[Thread mysteriously split into two ...] Consider the following construction of a continuously differentiable surface, with tetrahedral symmetry: (0) start from a regular tetrahedron of circumradius 1; (1) balloon each face out to a spherical cap of radius 1+r, bounded where it meets each of the other 3 faces extended; (2) balloon each vertex out to a cap of radius r, bounded similarly; (3) balloon each edge out the the unique Dupin cyclide tangent to all 4 adjacent spherical boundaries. Does this have constant width? Meissner's surface seems to be the case r = 0, which is not differentiable. A couple of recent references turned up via Google --- mathoverflow.net/.../are-there-smooth-bodies-of-constant-width www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/pub100.pdf One response under the first thread suggests a method of constructing arbitrary _analytic_ surfaces of constant width, but is so cryptic that I can make neither head nor tail of it. Fred Lunnon On 1/8/13, George Hart <george@georgehart.com> wrote:
Jim,
There are four shapes of rollers in the MoMath ride-on exhibit. Three are acorn-like surfaces of revolution and the fourth is a Meissner tetrahedron.
One of them is very similar to this very cool commercially available metal set:
http://www.grand-illusions.com/acatalog/Solids_of_Constant_Width.html
The second has a point at the apex but is differentiable everywhere else, and the third is based on a regular pentagon (extended with radii and rotated to a surface of revolution).
(I also wanted to have one sphere in the mix, with a different color, to emphasize that all the others aren't spheres, but that was vetoed...)
George http://georgehart.com/
On 1/8/2013 5:35 PM, James Propp wrote:
Does anyone know which sort of rollers the MoMath exhibit "Coaster Rollers" uses?
Jim Propp
On Tue, Jan 8, 2013 at 6:12 AM, Tom Karzes <karzes@sonic.net> wrote:
I wonder if Meissner tetrahedra (true constant width) would work better or worse than Reuleaux tetrahedra (approximate constant width) for this purpose.
Tom