Klein bottle

The Klein bottle (even small shear tube ), named after the German mathematician Felix Klein, is a geometric object. Formulated Colloquially, it has the property that the inside and outside can not be distinguished, or in other words, that it has only one side that is inside and outside simultaneously. This is called in mathematics a non- orientable surface.

The name " Klein bottle " said to have originated from a mistranslation into English. Originally supposed to have been called this object in German Klein surface, and have been translated by a confusion between bottle and Area as a Klein Bottle. After this descriptive name has prevailed, the term bottle is now also used in German.

Description

Like the Möbius strip is the Klein bottle, a two-dimensional differentiable manifold which is not orientable. In contrast to the Möbius strip, the Klein bottle can be embedded with self-intersection in the three-dimensional Euclidean space. Without self but this is possible and for the higher dimensions.

A Klein bottle can be represented by the following equations for and in:

Being. is the approximate width of the approximate height of the figure. Typical values ​​:.

Note: The Klein bottle can divide so that two Möbius bands arise from (see the figure on the right ).

Construction

You start with a square and glue the corners and edges together with the appropriate colors, so that the arrows match. This is shown in the following diagram. Said Formally, the Klein bottle is described by the quotient topology of the square [0,1] × [0,1] with edges which satisfy the following relations: (0, y) ~ (1, y) for 0 ≤ y ≤ 1 and (x, 0) ~ (1-x, 1 ) for 0 ≤ x ≤ 1

The square is a fundamental polygon of the Klein bottle.

Note that this description means the " sticking" in an abstract sense, which attempts to construct the three-dimensional Klein bottle with yourself about intersecting edges. In fact, the Klein bottle has no intersecting edges. Nevertheless, there is a way to illustrate this object in its construction.

You glue the red arrows of the square together (left and right edges ) so that one obtains a cylinder. You pull the cylinder apart slightly and continue to glue the ends together so that the arrows fit on the circle. The circular area of ​​a cylindrical surface is shifted through those of the other. Note that this process leads to the crossing of edges. This is called immersion of the Klein bottle in three dimensional space.

Step 2

Step 3

Step 4

Step 5

Step 6

When you place the Klein bottle in the four-dimensional real space one, a self-intersection can be avoided. Clearly, this is done as follows: take the immersion pictured above in the three-dimensional space and initially leaves the fourth coordinate at zero. Near the self-penetration increasing the value of the fourth coordinate for one of the (local) components steadily to one, and then lowers it again. Graphically, the fourth coordinate can be illustrated by a different color choice.

Topological properties

The fundamental group of the Klein bottle has the presentation

The homology groups are

The Klein bottle is a non- orientable closed surface of genus 1

There is a 2- sheeted covering of the Klein Bottle by the torus.