Klein's bottle: what she is

A geometric object that subsequently receivedThe name “Klein bottle” was first described in 1882 by the German mathematician Felix Klein. What is he like? This object (or rather, a geometric or topological surface) simply cannot exist in our three-dimensional world. All the models that are put up for sale in the souvenir shops have the appearance that only gives a distant idea of ​​what a Klein bottle is.

For greater clarity, it is described as: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 out through the hole in the bottom. The resulting object will be a projection of the object of four-dimensional space, which is the real Klein bottle, in our three-dimensional space.

Описание бутылки Клейна языком математических terms or formulas will not say anything to the non-professional. How many will satisfy such a definition: a Klein bottle is a non-orientable manifold (or surface) with a number of properties. After the word "properties" you can build a long line 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 the bottle in three-dimensional space is.

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

For many cultural figures (first of allscience fiction writers) turned out to be attractive the term "bottle of Klein". Its use as an attribute, and sometimes the main actor, became a sign of intellectual fiction. Such, for example, is the story "The Last Illusionist," written by Bruce Eliot. In the story the assistant magician deals with his patron, who did tricks with a four-dimensional bottle of Klein. The illusionist who has got into the bottle remains half immersed in it. According to the author, this bottle can not be broken without damaging the contents. Is it really so - no one can say. At least, the mathematicians who, perhaps, could have answered this question, did not bother them, for science this is irrelevant.

Occasionally in specially made Klein bottlesfor promotional purposes spill wine. True, to make such a bottle of glass is technically difficult, for this you need an extra-grade glass blower. Therefore, it has a fairly high cost and is used infrequently. And working out the technology and staging the production of such bottles for a stream does not make sense, because this would require working out the technique of filling the bottle with liquid (there are also difficulties). And the feeling of unusualness and novelty will be quickly superseded by the inconvenience of spilling wine from such a bottle in wine glasses.