Talk:Umbilic torus

So it would seem to me that the "umbilic torus" would somehow be related to the Mobius strip, a guess that is borne out by the reference to Mobius strip in the links. If this is indeed the case, how so? Druiffic (talk) 08:52, 24 January 2009 (UTC)

The obvious similarity is that, like the Mobius strip, it only has one side. Other than that, I'm not sure. I'll be updating this page in a month or so when the giant sculpture is dedicated, so expect more information then. Danski14^(talk) 15:53, 5 September 2012 (UTC)

It'd really just an improved Klein bottle. — Preceding unsigned comment added by 98.100.99.130 (talk) 19:15, 6 December 2013 (UTC)

Both the Mobius strip and the umbilic torus have a single edge and a single external face (the Mobius strip has no internal face, but the umbilic torus also has a single internal face). The cross-section of the Mobius strip is a segment, while the cross section of the umbilic torus is a triangle (actually a deltoid: an hypocycloid with 3-fold rotational symmetry). Now, I'm hinting that we could generalize this idea with cross-sections which are polygons with an odd number of sides, am I right? Actually, the cross-sections would have to be k-sided polygons (or hypocycloids with k-fold rotational symmetry) where k is a noncomposite number (i.e. one or a prime number)...

Here's with pentagonal cross-sections: http://pub.ist.ac.at/~edels/Tubes/tori/penta/