Hi Fred, I suspect the Minkowski mean of Meissner tetrahedra is the same constant width solid Roberts describes. If so, your argument below must have a flaw. When you say
rescaling, the mean comprises two arcs of radius (1 + e)/2 , touching the summands where they meet, and capped by a third arc of radius e/2 . Since e = 1 - rt3 / 2 is nonzero, the mean section is non-circular; whereas the
you seem to assume that the "arc" length of radius (1 + e)/2 in your cross section is nonzero. I believe it must be zero. If you think it's nonzero, look carefully at the diameters normal to that arc segment, and where the trajectory of the opposite end of those diameters would have to lie. Best, - Scott
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of Fred lunnon Sent: Monday, January 14, 2013 9:05 AM To: math-fun Subject: Re: [math-fun] Low Rollers: Bodies with Minimal Constant Width and Tetrahedral Symmetry
Despite initially having been hazy about the details of Minkowski sums, it turns out that my intuition was not quite so wonky as earlier appeared!
Lemma: The Minkowski sum of a surface and a sphere of radius r is (congruent to) the parallel surface at distance r ; proof elementary.
Consider the mean M_T = (M_V (+) M_F)/2 , the summands positioned so that all three coincide in the spherical caps over each tetrahedral face, and the mean has the symmetry of a regular tetrahedron. Section it along a plane of symmetry: over each edge, the mean composes a circular arc of small radius e from one summand, with a pair of arcs of unit radius from the other meeting in nodal line; at their free ends, the small and unit arcs touch tangentially.
* * * * * * * * * * * * * * * * * * * * * * * * *
By the lemma, the sum is parallel to the unit arcs at distance e : rescaling, the mean comprises two arcs of radius (1 + e)/2 , touching the summands where they meet, and capped by a third arc of radius e/2 . Since e = 1 - rt3 / 2 is nonzero, the mean section is non-circular; whereas the corresponding portion of the Roberts section is a single arc of radius d = (2 - rt2) / 8 .
Finally, the Minkowski and Roberts bodies are both minimal constant-width, with tetrahedral symmetry, but incongruent.
The screed is already in need of updating ... comments are invited. While it suffers from an absence of diagrams, that on page 2 of Roberts report provides a partial substitute. In that connection I have already spotted one misprint:
Stage (A): for "d = 0" read "d = c"
Fred Lunnon
On 1/12/13, Fred lunnon <fred.lunnon@gmail.com> wrote:
... Hence it looks a safe bet that Roberts = Minkowski holds, as conjectured by Dan Asimov, and earlier dismissed by myself with gratuitous contempt. A more detailed investigation along the lines above would not come amiss ... But now more generally, does minimal constant-width with tetrahedral symmetry uniquely determine the body modulo similarity --- why should no others exist?
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