Klein bottle

In mathematics, the Klein bottle is a certain non-orientable surface, i.e., a surface (a two-dimensional topological space) with no distinction between the "inside" and "outside" surfaces. 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. The initial name given was "Klein Fläche" (Fläche = Surface); however, this was wrongfully interpreted as "Klein Flasche" (Flasche = bottle), which ultimately, due to the dominance of the English language in science, led to the adoption of this term in the German language, too.

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 without touching the surface (an act which is impossible in three-dimensional space), and extend the neck down inside the bottle until it joins the hole in the bottom. A true Klein bottle in four dimensions does not intersect itself where it crosses the side.

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").

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:

This square is known as the fundamental polygon of the Klein bottle.

The Klein bottle can be seen as a fiber bundle as follows: one takes this square to be E, the total space, while the base space B is given by the unit interval in x, and the projection $\pi$ is given by $\pi (x,y)=x.$ Since the two endpoints of the unit interval in x are identified, the base space B is actually the circle $S^{1}$ , and so the Klein bottle is the twisted $S^{1}$ -bundle (circle bundle) over the circle.

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.

File:Kleinbot2.jpg

a handblown Klein Bottle made by Mitsugi Ohno

The Klein bottle can be constructed (in a mathematical sense, because it cannot be done without allowing the surface to intersect itself) 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."

It can also be constructed by folding a Möbius strip in half lengthwise and attaching the edge to itself.

Six colors suffice to color any map on the surface of a Klein bottle; this is the only exception to the Heawood conjecture, a generalization of the four color theorem, which would require seven.

Dissection

Dissecting the Klein bottle results in Möbius strips.

The "figure 8" immersion of the Klein bottle.

Dissecting a Klein bottle into halves along its plane of symmetry results in two mirror image Möbius strips, i.e. one with a left-handed half-twist and the other with a right-handed half-twist (one of these is pictured on the right). Remember that the intersection pictured isn't really there. In fact, it is also 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:

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

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

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

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

Generalizations

The generalization of the Klein bottle to higher genus is given in the article on the fundamental polygon.

Trivia

File:Kleins-beer (futurama).jpg

Klein's Beer bottles in Futurama

A mounted Klein bottle is the trophy for the BASIC WonderCup Challenge.
The TV series Futurama shows a brand of beer named Klein's on a shelf – in a Klein bottle.
The British Science Museum has on display a beautiful collection of hand-blown glass Klein bottles, exhibiting many variations on the same topological theme. The bottles date from 1995 and were made for the museum by Alan Bennett. [1]
Russell Hoban's 2001 novel Amaryllis Night and Day makes extensive use of the Klein bottle as a metaphor. The display of bottles at London's Science Museum, and Alan Bennett himself, also feature in the book.
In the computer game Nethack, attempting to dip a potion into itself gives the message, "That is a potion bottle, not a Klein bottle!"
Joe Strummer and the Mescaleros have a song entitled "Mega Bottle Ride" which describes a journey "into the fourth dimension" via the "Banchoff-Klein Mega Bottle Ride".
In the Janine Melnitz, Ghostbuster episode of the TV show The Real Ghostbusters, Ray mentions adding another Klein bottle to the Containment Unit.
In the book "Visitors From Oz", the characters construct a Klein bottle to travel from Oz to Earth.
In the Infocom game Trinity, a giant Klein bottle figures prominently, and is used to help solve one of the puzzles.
Clifford Stoll, author of The Cuckoo's Egg manufactures Klein bottles and sells them via the Internet at Acme Klein Bottle.

References

Weisstein, Eric W. "Klein Bottle". MathWorld.

External links