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.