The Klein Bottle

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:

The result is not a true picture of the Klein bottle, since it depicts a self-intersection which isn't really there. The Klein bottle can be realized in 4-dimensional space: one lifts up the narrow part of the tube in the direction of the 4-th coordinate axis just as it is about to pass through the thick part of the tube, then drops it back down into 3-dimensional space inside the thick part of the tube.

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:

You need a JAVA enabled browser to view this, such as Netscape Navigator v3.0 for the Macintosh (or v2.0 for Windows)

Note that the animated crab's left and right sides get interchanged as it moves around the Klein bottle, showing that it is an unorientable surface. Note also that unlike the Möbius strip, the Klein bottle has no boundary -- it is a closed surface: the moving crab never encounters any barriers to its motion.

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.

Here is the Java source code for the above animation.

For an alternative take on some of these topics, check out this innovative multi-disciplinary project by a group of Yale undergraduates: