Oh, man, I cannot rigorize this at all, but I have a *strong* intuition that the answer is "no". It really feels like there is a simple topological fixed-point argument that would make it clear that this is impossible. Maybe start by finding a vector around which the winding number of x(s) is 1; then look at the vector y'(t) dotted with that vector. Surely this must change sign at least once. Here I trail off. On Tue, Sep 1, 2009 at 12:23 PM, Veit Elser <ve10@cornell.edu> wrote:
On Aug 31, 2009, at 6:25 PM, Fred lunnon wrote:
Looks like we're the only two left, Mike --- did we miss Armageddon? WFL
Yes, and it's called "start-of-the-semester".
But here's something for torus fans:
Can the Minkowski sum of two smooth closed curves in R^3 ever be a smooth torus?
If you don't know what a Minkowski sum is, let x(s) and y(t) be smooth closed curves in R^3 parameterized on the unit interval. Now consider the set
z(s,t) = x(s) + y(t), (s,t) in [0,1]^2.
Question: can you find an x(s) and y(t) so z(s,t) is a smooth surface for all (s,t) in [0,1]^2 ? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun