In mathematics, the Klein bottle (pronounced /ˈklaɪn/) is a nonorientable surface, informally, a surface (a twodimensional manifold) with no identifiable "inner" and "outer" sides. Other related nonorientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a twodimensional 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 is sometimes claimed that it was originally named the Kleinsche Fläche "Klein surface" and that this was incorrectly interpreted as Kleinsche Flasche "Klein bottle," which ultimately led to the adoption of this term in the German language as well.^{[1]}
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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 selfintersecting Klein bottle. The Klein bottle, proper, does not selfintersect. 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 selfintersection. This is an immersion of the Klein bottle in three dimensions.
By adding a fourth dimension to the three dimensional space, the selfintersection can be eliminated. Gently push a piece of the tube containing the intersection out of the original three dimensional space. A useful analogy is to consider a selfintersecting curve on the plane; selfintersections can be eliminated by lifting one strand off the plane.
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