Is it true that a real Klein bottle can t be made, and on what basis?

The real Klein bottle really can't be made, because it is a special non-directional surface in a four-dimensional space, and it needs four spatial dimensions to be fully displayed. It's not that it can't be built so far, but it can never be made (because the world we can operate only has three spatial dimensions, and there is still one dimension missing, which is physically impossible to break through, just like can you make a sphere on a plane?). Apparently impossible.

Many people's impression of the Klein bottle is a bottle, but in fact, its proposer, the German mathematician Klein, proposed a non-directional two-dimensional surface, so it should be called a Klein surface. This two-dimensional face has no front and back side, it has only one side ......

When you walk all the way along one direction of the Klein surface, you can walk from the front to the back and back again, which means that it doesn't really have a back at all, it only has one side ......

If you still don't understand this diagram, let's take a look at a**:

Can you see it this way? It has only one side.

So the question is, if it's just a two-dimensional plane, why can't we make it in three-dimensional space? Because to achieve the effect of a true Klein bottle, a part of the bottle must reach into the four-dimensional space. Have you found the bottle in the picture above, one of the sides of the junction between the bottle and the bottleneck is closed, that is, it cannot be passed, and you can easily see it from the above.

The real Klein surface has no section, so the bottleneck of the real Klein bottle needs to enter the four-dimensional space, and penetrate the bottle body from the four-dimensional space. This is why a two-dimensional surface requires a four-dimensional space to construct – because the surface is enclosed and already constitutes a three-dimensional geometric structure, and its geometric properties can only be fully represented in a four-dimensional space.

If you still can't understand why you can't create a special two-dimensional surface in three-dimensional space, let's consider dimensionality reduction. Do you see the line in the diagram? If you take it out, it's actually a Möbius strip.

The Möbius strip was proposed by the German mathematician Mobius, which is also a two-dimensional plane, the same only one face, the difference is that it is not closed, so it does not constitute a three-dimensional geometry, it is a two-dimensional plane with boundaries. However, it cannot be made ...... in two-dimensional spaceBecause its structure cannot be produced in normal two-dimensional space, whether it is in the space of a plane, a sphere or a saddle, it cannot be produced, it can only be made in three dimensions.

A bottle that can't even fill the four oceans! Can a Klein bottle really be made?

The origin of the name of the Klein bottle was actually made clear by the well-known French scholar Klein, who clearly stated the situation. Originally, when the name was chosen in German, it was called the Klein Flat, but later due to the German translation and later widespread problems, it was finally understood to be the Klein bottle。In the view of the industry analyzed in mathematics class, the Klein bottle actually refers to a plane with no definite boundary, like the two-dimensional space that everyone knows very well, and there is no boundary difference between the external environment.

According to the relevant concept, the Klein bottle is a plane with no boundaries, so it is of course difficult to fill the Klein bottle for an object with a fixed and immovable quality such as the sea.

First of all, from the composition and structure of the Klein bottle, we can imagine the Klein bottle as a bottle, but this bottle is different from the ordinary bottle, the bottom of the Klein bottle, there is a hole, we can connect the hole at the bottom of the bottle, from the inside of the bottle can immediately cross to the outside of the bottle. Therefore, under this structure, it can be said that the Klein bottle can be understood as never having the external environment in the middle of the bottle, and under the premise of the plane of precise positioning of this kind of indoor space, not to mention the whole sea, even on the earth, from the factual argument, it is all unsatisfactory with the Klein bottle.

The structure of the Klein bottle looks unique, but it is not too difficult to describe, if you first think of it as an ordinary bottle, first increase its shortcomings, distort it from the internal structure, and then make a hole in the bottom of the bottle, so that the part that widens into the bottle is connected to this hole, you can get the so-called Klein bottle. So looking at the Klein bottle, it is obvious that there is a wheel corridor, but in fact it has no "edge" at all, and its appearance is not easy to end, because it refers to an indefinite plane in the digital industry, which is neither a sphere nor a cylinder, in short, a data model that has no difference from the external environment.

The Klein bottle was discovered by a well-known mathematician, Felix Klein, who named it after it, and there is an interesting story about the origin of the name Klein bottle, which originally meant Klein plane in Russian, and in the case of English translation "plane" was transliterated as "bottle".Later, we had the name Klein bottle, although at that time this oolong tea caused by translation was also discovered by us, but many scholars feel that the Klein plane is like a bottle on the third level, and it is also interesting to call it a Klein bottle, so the original plane was mistaken into a bottle, which has continued to this day.

In the 19th century, the mathematician Klein invented the Klein bottle, the bottom of the Klein bottle has a hole, when the neck of the bottle is extended, it twists into the inside of the bottle, and finally connects the hole at the bottom of the bottle, so as to form a bottle without edges, because of the special structure, so when filling the bottle with water, it is impossible to fill the water at all.

It is indeed unsatisfying, because no matter what is put in this bottle, it will flow outside; This bottle was invented using four-dimensional space, and we can only imagine three-dimensional space now, so it seems that we can't fill it anyway.

Yes; The Klein bottle was invented by the 19th-century mathematician Klein, and the structure of this bottle is very special.

