Lawrence J. Peterson
University of North Dakota
Department of Mathematics
Here is a picture which attempts to describe the Klein bottle. The
narrow tube starts at the top of the bottle, extends sideways, and
then bends down. It then goes through the side of the bottle. The
tube then widens and is glued to the bottom of the bottle. If a fly
wanted to get inside the bottle, it could fly upward into the bottom
of the bottle, through the narrow tube, and then down into the main
part of the bottle from the top. The fly could not get out except by
flying back up to the top of the bottle, through the tube, and then
out through the bottom of the bottle.
The most interesting part of the Klein bottle is the way in which the
narrow tube passes through the side of the bottle. One way to view
this is to consider a television and comic book character from many
years ago. The character in question was called Superman. Superman
was able to walk through walls. The tube part of the Klein bottle
goes through the wall of the Klein bottle in much the same way as
Superman supposedly walked through cement walls. Recall the fly that
we discussed above. As this fly moves up through the tube portion of
the Klein bottle, it also passes through the side wall of the Klein
bottle; it does this in the same way as Superman walked through walls.
Superman, of course, was only a fictitious character. The Klein
bottle, on the other hand, is real. But when I say this, I must
quickly point out that the Klein bottle is not a physical object that
one can construct out of metal, plastic, or wood. Rather, it is an
abstract mathematical construct. It is just as real as the number
line and the set of real numbers. Unfortunately, I do not have time
to explain this in further detail now. If you would like to learn
more, find a book on differential topology!