What is a Klein bottle?

Author: klein4d Release time: 2024-05-24 13:17:14 View number: 46

The Klein bottle was first introduced by German mathematician Felix Klein in 1882 and was named after him.


Interestingly, it is not actually called the Klein bottle, but a misspelling of the German name, which is actually known as the Klein flat. The German word "Kleinsche Fläche" was incorrectly written as "Kleinsche Flasche".


The German word "Fläche" means flat, while "Flasche", which is missing two dots, refers to the bottle. This is where the name Klein bottle comes from, isn't it interesting?


Before talking about the Klein bottle we can briefly understand the Möbius strip, which can be said to be the elementary version of the Klein bottle. discovered and proposed by the German mathematician Möbius in 1858.


It is very simple to make a Möbius ring. After twisting a strip of paper 180 degrees, the two ends are glued together to form a ring.


If an ant with infinite power is placed on it, it can crawl in an infinite loop on the original two-sided strip of paper without obstacles, because the strip of paper now becomes only one side.


It is this ring which possesses special properties.





The Möbius ring shows an infinite expansibility on a two-dimensional plane, but the Möbius ring can only be made in three dimensions.


Then its upgraded version of Klein bottle shows the undirected plane, this bottle has no inside and outside, can be free and unhindered from inside to outside.

Just like the picture above, let the Möbius ring be derived, and it becomes the Klein bottle.


But the neck and body of a Klein bottle in three dimensions intersect, and a true Klein bottle requires an extra dimension of space to avoid this intersection.

So even if we are as skilled as we are, we cannot make a real Klein bottle in three dimensions.


It exists in four dimensions.



We are proud to be the unbounded, undirected, one-sided surface provider for anyone in the universe. Oh~! Models~!




You can click here to view an excellent article on Klein bottles written by Konrad Pollthier of the Zuss Institute in Berlin in Plus magazine (No. 26, September 2003).



We also found an even earlier article in 1975.




If you are interested you can read 《basic Topology》 M. A. Armstrong by M. A. Armstrong, a British topologist, published in 1981.