# Klein bottle

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KleinBottle-01.png
The Klein bottle immersed in three-dimensional space.

In mathematics, the Klein bottle is a certain genus-1 non-orientable surface, i.e. a surface (a two-dimensional topological space), for which there is no distinction between the "inside" and the "outside" of the surface. The Klein bottle was first described in 1882 by the German mathematician Felix Klein. It is closely related to the Mbius strip and embeddings of the real projective plane such as Boy's surface.

Picture a bottle with a hole in the bottom. Now extend the neck. Curve the neck back on itself, insert it through the side of the bottle (a true Klein bottle in four dimensions would not require this step, but it is necessary when representing it in three-dimensional Euclidean space), and connect it to the hole in the bottom.

Unlike a drinking glass, this object has no "rim" where the surface stops abruptly. Unlike a balloon, a fly can go from the outside to the inside without passing through the surface (so there isn't really an "outside" and "inside").

The name 'Klein bottle' seems to have arisen from a mistranslation of the German term 'flche' which means 'surface'. This was mistaken for the word 'flasche' which means bottle. Nevertheless, the name is appropriate.

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## Properties

Topologically, the Klein bottle can be defined as the square [0,1] × [0,1] with sides identified by the relations (0,y) ~ (1,y) for 0 ≤ y ≤ 1 and (x,0) ~ (1-x,1) for 0 ≤ x ≤ 1, as in the following diagram:

   ---->
^   ^
|   |
<----


Like the Mbius strip, the Klein bottle is a two-dimensional differentiable manifold which is not orientable. Unlike the Mbius strip, the Klein bottle is a closed manifold, meaning it is a compact manifold without boundary. While the Mbius strip can be embedded in three-dimensional Euclidean space R3, the Klein bottle cannot. It can be embedded in R4, however.

The Klein bottle can be constructed (in a mathematical sense) by joining the edges of two Mbius strips together, as described in the following anonymous limerick:

A mathematician named Klein
Thought the Mbius band was divine.
Said he: "If you glue
The edges of two,
You'll get a weird bottle like mine."

The Klein bottle is the only exception to the Heawood conjecture, a generalization of the four color theorem.

## Dissection

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KleinBottle-02.png
Dissecting the Klein bottle results in a Mbius strip.
Missing image
KleinBottle-Figure8-01.png
The "figure 8" immersion of the Klein bottle.

If a Klein bottle is dissected into halves along its plane of symmetry, the result is a Mbius strip, pictured right. Remember that the intersection pictured isn't really there. In fact, it is possible to cut the Klein bottle into a single Mbius strip.

## Parametrization

The "figure 8" immersion of the Klein bottle has a particularly simple parametrization:

[itex]x = \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \cos u[itex]
[itex]y = \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \sin u[itex]
[itex]z = \sin\frac{u}{2}\sin v + \cos\frac{u}{2}\sin 2v[itex]

In this immersion, the self-intersection circle is a geometric circle in the XY plane. The positive constant [itex]r[itex] is the radius of this circle. The parameter [itex]u[itex] gives the angle in the XY plane, and [itex]v[itex] specifies the position around the 8-shaped cross section.

## References

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy