# Klein bottle

#### Background Information

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In mathematics, the **Klein bottle** is a certain non- orientable surface, *i.e.*, a surface (a two-dimensional topological space) with no distinct "inner" and "outer" sides. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a two dimensional object with one side and one edge, a Klein bottle is a three dimensional object with one side and *no* edges. (For comparison, a sphere is a three dimensional object with no edges and two sides.)

The Klein bottle was first described in 1882 by the German mathematician Felix Klein. It was originally named the *Kleinsche Fläche* "Klein surface"; however, this was incorrectly interpreted as *Kleinsche Flasche* "Klein bottle", which ultimately led to the adoption of this term in the German language as well.

## Construction

Start with a square, and then glue together corresponding colored edges, in the following diagram, so that the arrows match. More formally, the Klein bottle is the quotient space described 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:

This square is a fundamental polygon of the Klein bottle.

Note that this is an "abstract" gluing in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle. The Klein bottle, proper, does not self-intersect. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions.

Glue the red arrows of the square together (left and right sides), resulting in a cylinder. To glue the ends together so that the arrows on the circles match, pass one end through the side of the cylinder. Note that this creates a circle of self-intersection. This is an immersion of the Klein bottle in three dimensions.

By adding a fourth dimension to the three dimensional space, the self-intersection can be eliminated. Gradually push a piece of the tube containing the intersection out of the original three dimensional space. A useful analogy is to consider a self-intersecting curve on the plane; self-intersections can be eliminated by lifting one strand off the plane.

This immersion is useful for visualizing many properties of the Klein bottle. For example, the Klein bottle has no *boundary*, where the surface stops abruptly, and it is non-orientable, as reflected in the one-sidedness of the immersion.

The common physical model of a Klein bottle is a similar construction. The British Science Museum has on display a collection of hand-blown glass Klein bottles, exhibiting many variations on this topological theme. The bottles date from 1995 and were made for the museum by Alan Bennett. Clifford Stoll, author of * The Cuckoo's Egg*, manufactures Klein bottles and sells them via the Internet at Acme Klein Bottle.

## Properties

The Klein bottle can be seen as a fibre bundle as follows: one takes the square from above to be *E*, the total space, while the base space *B* is given by the unit interval in *x*, and the projection π is given by π(*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 **R**^{3}, the Klein bottle cannot. It can be embedded in **R**^{4}, however.

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

A Klein bottle is equivalent to a sphere plus two cross caps.

## Dissection

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:

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.

The parametrization of the 3-dimensional immersion of the bottle itself is much more complicated. Here is a simplified version:

where

for 0 ≤ *u* < 2π and 0 ≤ *v* < 2π.

In this parametrization, *u* follows the length of the bottle's body while *v* goes around its circumference.

## Generalizations

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

## Klein surface

A **Klein surface** is, as for Riemann surfaces, a surface with an atlas allowing that the transition functions can be composed with complex conjugation one can obtains the so called dianalytic structure.