Klein bottle

A two-dimensional representation of the Klein bottle immersed in three-dimensional space
Structure of a three-dimensional Klein bottle

In mathematics, the Klein bottle is an example of a non-orientable surface; informally, it is a surface (a two-dimensional manifold) in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary (for comparison, a sphere is an orientable surface with no boundary).

The Klein bottle was first described in 1882 by the German mathematician Felix Klein. It may have been originally named the Kleinsche Fläche ("Klein surface") and then misinterpreted as Kleinsche Flasche ("Klein bottle"), which ultimately led to the adoption of this term in the German language as well.[1]

Construction

Start with a square, and then glue together corresponding coloured 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. Gently push a piece of the tube containing the intersection along the fourth dimension, 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.

Embedded Klein bottles in the Science Museum in London
A hand-blown Klein Bottle
Dissecting the Klein bottle results in Möbius strips.

The common physical model of a Klein bottle is a similar construction. The Science Museum in London 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.[2]

Properties

Like the Möbius strip, the Klein bottle is a two-dimensional 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 seen as a fiber bundle over the circle S1, with fibre S1, as follows: one takes the square (modulo the edge identifying equivalence relation) from above to be E, the total space, while the base space B is given by the unit interval in y, modulo 1~0. The projection π:EB is then given by π([x, y]) = [y].

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 limerick by Leo Moser:[3]

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 initial construction of the Klein bottle by identifying opposite edges of a square shows that the Klein bottle is a CW complex with one 0-cell P, two 1-cells C1, C2 and one 2-cell D. Its Euler characteristic is therefore 1-2+1 = 0. The boundary homomorphism is given by ∂D = 2C1 and ∂C1=∂C1=0, yielding the homology groups of the Klein bottle K to be H0(K,Z)=Z, H1(K,Z)=Z×(Z/2Z) and Hn(K,Z) = 0 for n>1.

There is a 2-1 covering map from the torus to the Klein bottle, because two copies of the fundamental region of the Klein bottle, one being placed next to the mirror image of the other, yield a fundamental region of the torus. The universal cover of both the torus and the Klein bottle is the plane R2.

The fundamental group of the Klein bottle can be determined as the group of deck transformations of the universal cover and has the presentation <a,b | ab = b−1a>.

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.

A Klein bottle is homeomorphic to the connected sum of two projective planes. It is also homeomorphic to a sphere plus two cross caps.

When embedded in Euclidean space the Klein bottle is one-sided. However there are other topological 3-spaces, and in some of the non-orientable examples a Klein bottle can be embedded such that it is two-sided, though due to the nature of the space it remains non-orientable.[4]

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.

Simple-closed curves

One description of the types of simple-closed curves that may appear on the surface of the Klein bottle is given by the use of the first homology group of the Klein bottle calculated with integer coefficients. This group is isomorphic to Z×Z2. Up to reversal of orientation, the only homology classes which contain simple-closed curves are as follows: (0,0), (1,0), (1,1), (2,0), (0,1). Up to reversal of the orientation of a simple closed curve, if it lies within one of the two crosscaps that make up the Klein bottle, then it is in homology class (1,0) or (1,1); if it cuts the Klein bottle into two Möbius strips, then it is in homology class (2,0); if it cuts the Klein bottle into an annulus, then it is in homology class (0,1); and if bounds a disk, then it is in homology class (0,0).

Parameterization

The "figure 8" immersion of the Klein bottle.
Klein bagel cross section employing a figure eight curve (the lemniscate of Gerono).

The figure 8 immersion

The "figure 8" immersion (Klein bagel) of the Klein bottle has a particularly simple parameterization. It is that of a "figure-8" torus with a 180 degree "Möbius" twist inserted:

\begin{align} x & = \left(r + \cos\frac{\theta}{2}\sin v - \sin\frac{\theta}{2}\sin 2v\right) \cos \theta\\ y & = \left(r + \cos\frac{\theta}{2}\sin v - \sin\frac{\theta}{2}\sin 2v\right) \sin \theta\\ z & = \sin\frac{\theta}{2}\sin v + \cos\frac{\theta}{2}\sin 2v \end{align}

for 0 ≤ θ < 2π, 0 ≤ v < 2π and r > 2.

In this immersion, the self-intersection circle (when v = 0, π) is a geometric circle in the xy-plane. The positive constant r is the radius of this circle. The parameter θ gives the angle in the xy-plane, and v specifies the position around the 8-shaped cross section. With the above parameterization the cross section is a 2:1 Lissajous curve.

In four dimensions this surface can be made non-intersecting by adding a little v dependent "bump" to the fourth w axis at the intersection point. E.g.

\begin{align} w & = \cos v \end{align}

4-D non-intersecting

Another non-intersecting 4-D parameterization is modeled after that of the flat torus:

\ \begin{align} x & = R\left(\cos\frac{\theta}{2}\cos v-\sin\frac{\theta}{2}\sin 2v\right) \\ y & = R\left(\sin\frac{\theta}{2}\cos v+\cos\frac{\theta}{2}\sin 2v\right) \\ z & = P\cos\theta\left(1+{\epsilon}\sin v\right) \\ w &= P\sin\theta\left(1+{\epsilon}\sin v\right) \end{align}

where R and P are constants that determine aspect ratio, θ and v are similar to as defined above. v determines the position around the figure-8 as well as the position in the x-y plane. θ determines the rotational angle of the figure-8 as well and the position around the z-w plane. ε is any small constant and ε sinv is a small v depended bump in z-w space to avoid self intersection. The v bump causes the self intersecting 2-D/planar figure-8 to spread out into a 3-D stylized "potato chip" or saddle shape in the x-y-w and x-y-z space viewed edge on. When ε=0 the self intersection is a circle in the z-w plane <0, 0, cosθ, sinθ>.

Bottle shape

The parameterization of the 3-dimensional immersion of the bottle itself is much more complicated. Here is a version found by Robert Israel:

Klein Bottle with slight transparency
\begin{align} x(u,v) &= -\frac{2}{15} \cos u (3 \cos{v}-30 \sin{u}+90 \cos^4{u} \sin{u} \\ &\quad -60 \cos^6{u} \sin{u}+5 \cos{u} \cos{v} \sin{u}) \\ y(u,v) &= -\frac{1}{15} \sin u (3 \cos{v}-3 \cos^2{u} \cos{v}-48 \cos^4{u} \cos{v}+ 48 \cos^6{u} \\ &\quad \cos{v}-60 \sin{u}+5 \cos{u} \cos{v} \sin{u}-5 \cos^3{u} \cos{v} \sin{u}-80 \\ &\quad \cos^5{u} \cos{v} \sin{u}+80 \cos^7{u} \cos{v} \sin{u}) \\ z(u,v) &= \frac{2}{15} (3+5 \cos{u} \sin{u}) \sin{v} \end{align}

for 0 ≤ u < π and 0 ≤ v < 2π.

Generalizations

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

In another order of ideas, constructing 3-manifolds, it is known that a solid Klein bottle is topologically equivalent with the Cartesian product: $\scriptstyle M\ddot{o}\times I$, the Mobius strip times an interval. The solid Klein bottle is the non-orientable version of the solid torus, equivalent to $\scriptstyle D^2\times S^1$.

Klein surface

A Klein surface is, as for Riemann surfaces, a surface with an atlas allowing the transition maps to be composed using complex conjugation. One can obtain the so-called dianalytic structure of the space.