It is a non-directional plane, that is, there is no distinction between inside and outside, some people will explain that what I see is a bottle how it is a plane, in fact, on the surface our eyes look like a closed curved surface like a ball, but it is different from the ball, a bee can fly directly from its inside to the outside It is more difficult to describe the Klein bottle in three dimensions, because the bottle is displayed in three dimensions, so everyone will think that the Klein bottle is a bottle, but in fact, this is a nested space, more precisely, The Klein bottle is not a bottle in the fourth dimension, but a space nested within each other.

This path goes up to three to four dimensions, and then it's the Klein bottle. Of course, this path is also one of the mathematical corollaries of the theory of relativity wormholes. The current mathematical explanation of that point is a white hole.

And the tail of the fish that connects this path, what we can see, is the black hole horizon. If you look at the ** of the Klein bottle, you will notice that its neck and bottle body intersect, and that some points of the neck of the bottle and some points on the wall of the bottle occupy the same position in 3D space. Is that really the case?

In fact, the Klein bottle is a curved surface that can actually be expressed in four-dimensional space.

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

Suppose there is a tube in two-dimensional space, and the head and tail of the tube are connected by passing through the three-dimensional space (that is, the space we are in). For creatures in two-dimensional space (assuming they exist), they will only see tubes in two-dimensional space.

It doesn't need to go through the wall of the bottle! But this is a space that is difficult to imagine, and itself constitutes a surface without boundaries, but it will not intersect with its own space, which cannot be realized in three dimensions, of course, before understanding this structure, we still have to understand the concept of dimension.

A Klein bottle refers to a non-directional plane, such as a two-dimensional plane, and there is no "inside" or "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).

After the neck of the Klein bottle is bent through the bottle wall, the bottle mouth is directly connected with the bottom of the bottle, this kind of bottle has no internal and external distinction, it is a non-directional plane, the interesting thing is to inject water into the bottle mouth, and it will never be able to fill the bottle, but with the current people's technology, this kind of bottle cannot be made, but it only exists in theory.

The biggest magic thing about the Klein bottle is that it has a hole in the bottom, and no matter how you fill it up, there's no way for it to stay in it forever. It's like an hourglass, it will eventually be lost bit by bit. Scientists have poured water into the bottle from various angles through various experiments, but they have not been able to fill it.

Is it true that Klein bottles can't be made? There is no solution for nearly a hundred years, just because it is a product of 4-dimensional space.

It can be made (the specific environment in which it is made is not specified), and this Klein bottle is very simple to make, and the method is as follows:

In the linear 3-for-1 software, a time slider is added, the shape and spatial position of the object are changed, and keyframes are added, and a controllable four-dimensional space-time "Klein bottle" is manufactured.

Depending on the cycle time, a four-dimensional bottle will form a series of retractable surface changes, which is a three-dimensional variant of the "Möbius ring" in a four-dimensional stretch and constriction cycle (twisted swimming circle only).

The bottleneck of the Klein bottle should not actually intersect with the bottle itself.

Therefore, the above picture is only a projection of the Klein bottle in our three-dimensional space, and the substitute for the real Klein bottle should be realized in the fourth dimension.

Just like a Möbius ring is a single-sided plane realized in three dimensions.

Actually, your ** is just a substitute for him in 3 dimensions, the real Klein bottle, the part that intersects in the middle of him is not intersecting, and the Klein bottle that penetrates the past in 4 dimensions is indeed like a bottle. But it doesn't have a bottom, its neck is elongated, then it seems to pass through the wall, and finally the neck is connected to the bottom ring.

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.

In other words, if you look at it from the front, the bottleneck is in front of the bottle, but it is inside the mouth of the bottle, and they are different dimensions.

What is being made now cannot reach the 4th dimension, so it cannot be made.

Because human beings don't know what 4-dimensional space is, they can't create 4-dimensional things if they use their eyes to observe from the perspective of three-dimensional space all the time.

Among the many scientific mysteries of the world, the most famous is the Klein bottle. This is because, according to the mathematician Klein, there should be a four-dimensional space in our universe, and this four-dimensional space should be a four-dimensional physical object according to Klein's conception, just like the special bottle that we often see named after the mathematician Klein. The special thing about this bottle is that no matter you pour water into it, the water will flow out of this bottle, so no one will be able to fill it up.

However, it is a pity that no one has made a real Klein bottle so far, and the long is a computer-generated model. This is so because it is likely to exist in a four-dimensional space, whereas our various existing manufacturing processes are only three-dimensional. Later, with the advancement of glass-making equipment and technology, humans also tried to make Klein bottles, but humans could not make them anyway.

Scientists say that it is because the existing technology in that part of the fourth dimension still cannot pass through, so it cannot be formed.

Scientists have carefully studied the structure of the Klein bottle, there is a black hole at the bottom of it, and then the mouth of the bottle is extended backwards, and after the distortion of the four dimensions, it leads to the black hole at the bottom of the bottle again, so no matter what is thrown into the Klein bottle, it cannot be preserved in it. Scientists used computer simulations to "throw" an ant into the Klein bottle, and as a result, the ant eventually walked out of the Klein bottle, and it didn't even need to pass through the surface of the cup.

The Klein bottle has been proposed for a long time as a strange space of plane geometry, but until now, we have not been able to make a physical object that meets the vision. After countless failures, scientists had to use computer simulation technology to create a four-dimensional model, which may be a little consolation for Klein.

If in the future, humanity's ability to comprehend does allow people to create Klein bottles, then it will be easy for humans to enter and exit the fourth dimension. But I wonder when humans will be able to understand the fourth dimension?