The Projective Plane

I have been thinking some more about the famous Möbius strip (see also my post on the Klein bottle). The so-called “Sudanese Möbius Band” in the video above is a Möbius strip stretched so as to make the boundary perfectly circular (it is not named after the country, but after the topologists Sue E. Goodman and Daniel Asimov, and you can purchase a plastic one here).

If we glue two of these Möbius strips together (not actually possible in 3 dimensions), we get a Klein bottle. If we glue one to a disc (also not possible in 3 dimensions), we get a projective plane.

Just for fun, the video below shows a Game of Life glider on the projective plane. The top and bottom of the square are considered to be joined, as are the left and right sides. In both cases, there is a reversal of orientation (a manoeuvre not really possible in 3 dimensions). The glider changes colour as it changes orientation.

Video produced using the R animation package.