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Klein bottle

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In mathematics, 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 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 (without touching, though this is physically impossible for a real three-dimensional bottle), 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, but it is necessary when depicting it in three-dimensional Euclidean space.

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").

Contents

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:

The Klein bottle can be seen as a fiber bundle, that 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.

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

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:

[x = \left(r + \cos\frac\sin v - \sin\frac\sin 2v\right) \cos u]
[y = \left(r + \cos\frac\sin v - \sin\frac\sin 2v\right) \sin u]
[z = \sin\frac\sin v + \cos\frac\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.

Trivia

See also

References

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