Klein Bottle

kleinbottle_fullThe Klein bottle is the next step up from a Moebius strip.  A Moebius Strip is a two-dimensional object in three-dimensional space, and a Klein Bottle is a one-dimensional object in three-dimensional space. Therefore, it has to intersect itself in order to be representable in three dimensions. It is a non-orientable surface with Euler characteristic equal to 0.

A Klein bottle can be made from a rectangular piece of the plane by identifying the top and bottom edges using the same orientation, but identifying the left and right edges with opposite orientation (as in the formation of a Möbius band). The first step forms a tube, but the second step can not be carried out without causing self-intersection: the tube must pass through itself in order to attach the ends correctly.

Another way to make the Klein bottle is to take two Möbius bands and join them along their boundaries (each band has a single boundary curve). Finally, the Klein bottle is the connected sum of two real projective planes, since the projective plane minus a disk is just a Möbius band.

The Klein bottle can not be embedded in three-space, but it can be represented there.

Most containers have an inside and an outside, a Klein bottle is a closed surface with no interior and only one surface. It is unrealizable in 3 dimensions without intersecting surfaces. It can be realized in 4 dimensions. The classical representation is shown below.

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  1. la banda de moebius es bilatera aniquila el quinto postulado de euclides pero el crosss cap y l la bortella de klein siendo asfericos necesitan del corte de autopenetracion segun stephen barr en su texto de ejercicios topologicos ,marc darmon y vapepereau solucionan la cosa con el hors point et ligne point siguiendo lo indicado por pierre soury. trabajo la topologia con el tetraedro y exadicos de rene lew aplanando los nudos para resolver el problema de doble retorsion intrinsica antihoraria levogrica que ello implica. quisiera poder enviar mis disenos soy el director de espacios topologicos del ecuador


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