When I was little, friends gave me as a birthday present a home made flip book that would show the deformation of the catenoid to the helicoid. That was a lot of work back then when you had to program all the 3D-graphics by hand, including hidden line algorithms. But I liked it to a have a physical object that would allow me to run my own little movie.
Daumenkinos – Thumb Theaters, are they called in German. Today we see some snapshots from a high tec version of such a Daumenkino, attempting to get to the core of Boy’s surface (an immersion of the projective plane), about which I have written briefly before.
The goal is to put a lid on a Möbius strip. The one we start with you will note is not just once twisted, but three times. I don’t know how essential this is to get an immersed projective plane at the end. I suppose it’s not, but makes things easier. Note that the strip has a single boundary curve, as expected.
The first two images show that Möbius strip, growing slowly. Below the first crucial step has happened: The growing strip has created a triple point, and intersection like that of three planes. But there still is only one boundary curve…
We keep growing
and growing:
Another critical event: The boundary curve emerges completely into free air, i.e. doesn’t pierce through the surface anymore. Now it’s easy to close the lid:
The Inner Frame 