## The Flat Surfaces

Both the torus and the Klein Bottle can be
obtained by rolling up a square into a tube and fastening the ends.

TORUS

KLEIN BOTTLE

important: the arrows show how to glue the sides!

Pause and digest!

### The Band Trick!

Now you just did this in reverse:

Note this is exactly the same as what we did at the top of this page,
except the centers of the squares have been removed. Thus:** A torus:**

And a **Klein Bottle:** We know how to physically make a
torus and a Klein bottle with only one little square taken out!

to outline

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

(c)1992-1994 Chaim Goodman-Strauss

Original paper masters available on request.