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# 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*

(c)1992-1994 Chaim Goodman-Strauss

Clean paper masters available on request.