Lots of Views of the Klein Bottle

We begin by recalling that a Klein bottle can be made from a square with its sides glued together in a particular way:

How can the Klein bottle be seen as a pair of bands sewn to a square disk along their single edge?

Lets see:

Here we see a Klein bottle drawn as the union of a pair of bands (Left) and a disk (middle).

note this is the same as fattening the curves on the left on the klein bottle when it is formed from a square.

A Klein bottle is also two Mobius bands stitched together along their single edge!

In the gluing-up-the-square diagram, these Mobius bands look like:

Puzzle:

Find this band in the Klein Bottle!

Is it one sided or two sided?
What are the remaining pieces of the Klein bottle when this band is removed?

Another thing:
After you cut a Mobius band down the middle, you can arrange the result to look like the above! How many twists does this band have?

BONUS

Extra Bonus: classify all surfaces!


  Chaim Goodman-Strauss
  Dept. Mathematics
  Univ. Arkansas
  Fayetteville, AR 72701
  cgstraus@comp.uark.edu
  501-575-6332