The Final Circle:

Finally, the curve returns to a convex circle, as required for the top sets of a tight immersion

. Note that the polyhedral circle stars large, gets smaller toward the critical level, then grows larger again for this final step; this is in keeping with the fact that this projective plane forms the central M- region of the tight immersion

and should have "negative curvature" everywhere; that is, it should have the basic shape of a hyperboloid of one sheet: wide at the top and bottom and thin in the middle.

Movie showing the complete sequence of levels

Level 7: No more self-intersection

Comparison to Kuiper's level sets

Other pictures

9/29/94 dpvc@geom.umn.edu -- The Geometry Center