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A two-dimensional representation of the Klein bottle immersed in three-dimensional space. In mathematics, the Klein bottle (/ ˈ k l aɪ n /) is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down.
In the Klein bottle diagram, a goes round one way and −a goes round the opposite way. If a is thought of as a cut, then −a can be thought of as a gluing operation. Making a cut and then re-gluing it does not change the surface, so a + (−a) = 0. But now consider two a-cycles.
In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle. [ 1 ] It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder D 2 × I {\displaystyle \scriptstyle D^{2}\times I} to the bottom disk by a reflection across a diameter of the disk.
Date/Time Thumbnail Dimensions User Comment; current: 02:58, 16 February 2007: 150 × 150 (4 KB): Inductiveload {{Information |Description=One of a series of diagrams illustrating the folding of a rectiangle into a Klein Bottle |Source=Own drawing |Date=16/02/2007 |Author=Inductiveload |Permission={{PD-self}} |other_version
One of a series of diagrams illustrating the folding of a rectangle into a Klein Bottle: Date: 16 February 2007: Source: Own work (Own drawing)
Gluing the circles together will produce a new, closed manifold without boundary, the Klein bottle. Closing the surface does nothing to improve the lack of orientability, it merely removes the boundary. Thus, the Klein bottle is a closed surface with no distinction between inside and outside.
The Klein bottle (fundamental polygon with appropriate edge identifications) decomposed as two Möbius strips A (in blue) and B (in red). A slightly more difficult application of the Mayer–Vietoris sequence is the calculation of the homology groups of the Klein bottle X.
English: Save this script (slightly modified from - see for the page containing this model) to file "bottle" then run the command line "gnuplot bottle", you will get "bottle.svg", if you have well installed gnuplot 4.0 or later.