From the square to the Moebius strip
This animation shows that, by pasting together two opposite sides of a square, we can get a cylinder or a Moebius strip. A slider controls the animation, while another slider, that can be moved along the circumpherence, controls the number of twists that have to be given to the band before pasting it. We obtain the Moebius strip with half a twist, the cylinder with no twist, but also other surfaces with 2, 3, 4, ... half twists. They are cylinders if the number of half twists is even, or Moebius strips if number is odd; in any case, they will be differently embedded in the three-dimensional space.
animazione di Gian Marco Todesco