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?
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.
In the gluing-up-the-square diagram, these Mobius bands look like:
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?
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?
Extra Bonus: classify all surfaces!
Chaim Goodman-Strauss Dept. Mathematics Univ. Arkansas Fayetteville, AR 72701 email@example.com 501-575-6332
(c)1992-1994 Chaim Goodman-Strauss
Clean paper masters available on request.