-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of Fred lunnon Sent: Thursday, January 10, 2013 2:17 PM To: math-fun Subject: Re: [math-fun] rounded tetrahedron
Apologies: I had completely forgotten that the Meissner construction replaces only _three_ of the six edges. So yes, you (and he) are correct: it does have constant width, but not tetrahedral symmetry.
The sectional curve along each edge which would achieve full symmetry must lie somewhere between the single circular arc used by Meissner, and the pair of arcs resulting from continuation of the spherical caps over the faces ("Releaux"). It's surprising that nobody seems to have explicitly given an analytical form for either implicitly or parametrically.
Fred Lunnon
See http://www.xtalgrafix.com/Spheroform.htm and its links, including http://www.xtalgrafix.com/Reuleaux/Spheroform%20Tetrahedron.pdf 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.
On 1/10/13, Dan Asimov <dasimov@earthlink.net> wrote:
P.S. Here's a nice write-up about the Meissner tetrahedron: (published in Math Intelligencer v.33 no.3 2011:
<http://www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/pub100.pdf>
--Dan