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