Klein's bottle: what she is

The geometrical object, later called "Klein's bottle", was first described in 1882 by the German mathematician Felix Klein. What does he represent? This object (or rather, a geometric or topological surface) in our three-dimensional world can not exist. All models that are put up for sale in souvenir shops look like they only give a remote idea of what a Klein bottle is. For greater clarity, it is described as follows: imagine a bottle with a very long neck. Then mentally make in it two holes: one in the wall, and the other in the bottom. Then bend the neck, insert it into the hole in the wall and pull it out through the hole in the bottom. The resulting object will be the projection of the object of the four-dimensional space, which is the real bottle of Klein, in our three-dimensional space.

The description of Klein's bottle by the language of mathematical terms or formulas will not tell a non-professional anything. How many will be satisfied with such a definition: a Klein bottle is a non-orientable manifold (or surface) possessing a number of properties. After the word "properties" you can build a long series consisting of trigonometric functions, numbers and Greek and Latin letters. But this can only confuse an unprepared person who has already received an idea of what the projection of a bottle in three-dimensional space is.

Interesting fact: the name "Klein bottle" this object was most likely due to an error or an interpreter's mistake. The fact is that Klein in his definition used the word Fläche, that is, the "surface" in German. When "traveling" from Germany to other countries, this word was transformed into a Flasche (bottle) similar to the one written. Then the term returned to the country of origin already in a new, modified form, and so it remained forever.

For many cultural figures (especially science fiction writers), the term "Klein bottle" itself was attractive. Its application as an attribute, and sometimes even the main acting "person", became a sign of "intellectual" fiction. Such, for example, is the story "The Last Illusionist", which belongs to the pen of Bruce Eliot. According to the plot, the magician's assistant straightens out his patron, who did the tricks with Klein's four-dimensional bottle. The illusionist who has got into a bottle and remains half immersed in it. According to the author, this bottle can not be broken without damaging the contents. Is it really so? Nobody can say. At least mathematicians, who might have been able to answer this question, did not puzzle them, for science it is irrelevant.

Occasionally, wine is bottled in specially made Klein bottles for promotional purposes. True, it is technically difficult to make such a bottle of glass, this requires an extra-class glass blower. Therefore, it has a fairly high cost and is rarely used. And the testing of technology and setting production of such bottles on the flow does not make sense, because for this it will be necessary to practice the technique of filling the bottle with liquid (here too, there are complications). And the feeling of unusualness and novelty will be quickly replaced by inconvenience when bottling wine from such a bottle by the glass.