Research Log: Week 5 |
| Monday, 6/13/05 |
| We typed the proof of straight +straight=straight in Latex, and worked on the ribbon curve hypothesis. The ribbon curve hypothesis says that you can take any space curve and attach a surface to it in such a way that the surface can be developed into a ribbon, or, in otherwords, a rectangle in the plane. (A rectangle of arbitrarily small height that is.) Perhaps, however, the conjecture should be that all plane curves have this property but as of yet unknown type of space curves can never be developed in such a way. |
| Tuesday, 6/14/05 |
We've concluded that only plane curves can form ribbons by connecting a parallel copy of the plane curve to the original curve. A space curve connected in such a way will form a generalized cylander, but the curve will no longer unroll to the geodesic edge of a ribbon. It will unroll to a curvy thing. Another possible method of forming the surface would be to crawl a unit height along the binormal vector to the original curve. This gives a parametrization for a ruled surface with the binormal vectors as the rulings. We know that any isometry of the plane must be a ruled surface, and thus have the standard form of a ruled surface. We also know that any isometry of the plane has the same First Fundamental Form as the plane, which is the identity matrix. This puts restrictions on the partial derivatives of the ruled surface, namely that the velocity in each parameter direction is unit speed. |
| Wednesday, 6/15/05 |
| Joe left for NSF and we did very little today. We worked on our webpages, and Duc worked on her paper for Smith Summer Science. |
| Thursday, 6/16/05 |
By assuming that the First Fundamental Form of our bent piece of paper is the identity matrix, and that the bent piece of paper can be modeled by the standard parametric form of a ruled surface, I seem to concluded that the rules of the surface must be perpendicular to the tangent vectors of both defining curves. Duc has pointed out that this does not appear to be true looking at our examples of bent paper. I agree with Duc's observation, but can't find the flaw in my computations. I've also shown, it seems, that the idea of extending the binormal only works for plane curves. Of course, I've shown this assuming the First Fundamental Form is the identity matrix, and seeing that my above computations using the First Fundamental Form are dubious, we must resolve the inconsistancy above before believing the results of this proof. One word of caution: the racetrack d-forms don't follow the rules for the form of a ruled surface that I have been using. Namely, the equation I've been using requires that the rules are never parallel to the tangent vectors of the defining curve, which is not true for d-forms. Therefore, looking at d-forms, one cannot tell whether my conjecture above is true or false. |
| Friday, 6/17/05 |
Today was Duc's last day and we mostly partied and ate sushi with the logicians. |