A small bottle has no volume

The Klein bottle

This year we have chosen a picture of half a Klein bottle as a distinguishing mark for the “Day of Mathematics” (see above right). In the following, Elmar Vogt, Professor of Mathematics at the Free University of Berlin, explains what the Klein bottle is all about.

For example, the introduction to an article by August Ferdinand Möbius from 1865, in which, probably for the first time, non-orientable surfaces without borders were described and treated. Areas without self-penetration in three-dimensional space always have an exterior and interior, so that one can determine the volume of the interior. If self-penetration is allowed, Möbius was surprised to find that there are surfaces without borders, from one side of which one can get to the other without having to penetrate the surface. So you have no interior at all.

In Möbius' essay we also find a description of the so-called Möbius strip, which is obtained by gluing an elongated rectangular strip of paper to its opposite short strip after rotating it through 180 degrees. But this was much earlier z. It has been described, for example, by Listing, and I would not be surprised if it was not considered centuries ago, without attaching great importance to it.

Möbius, however, specifies areas without a border, but with self-penetration. An explicit example was a surface that is created from the Möbius strip by attaching a circular disc to its edge. He notes that this is probably the simplest "one-sided" surface. This area is the projective plane. However, the Möbius has not noted. It's a little more difficult to visualize than the logo for this year's Mathematics Day. Our surface is the Klein bottle named after Felix Klein, a one-sided surface that is created by gluing two Möbius strips along their edges.

We see half of the Klein bottle at the top right, that is, we see one Möbius strip, but with self-penetration. The second is simply glued on top in a mirror image. The picture clearly shows how you can get from one side to the other along a closed curve.

Without self-penetration we can imagine the Klein bottle if we treat ourselves to four dimensions. By the way, Klein already knew that too. Just as points in three-dimensional space are described by three real numbers, the x, y and z coordinates, points in four-dimensional space are coded by four real numbers. The three-dimensional space is then described by the points whose fourth coordinate is zero. When constructing the Klein bottle by bending, pushing through and stapling the ends of a tube (see picture above), one only has to lift the end that we are moving briefly into the fourth dimension while allowing the tube to penetrate itself in three-dimensional space, So raise the fourth coordinate for all points of the hose end evenly a little and after exceeding the penetration lower it to zero again, now back in three-dimensional space, then staple the ends as before.

With a little practice you can imagine that to some extent. If you don't want to believe it, we can easily convince you with precise equations in four-dimensional space.


We will also raffle small small glass bottles as a special prize among all competition participants who take part in the award ceremony on May 5, 2012.


Status: April 18, 2012