The Klein Bottle

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Introduction:

The Klein bottle is a well-known and interesting surface which, like the Möbius strip, is non-orientable. There are actually two forms of Klein bottles; the one above (parameterized by the equations below) is defined much like a Möbius strip, while the one pictured below, which is defined more topologically, is the variety first proposed by C.F. Klein, for whom the surface is named.

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As your intuition might lead you to believe, these two Klein bottles are distinct when viewed as surfaces in R3; a neighborhood of the self-intersection curve of the first Klein bottle is non-orientable (like a Möbius strip made out of an X instead of a line), while a neighborhood of the self-intersection curve of the second Klein bottle is orientable.

One other way to represent a Klein bottle is using the box-and-arrow notation used to describe the Möbius strip. First, recall the representations for a cylinder and a Möbius strip:

Cylinder Möbius strip

 

The final surface is formed by bending the surface so that the two arrow heads intersect. (This can actually be done with a strip of paper; use a long strip with the arrows on the short ends.) A natural question arises: What would happen if we put arrows on the other two sides? We can now define a torus, a Klein bottle, and another surface known as a real projective plane:

Torus Klein bottle Projective plane

 

Definition:

Note the similarity of the above definition to the definition given for a Möbius strip. In fact, the definition for the first Klein bottle pictured above is identical to that of a Möbius strip, except that a figure eight is rotated about the circle instead of a line.

The second Klein bottle pictured above is defined topologically in terms of bending a tube over on itself in space. We begin with a hollow tube that has one large end and one smaller end. We twist the tube around at the smaller end, and poke it through the surface at the larger end. The surface is then completed by placing a half-torus on top, such that its inner edge coincides with the smaller end of the tube, and its outer edge joins with the larger end of the tube. Although this surface can be piecewise parameterized, the parameterization itself means very little.

Properties:

Gaussian Curvature:

K[u, v] = [click for formula]

Gaussian curvature of the surface.
Surface colored by Gaussian curvature.

Mean Curvature:

H[u, v] = [click for formula]

Mean curvature of the surface.
Surface colored by Mean curvature.