From ancient times to modern society, many things that people thought impossible have been realized one by one. Before Edison invented the light bulb, everyone dreamed that the night was as bright as the day. In ancient times, when trams, motorcycles, cars and trains were not invented, people once thought that Wan Li Road could be used. I didn't expect it to come true at last. When people no longer just fantasize that they can fly to the sky. When people no longer just imagine that they can have thousands of legs. When everything becomes the world, do you still think there will be something that will not happen in the future?

What does Klein bottle mean? Klein bottle? But it's always in the head of the great mathematician Mr. Klein? Fiction? I can't do it at all. Mathematicians in many countries always want to give it as a gift to the International Congress of Mathematicians. However, what awaits them is failure again and again.

In topology, Klein bottle is defined as square area [0, 1]? [0, 1] modular equivalence relation (0, y)~( 1, y), 0? y？ 1 and (x, 0)~( 1-x, 1), 0? x？ 1。 Similar to Mobius belt, Klein bottle can't be oriented. But Mobius band can be embedded in R^ 3, and Klein bottle can only be embedded in four-dimensional (or higher-dimensional) space, which explains mathematically why Klein bottle can't be made in reality.

Today, we are here to deliver what everyone thinks is impossible. When you get a bottle of water, you will consider how much water it is. When you consider what it is, it is a unit of measurement. In your subconscious, you also think that all the water will be filled and put into a container. Even the ocean is in a container. Because it is surrounded by the mainland, we can regard it as being contained in a depression. But have you ever thought that a bottle of water is not enough in the world? Even if all the four oceans are put in, the water from the Arctic Ocean, the Atlantic Ocean and the Indian Ocean will be put in. You are still not satisfied.

Then you should ask questions again. Is there really such a container in the world? Yes, it's true. Because we often can't imagine the three-dimensional graphics in our minds. There is such a bottle in mathematics called Klein. This is our hero today. He can put all the water on the ground in, but he still pretends to be dissatisfied. Klein is not a bottle on the plane we imagined, it is three-dimensional. The circulating bottle is also an important reason why it can put all the water in. But this is just a concept we are talking about. This is a complex spatial image.

Germans call it the Klein plane. We can call it Klein bottle, because it is a space concept full of dissatisfaction anyway. Mathematically speaking, someone should have touched the surface of Klein bottle. He can't go to the end indefinitely. Some scientists made a toy based on his model, because it is impossible to make anything real in reality. It can only exist on an imaginary level, just like that sentence, if you give me a fulcrum, I can pry up the whole earth.

But you can't think he is fake just because he was not created. In fact, what can be introduced in theory before will be invented in the future world. Just like 3d printing now, we thought it was impossible before. So imagine everything and realize it. Only imagination is the greatest. She can create an object to create happiness. Many scientists have made the impossible possible, so we should also believe that some impossible things in this world can exist in the future.