The Klein Bottle

To create a Klein Bottle we make one identification in the straightforward way, but the other identification is made "with a twist". An object moving around in a "Klein bottle" universe gets flipped if it completes a cycle in one direction. If you had a left-handed friend who walked away from you in the twisted direction, he might return right-handed! The animation shows an object moving in a Kleinian universe. As it moves across the side boundaries it comes back on the other side -- and moving in the same direction. As it crosses the top or bottom boundary, it comes back on the opposite side with its position, momentum and handedness flipped.