Klein bottle

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 Möbius 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 'fläche' which means 'surface'. This was mistaken for the word 'flasche' which means bottle. Nevertheless, the name is appropriate.

Contents

1 Properties 2 Dissection 3 Parametrization 4 See also 5 References 6 External links

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 Möbius strip, the Klein bottle is a two-dimensional differentiable manifold which is not orientable. Unlike the Möbius strip, the Klein bottle is a closed manifold, meaning it is a compact manifold without boundary. While the Möbius 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 Möbius strips together, as described in the following anonymous limerick:

A mathematician named Klein

Thought the Möbius 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

If a Klein bottle is dissected into halves along its plane of symmetry, the result is a Möbius 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 Möbius strip.

Parametrization

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

<math>x = \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \cos u<math>

<math>y = \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \sin u<math>

<math>z = \sin\frac{u}{2}\sin v + \cos\frac{u}{2}\sin 2v<math>

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

See also

• Topology
• Algebraic topology
• Surface

References

• Template:MathWorld

External links

Template:Commons

• Acme Klein Bottles (http://www.kleinbottle.com/) - Clifford Stoll sells blown-glass immersed Klein bottles.
• Paper Strip Activities (http://www.cut-the-knot.org/do_you_know/paper_strip.shtml)
• Slider puzzles (http://www.cut-the-knot.org/Curriculum/Combinatorics/Sliders.shtml) (Slider puzzles on the plane, cylinder, torus, Klein's bottle and Projectie plane)
• Klein bottle construction (http://www.cut-the-knot.org/shortcut.shtml) (an avi movie)
• Andrew Lipson's Mathematical LEGO Sculptures (http://www.lipsons.pwp.blueyonder.co.uk/mathlego.htm) - Lego constructions of Möbius strip and Klein bottle structures. This site also shows the dissection of the Klein bottle.
• Klein Bottle Images by John Sullivan (http://www.math.uiuc.edu/~jms/Images/klein.html)
• The Klein bottle (http://www.geom.uiuc.edu/zoo/toptype/klein/)
• A modular origami model of the Klein bottle (http://hektor.umcs.lublin.pl/~mikosmul/origami/misc.html)de:Kleinsche Flasche

es:Botella de Klein fr:Bouteille de Klein io:Klein-botelo ja:クラインの壺 pl:Butelka Kleina