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KLEIN BOTTLE
link to a manipulable figure
Parametrization without singular points in :
Parametrization giving the classic form in (hence with selfintersection): First part: Second part: , where , with a = 3, b = 4, c = 2. Parametrization as a sinusoidal torus (3D projection of the 4D surface above): . Rotation of a figureeight with a twist:. Polynomial Cartesian equation of Ian Stewart: . See here a general method to create a Klein bottle. 
The notion of Klein bottle, written , refers to any topological space homeomorphic to that obtained by identification in a full square of the opposite sides with a change of direction for one of the pairs. 

If we begin by identifying the two sides with same direction (in concrete terms, in we sew them together), we get a trunk of cylinder; the Klein bottle is therefore a trunk of cylinder the two boundaries of which are identified with an inversion of direction. 

If we begin by identifying the two sides with opposite directions, we get a Möbius strip. The Klein bottle is therefore a Möbius strip the two halfboundaries of which are identified in the same direction. (If they are identified with an inversion of direction, we get a projective plane). 

By using the triangular representation of the Möbius strip, and identifying 2 lower halfbases of the triangle in the same direction, we get the opposite representation of the Klein bottle...  
...which shows that the Klein bottle is also the surface obtained by sewing two Möbius strips together.  
View showing that a half Klein bottle is indeed a Möbius strip.


Moreover, since the Möbius strip is a punctured projective plane, the Klein bottle is also the connected sum of two real projective planes.
We therefore get another representation of it by adjoining two open crosscaps:

This is also a Klein bottle! 
The Möbius strip is a onesided surface (with one face), therefore is not orientable, of genus 2, zero Euler characteristic, and chromatic number equal to 6 like the torus:(and not 7 like the Heawood formula would indicate) :
It cannot be embedded in R^{3}, but only immerged with a selfintersection.
Much as the classic representation does indeed look like a bottle (into which one should avoid to pour liquid!),
The 3 following representations of the Klein bottle using this method are more symmetrical than the classic representation, but have the inconvenient of not having a welldefined tangent plane at some points:



Collage of a half torus with a half Bohemian dome.
There are two cuspidal points. 
Sine torus.
Two cuspidal points too. 
Tube with a bore shaped like a piriform quartic.
There is a cuspidal line. 
Rotation of a figureeight curve with a halftwist; we get this time a real immersion of the Klein bottle, not at all shaped like a bottle! 


The first representation allows to see how the Klein bottle is obtained by identification of two halfboundaries of a Möbius strip: 
The second representation allows to see how the Klein bottle is obtained by sewing two Möbius strips together.


In , the movement of a circle with a twist can be carried out without the resulting surface presenting selfintersections (see the parametrization in the above box).
Finally, here is a polyhedral model of the Klein bottle:
If we paint the Klein bottle, the layer of paint obtained (which is in one piece since the surface is onesided) is an immersion of the torus (in other words, the twosheeted covering is the torus); it is the reason why the Klein bottle was used as a central step for turning the torus inside out: see for example this text in Pour la Science.
The three utilities problem can be solved on the Klein bottle, whereas it cannot be solved on the plane. In other words, the bipartite graph can be traced on the Klein bottle without edges crossing.
The same holds for the complete graph with 5 vertices that the reader will trace easily. 

The punctured Klein bottle (characterized by the fact that it is a onesided surface of genus 2 with a boundary) can be represented by the Möbius shorts: we get it by connecting a closed strip with no twists by a band like below:
These shorts, impossible to put on, are a punctured Klein bottle! 

Just as the Möbius strip can be generalized into a strip with n halftwists (and we topologically get a Möbius strip only when n is odd), the Klein bottle can be generalized into a bottle with n twists, which is topologically a Klein bottle only when n is odd.
Opposite, the multiple bottles were created from a tube with variable section the bore of which is an epicycloid. 

The Klein bottle enables computer graphics designers to display their talent:
LINKS:
www.math.ohiostate.edu/~fiedorow/math655/Klein2.html
www.math.uiuc.edu/~jms/Images/klein.html
www.geom.uiuc.edu/~banchoff/Klein4D/Klein4D.html
Games on a Klein bottle: geometrygames.org/TorusGames/index.html
To buy glass Klein bottle and even a Klein cap!: www.kleinbottle.com
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© Robert FERRÉOL, Jacques MANDONNET 2017