Today, I wrote my Women in Science paper.

We realized that the endpoints of the paper whose edge lies in a plane don't behave the same as the middle points. The rules coming from these singular endpoints have much more freedom to rotate in space. For this reason it might be a good idea to look at closed plane curves first, since the properties of the rules attached to interior points should be the same as those attached to the points on a closed curve.

We also graphed the smooth version of the tennis ball curve today. Since the original tennis ball curve had step function curvature that oscillated between zero and one, I decided that the easiest way to smooth the curvature graph out was to use a shifted sine function (actually, I used cosine first). I ran the code that Joe had taken from Gray's differential geometry with Mathematica book, and obtained a curly plane curve. Then Joe came in and modified the shifted sine function so that it had intervals with zero curvature (he did this by piecewise defining the curve). With a few modifications, we obtained a graph that looked just like the original tennis ball curve only, by our construction, it is smooth. Provided that we can prove that the generalized cylinder is the only ribbon of a plane curve, this smooth tennis ball curve should be a counter-example to the theorem, every C2 space curve is a ribbon curve.

We have an intuitive argument about the way isometries of the plane work, and this argument implies that if a plane curve is somewhere non-singular (it isn't everywhere a straight line), it's ribbon must be locally perpendicular to the plane that contains the plane curve. The idea goes as follows:

Lay a grid on your piece of paper and triangulate it by drawing the two diagonals through each square in your grid. The paper is flexible along all the lines on your paper, namely along the edges of each square and the two diagonals of each square. A paper that is bent along these lines is an approximation of an isometry of the plane, where bending is allowed at any point on the plane. (no stretching.) If we did not allow bending along the diagonal, we would see that any angle between two squares joined at an edge must be constant along the line in which the edge is contained. Bending along one line through a vertex implies no bending through the perpendicular line through the same vertex by an angle deficit argument.

Since the actual piece of paper may bend along lines other than the diagonals of the squares (i.e. they may bend at other angles than 45 degrees) this model is not a good one. We returned to the argument of using the First Fundamental form.