The Klein Bottle: Non-Orientable Surface

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  • Topic: Manifold, Orientability, Klein bottle
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  • Published : December 3, 2010
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In mathematics, the Klein bottle ([klaɪ̯n]) is a non-orientable surface, informally, a surface (a two-dimensional manifold) with no identifiable "inner" and "outer" sides. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a two-dimensional 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.

In physics/electro-technology:
* as compact resonator with the resonance frequency which is half that of identically constructed linear coils * as inductionless resistance.
* as superconductors with high transition temperature
In chemistry/nano-technology:
* as molecular knots with special characteristics (Knotane) * as molecular engines
* as graphene volume (nano-graphite) with new electronic characteristics, like helical magnetism. * In a special type of aromaticity: Möbius aromaticity * Charged particles, which were caught in the magnetic field of the earth, can move on a Möbius band. * The cyclotide (cyclic protein), active substance of the plant Oldenlandia affinis, contains Möbius topology for the peptide backbone.

In mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent (i.e. homeomorphic) to a Möbius strip. The term ‘cross-cap’, however, often implies that the surface has been deformed so that its boundary is an ordinary circle.

A cross-cap that has been closed up by gluing a disc to its boundary is an immersion of the real projective plane. Two cross-caps glued together at their boundaries form a Klein bottle.

In chemistry, a molecular knot, or knotane, is a mechanically-interlocked molecular architecture. Examples of naturally formed knotanes are DNA and certain proteins. Lactoferrin has an unusual biochemical reactivity compared to its...
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