What is a Klein bottle? What is the principle of the Klein bottle

Why is the Klein bottle not satisfied? The whole animation is displayed, just look at the internal structure.

In the field of mathematics, the Klein bottle refers to a non-directional plane, such as a two-dimensional plane, and there is no "inside" or "outside". In topology, the Klein bottle is a non-orientable topological space.

The Magic Klein Bottle, what is it?

As we all know, the dimensions of our lives are three-dimensional, but there are also high-latitude spaces and objects in the world, such as the mysterious Klein bottle.

In the field of Klein Bottle Mathematics in three-dimensional space, the Klein bottle refers to a non-directional plane, such as a two-dimensional plane, and there is no distinction between "inside" and "outside". The concept of the Klein bottle was originally proposed by the German mathematician Felix Klein. The Klein bottle is very similar to the Möbius belt.

The structure of the Klein bottle is very simple, with a hole in the bottom of a bottle, which now extends the neck of the bottle and twists into the inside of the bottle, and then connects with the hole at the bottom. Unlike the cups we usually use to drink water, this object has no "edges" and its surface does not end. It is also not similar to a balloon, where a fly can fly directly from the inside of a bottle to the outside without having to cross the surface (so there is no distinction between inside and outside).

What is the structure of a Klein bottle?

What is the principle of the Klein bottle? A bottle that the four oceans can't fill it up.

Through a plane division, there is no distinction between inside and outside.

In the past, the German mathematician Klein put forward the "impossible" idea, that is, the great monster of topology - the Klein bottle. There is no distinction between inside and outside of this bottle, no matter where it penetrates the curved surface, the place where it reaches is still on the outside of the bottle, so it is essentially a strange thing that "has an outside and no inside".

Although the modern glass industry has developed very advanced, the so-called "Klein bottle" has always been a "fiction" in the mind of the great mathematician Mr. Klein, and it cannot be made at all. Mathematicians in many countries have always wanted to make one of them as a gift to the International Congress of Mathematicians. However, what awaits them is one failure after another.

Some people also think that even if you can't make glass, it's good to be able to make a paper model. If this problem is solved, it will be a big gain!

Therefore, in the past, it was generally believed that it was impossible for a Klein bottle to be embedded in three-dimensional space. In three-dimensional space, the Klein bottle must intersect with itself, and in mathematical terms, the realization of the resulting Klein bottle in three dimensions is the immersion of the Klein bottle in three-dimensional space.

What is the principle of the Klein bottle? A bottle that the four oceans can't fill it up.

The principle of the Klein bottle is explained as follows:The Klein bottle refers to an unoriented plane that has no edges and its surface does not end. It is not the same as a sphere, it has no distinction between inside and outside, its bottleneck is connected to the bottom of the bottle through the fourth dimension, and it does not need to pass through itself, and naturally it does not intersect.

Because of this, the Klein bottle can never be filled with water.

This bottle was invented by a scientist named Klein because of its complex structure. The materials should also be very special. So so far, we haven't made a physical finished product, it's a very strange bottle.

It is a caliber that sticks out from the bottom and then extends upwards, and the whole bottle is one, and no matter what you put in it, it cannot be preserved inside. Even if a small insect enters, it will eventually come out without even going inside. Mathematically, we generally refer to this phenomenon as non-orientable topological spaces.

The Klein bottle is unattainable in real life, it is essentially a "outside and inside" weird thing. In the language of mathematics, the realization of the resulting Klein bottle in three dimensions is the immersion of the Klein bottle in three-dimensional space. If we cut the Klein bottle along its line of symmetry, we get two Möbius strips.

The horror is that there is no distinction between inside and outside, so there is no boundary, but there is a boundary in visual inspection, and this is the horror of the place.

A bottle has a hole in the bottom that now extends the neck of the bottle and twists into the inside of the bottle, then connects to the hole in the bottom. Unlike the cups we usually use to drink water, this object has no "edges" and its surface does not end. Unlike a sphere, a fly can fly directly from the inside of the bottle to the outside without crossing the surface, i.e. it has no distinction between inside and outside.

The Klein bottle is a non-orientable two-dimensional tight manifold, whereas a spherical or tread face is a two-dimensional tight manifold that is orientable. One thing that seems confusing if you look at the Klein bottle is that the neck of the bottle and the bottle body intersect, in other words, some points on the neck and some points on the wall of the bottle occupy the same position in three-dimensional space.

Item of the same principle: Möbius Strip.

Twist one section of a paper tape 180° and glue it to the other end to get a model of a Möbius tape. This is also a curved surface with only a Möbius strip and one face, but unlike a sphere, a tire surface, and a Klein bottle, it has an edge (note that it only has one edge).

If we glue two Möbius strips along their only edges, you get a Klein bottle (don't forget, of course, we have to be in four dimensions to really make it possible to do this, otherwise we'll have to tear the paper a bit). Similarly, if we cut a Klein bottle properly, we get two Möbius strips.

In addition to the appearance of the Klein bottle we saw above, there is also a lesser-known "figure-eight" Klein bottle. It looks completely different from the surface above, but in the fourth dimension they are actually the same surface Klein bottle.

The Klein bottle is a conceptual model in the field of mathematics, a non-directional plane, there is no distinction between "inside" and "outside", just like a two-dimensional plane, if it is divided into two halves of equal size, it is two Möbius strips. Due to the special structure, it is not possible to construct such a surface in three-dimensional space, but it is possible to construct a model, that is, to plug the thin end of an open conical tube into the thick end (in the same direction of the opening), and then seal each other.

At our level, we can't make a perpetual motion machine, and there is no way to make a Klein bottle, because the Klein bottle is more like a product of the four-dimensional space, and the space we are in is a three-dimensional space, and the product of the four-dimensional space cannot come to the three-dimensional space, and this is the reason why we can't make a Klein bottle. But now there are a lot of designs that imitate Klein bottles, but this is also a copy of its shape, not a real Klein bottle.

As we all know, the dimensions of our lives are three-dimensional, but there are also high-latitude spaces and objects in the world, such as the mysterious Klein bottle.

What exactly is a Klein bottle? What does it have to do with the Möbius belt, after reading it, I have read it!

I don't agree with the idea that it can't be made, I think it's a way to cut the appliance into two halves equally, and then make the two halves separately and then put them together, and as for Zhang to do it seamlessly, he has to find another way.

The geometric dimension of the space we live in is three-dimensional, and the Klein bottle can only exist in the four-dimensional space and higher, and theoretically cannot be made in our space, just as a sphere cannot be made in a two-dimensional plane. If you've looked at the model of the Klein bottle, you'll see that the neck of the Klein bottle intersects the bottle, which is a compromise.

If it can be made, it can prove that the four-dimensional space really exists xd, and if a person comes from crossing the Klein bottle mouth to the outside of the bottle, then he will be reversed from the inside out, not to mention it, it's too disgusting.