Klein bottle

The Klein bottle is a certain 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 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's necessary to represent it in three dimensions), 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").

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."

See also: topology, algebraic topology

External links

• Acme Klein Bottles (http://www.kleinbottle.com/) - Actual Klein bottles! (Or at least 3D projections of them).
• Animated 3D-picture of a Klein bottle (http://www.geom.umn.edu/~banchoff/Klein4D/Klein4D)
• Andrew Lipson's Mathematical LEGO Sculptures (http://www.lipsons.pwp.blueyonder.co.uk/mathlego.htm) - Lego constructions of Möbius strip and Klein bottle structures.
• Klein Bottle Images by John Sullivan (http://www.math.uiuc.edu/~jms/Images/klein)