The Klein bottle is another unorientable surface. It can be constructed by
gluing together the two ends of a cylindrical tube with a twist. Unfortunately
this can't be realized physically in 3-dimensional space. The best we can
do is to pass one of the ends into the interior of the tube at the other end
(while simultaneously inflating the tube at this second end) before gluing the
ends. The resulting picture looks something like this:
However there is no need to go through the mental contortions of visualizing
the Klein bottle in 4-dimensional space, if we adopt the intrinsic point of
view we developed for dealing with the Möbius strip. We do not attempt to
physically realize the gluing described above, but rather think of it as an
abstract gluing, imagining how the resulting space would look to a 2-dimensional
crab swimming within the surface of the Klein bottle. This leads us to the
following convenient model of the Klein bottle:
To relate this to our previous description of the Klein bottle, note that the gluing instructions tell us to glue the top and bottom edges of the rectangle. The result is a cylindrical tube with the left and right edges forming the two circular ends of the tube. The gluing instructions then tell us to glue the two ends of the tube with a twist. (Note that the gluing instructions tells to glue all four corners of the rectangle into a single point.) The following computer animation shows our 2-dimensional crab moving within the resulting space given by these abstract gluing instructions:
The Klein bottle is named after the German mathematician
Felix Klein (1849-1925).
Let's now use the idea of abstract gluing to study the boundary of the hypercube.
You might now want to move on to the classification of closed surfaces.
Here's a fancier picture of the Klein bottle, part of the Surfaces Beyond the Third Dimension internet art exhibit.