V ⊕ (½)M_F mentioned in the Intelligencer article.
On 2013-01-10, at 4:20 PM, Fred lunnon wrote:
On 1/10/13, Huddleston, Scott <scott.huddleston@intel.com> wrote:
See http://www.xtalgrafix.com/Spheroform.htm and its links, including http://www.xtalgrafix.com/Reuleaux/Spheroform%20Tetrahedron.pdf
Looks like that probably wraps Dan's question up, albeit at some length!
Also, p.4 of http://www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/pub100.pdf says that a Meissner-like tetrahedron with congruent "edge" parts is realizable as a Minkowski sum of the two Meissner tetrahedra.
That was obviously true --- but there seems no reason to expect that the width would be constant elsewhere.
WFL
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