What is a Klein Bottle?

In the field of mathematics, Klein bottle refers to a non-directional plane, such as a two-dimensional plane. There is no distinction between "inside" and "outside". In topology, Klein Bottle is an unorientable topological space. The Klein bottle was originally proposed by Felix Klein, a German geometry scientist. In 1882, the well-known mathematician Felix Klein discovered the famous "Bottle" that was later named after him. The structure of a Klein bottle can be expressed as: a bottle has a hole at the bottom, now the neck of the bottle is extended, and twisted into the inside of the bottle, and then connected to the hole at the bottom. Unlike the cups we usually use to drink water, this object has no "edge" and its surface will not end. Unlike a sphere, a fly can fly directly from the inside to the outside of the bottle without passing through the surface, ie it has no distinction between inside and outside.

The Klein bottle is a non-orientable two-dimensional compact manifold, and

Klein bottle is defined as a square area [0,1] × [0,1] modular equivalent relationship (0, y) ~ (1, y), 0y1 and (x, 0) ~ (1 -x, 1), 0x1. Similar to the Mobius Band, Klein bottles are not orientable. But Mobius bands can be embedded

While Klein bottles can only be embedded

Twist a section of a piece of paper tape 180 ° and stick it to the other end to get a strip

Felix Klein

Klein

(Felix Christian Klein, April 25, 1849 to June 22, 1925), a German mathematician, born in Dusseldorf, Germany. He was a professor in Erlangen, Munich and Leipzig, and finally arrived in Argentina to teach mathematics. His main topics are non-Euclidean geometry, group theory, and function theory. The publication of his Irish Root Program, which classifies various geometries with their basic symmetry groups, was far-reaching: it was a synthesis of many mathematics at the time. His work includes Elementary Mathematics from a High Perspective, and he died in Göttingen.

In 1885, Klein was selected as a foreign member by the Royal Society and was awarded a Coplerian prize.

In 1908 Klein was elected chairman of the Congress of Mathematicians in Rome by the International Mathematical Society.

He has made many contributions in topology and geometry. He recognized the importance of group theory and applied the concept of group to many branches of mathematics. In 1872, the famous Erlangen Program was published. He proposed the idea of classifying geometry according to the properties that remain unchanged under transformation groups, and unified geometry with group theory. It has a profound impact on the development of modern geometry, and has prepared conditions for the establishment of special relativity. After 1886, he taught at the University of Göttingen for a long time and was an outstanding representative of the Göttingen School during its heyday. His views on the unity of mathematics had a great influence on Hilbert. He also suggested that mathematics should be closely linked with reality. He has organized many seminars on mathematics, and through teaching activities, students have a comprehensive understanding of mathematics as a whole. When he taught the theorem, he only talked about the outline of the proof, and left the proof to the students to complete it. He first advocated reforming the mathematics content of secondary education, which has an important influence on modern mathematics education. ^[2]