The Unapologetic Mathematician

Mathematics for the interested outsider

More on the C-G Eversion

Some people had trouble grabbing the whole 50MB file that I posted, so Scott Carter broke it into pieces. He also included these comments:

The red, blue, and purple curves on the large (distorted) spherical objects at the bottom of each page of the eversion are the preimages of the the folds (color coded of course) and the double decker sets. Since at each time the sphere is immersed it may have double and triple points. Each arc of double points lifts to a pair of arcs on the ambient sphere, and each triple point lifts to three points on the ambient sphere. These lifts are the “decker sets.”

They are obtained via Gauss-Morse codes. Pick a base point and orientation on each curve in a movie. These are chosen
consistently from one still to the next. Label the double points and the optima and read the labels as they are encountered upon a single journey around the curve. The labels, too, are chosen consistently from one still to the next. Write these down for each curve in a movie, and connect the letters in the words as the curves change according to the basic changes that occur in each of the movie scenes.

These curves then are instructions on how to immerse the ambient sphere to create the illustrations.

Sarah’s thesis computes that the fold set is an annulus, the double point set is the connected sum of three projective planes, and the double decker set is the connected orientation double cover: a genus 2 surface.

So here are the pieces:

  1. Immersed spheres as movies (2.2 MB)
  2. The basic movie moves (3.4 MB)
  3. The eversion from the red side to the quadruple point (19 MB)
  4. Half of the eversion from the quadruple point halfway to the blue side (24 MB)
  5. The other half of the eversion from the quadruple point halfway to the blue side (17 MB)

There’s a glitch in part 4, so I’ll post that as soon as I can.

July 10, 2008 Posted by | Category theory, Knot theory, Topology | 1 Comment

   

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