You cannot just untwist it while confined to the surface of a sphere, but maybe my phrasing: "at all stages of the deformation the tangent directions vary continuously" was less than clear. I mean that as the deformation proceeds in "time", the tangent directions of the various curves of the deformation evolve continuously. So for instance you can't pull a kink in a curve tight to make it disappear, since that would create a discontinuity of the tangent directions at the last moment of time. —Dan
On Jul 10, 2015, at 8:52 PM, rwg <rwg@sdf.org> wrote:
On 2015-07-10 16:07, Dan Asimov wrote:
Let a C^oo closed curve in R^3 be called a "curly loop" if its curvature is nowhere vanishing. Two curly loops are for instance A) the unit circle in the plane and B) the unit circle in the plane traversed twice around. PUZZLE: Can A and B be continuously deformed one into the other — through curly loops — so that at all stages of the deformation the tangent directions vary continuously? —Dan Can't you just untwist it while confined to the surface of a sphere? --rwg