The Klein bottle reference article from the English Wikipedia on 24-Jul-2004
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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 (a true Klein bottle in four dimensions would not require this step, but it is necessary when representing it in three-dimensional Euclidean space), 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").

Two views of a Klein bottle immersed in three-dimensional space

Two views of a Klein bottle immersed in three-dimensional space.

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."

If a Klein bottle is dissected into halves along its plane of symmetry, the result is the surface shown in Figure 1. Note that the tube part that is coming out the side isnt shown dissected in half nor is the top part of the bottle.therefor the bottle isnt dissected along its plane of symmetry.

Figure 1(a): Dissection of a Klein bottle. Figure 1(b): ROT13 cipher inscribed along perimeter of dissection.

In Figure 1(b), Twenty six points on the dissection's perimeter (the blue curve) have been labeled with the twenty six letters of the alphabet. But the dissection is a surface, not a curve. The red lines show how the surface is subtended by the perimeter.

Figure 2 shows a Möbius strip. Figure 2. A Moebius strip

Figure 2. A Möbius strip.

The strip is a surface: its perimeter is shown as a blue curve, and the red lines show how the surface is subtended by the perimeter.

In both the dissected Klein bottle and Möbius strip, the red lines connect letters which are related mutually in the ROT13 cipher. This helps to illustrate that half a Klein bottle is homeomorphic to a Möbius strip.

It is also possible to perceive directly that Figure 1 is a Möbius strip, by imagining that the narrower, re-entrant part of the bottle no longer intersects line segment DB after the dissection is performed, but that it becomes loose from dissecting plane and Figure 1 is actually three-dimensional, with line segments VW and IJ hovering above line segment DB. Then, suddenly, Figure 1 looks like a roller coaster, and by imagining the motion of a rail car along the blue rails of this roller coaster, one perceives that this roller coaster is non-orientable.

Parametrizing the Klein bottle

Since the Klein bottle is a topological surface, there can be several ways of describing it parametrically. One way is the following:

where parameters ψ ranges from 0 to and φ ranges from 0 to ; δ\ is the
Dirac delta function and tanh is the hyperbolic tangent function. Variables X, Y and Z in the last three equations are the actual coordinates in of a point on the Klein bottle whose parameters are φ and ψ.

A Klein bottle which is defined by the set of parametric equations above.

Equations (2), (4) and (6) are derivatives with respect to φ of equations (1), (3) and (5) — respectively. Functions x(φ) and y(φ) are the components of a position vector function which defines the locus of points of a plane curve which has one cusp at the origin. This curve might be called the "spine" of the Klein bottle, but this spine is not itself a subset of the Klein bottle.

Equations (3) and (5) not only describe a plane curve, they also describe the movement of a point along that plane curve. The cusp is a special point where the curve abruptly reverses direction by 180°. The movement of the point along the "spine" has been defined in such a way that the point slows down as it gets closer to the cusp, stops when it reaches the cusp, and then begins to accelerate in the opposite direction.

Functions x'(φ) and y'(φ) are the components of a vector function which is the tangent vector of spine. Functions xN(φ) and yN(φ) are the components of a vector function which is the unit normal vector of the spine (see Frenet-Serret formulas). The binormal unit vector always points in the +z direction — since it is a curve on the x-y plane — but is not necessary for parametrizing the Klein bottle. The unit normal was obtained from the tangent by rotating 90° — that is, by changing (x′, y′) to (−y′, x′) — and then normalizing. The Dirac delta terms were added to Equations (7) and (8) in order to fix discontinuities at the pinch point.

Equations (9) and (10) describe the "amplitude constant" α and the "amplitude function" w(φ). Both of these are used to modulate the "amplitude" — i.e. magnitude — of the normal vector in Equations (11), (12) and (13). The points on the Klein bottle are defined by rotating the amplified normal vector 360° around the position vector's point on the spine (the tail of the normal vector is at the head of the position vector; the point on the head of the normal vector belongs to the Klein bottle; the position vector moves along the spine). Parameter φ specifies a point on the spine, and parameter ψ specifies the degree of rotation of the normal around the spine.

See also: topology, algebraic topology

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