The geometric object, later called the "Klein 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) simply cannot exist in our three-dimensional world. All models that are for sale in souvenir shops have a look that only gives a vague 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 two holes in it: one in the wall, and the second in the bottom. Then bend the neck, insert it into the hole in the wall and bring it out through the hole in the bottom. The resulting object will be a projection of a four-dimensional space object, which is the real Klein bottle, in our three-dimensional space.
Description of the Klein bottle in the language of mathematical terms orformulas will not say anything to the layman. Will such a definition satisfy many people: a Klein bottle is a non-orientable manifold (or surface) that has 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 got an idea of what the projection of a bottle is in three-dimensional space.
Interesting fact: the name “Klein bottle” was given to this object, most likely, due to a mistake or typographical error of the translator. The fact is that Klein in his definition used the word Fläche, that is, “surface” in German. When "traveling" from Germany to other countries, this word was transformed into a similar spelling Flasche (bottle). Then the term returned to the country of origin in a new, modified form, and remained so forever.
For many cultural figures (primarily science fiction writers), the very term "Klein bottle" turned out to be attractive. Its use as an attribute, and sometimes the main character, has become a sign of "intellectual" fiction. Such, for example, is the story "The Last Illusionist", written by Bruce Eliot. In the story, a magician's assistant cracks down on his patron, who was doing tricks with a four-dimensional Klein bottle. The illusionist who climbed into the bottle remains half immersed in it. According to the author, this bottle cannot be broken without damaging the contents. Is it really so - can not saynone. At least, mathematicians, who could possibly answer this question, were not puzzled by it, for science this is irrelevant.
Sometimes specially made Klein bottles are filled with wine for promotional purposes. True, it is technically difficult to make such a glass bottle; this requires an extra-class glass blower. Therefore, it has a rather high cost and is used infrequently. And the development of technology and the production of such bottles on a stream does not make sense, because for this it will be necessary to work out the method of filling the bottle with liquid (here, too, there are difficulties). And the feeling of unusualness and novelty will be quickly replaced by the inconvenience of pouring wine from such a bottle into glasses.
