Why can't the Klein bottle be made in the real world?

In the market, we often see this bottle, its name is Klein bottle.

But in fact, this is not a real Klein bottle, and there is never a real Klein bottle in the world we live in. Because the Klein bottle can never be made in real life.

What is 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, the Klein bottle is a non-orientable topological space. It was first proposed by the famous mathematician Felix Klein, but Klein’s original intention was, "Kleinsche Fläche", which is the Klein plane, which has no internal and external distinctions, but later it became Klein Rhein bottle.

The structure of the 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 with the hole at the bottom. Unlike the cup we usually use to drink water, this object has no "edge" and its surface will not end. It is different from a spherical surface. A fly can fly directly from the inside of the bottle to the outside without passing through the surface, that is, it has no distinction between inside and outside. Because of this, the Klein bottle is never filled with water.

Klein bottle is impossible in real life, but the Klein bottle we see now is actually They are all fakes. You will find that the Klein bottle we see must intersect with itself. In mathematical terms, the realization of the Klein bottle obtained in this way in three dimensions is the immersion of the Klein bottle in three-dimensional space ( immersion).

But in fact, the bottleneck of the real Klein bottle is connected to the bottom of the bottle through the fourth dimension, and does not need to pass through or intersect with itself. We can see its definition "a non-directional plane". This kind of bottle has no distinction between inside and outside. No matter where it penetrates the curved surface, the place it reaches is still outside the bottle. It is essentially a The weird thing of "Outside No Inside".

This is a three-dimensional concept object, so it can only exist in four-dimensional space, it is impossible to embed it in three-dimensional space If we must show it in three-dimensional space, its bottleneck can only pass through itself. This is a compromise that three-dimensional space cannot express it smoothly, but this violates the definition of Klein bottle, so it is on the market All Klein bottles are fake.

Since the real Klein bottle is connected to the bottom of the bottle through the fourth dimension, it will never be possible It is full, even if you put the sea in, it is the same.

If you cut the Klein bottle along its symmetry line, you would get two Mobius rings. In 1858, the German mathematician Mobius (1790～1868) and John ·Listing found that after twisting a piece of paper by 180°, the two ends are bonded together to make a paper loop with magical properties. This is the Mobius ring.

In our three-dimensional world, a piece of paper has two sides, but for creatures in the two-dimensional world, it is on a flat surface, without two sides.

In our eyes, the ant will crawl over the two connected faces to return to the origin. At this time, the ant doesn't need to climb over the edge of the paper, the distance it can crawl has doubled.

But for a two-dimensional ant living inside a plane, its space did not increase, but it was distorted and then connected.

So, this two-dimensional ant only needs to crawl for a week before it can go back to its original place. Understand why the crawling distance of ants in the three-dimensional world has doubled.

If the Mobius belt can perfectly show a "one-dimensional infinitely expandable space model in a two-dimensional space", the Klein bottle can only be used to show a "two-dimensional infinitely expandable space model in a three-dimensional space". Reference to “Spatial Model”.

So for "creatures" in two-dimensional space, they cannot understand the fact that space is distorted, of course For human beings in our three-dimensional space, we also cannot understand that our space is connecting to itself through additional dimensions.

What is a four-dimensional space?

Four-dimensional space and four-dimensional space-time are not a concept. Four-dimensional space generally refers to standard Euclidean space; four-dimensional space-time refers to Minkowski space A misunderstanding of the concept. As a three-dimensional object, human beings can understand four-dimensional space-time (three spatial dimensions and one time dimension) but cannot recognize and exist in four-dimensional space, because human beings belong to the third spatial dimension.

The zero dimension is a point, without length, width and height. One dimension is a line composed of countless points, with only length, without width and height. Two-dimensional is a surface composed of countless lines, with length, width and no height. Three-dimensional is a body composed of countless faces, with length, width, and height.

Because human eyes can only see two dimensions, two-dimensional creatures have only one line to see each other. Human eyes see two two-dimensional projections, which are processed by the brain to form a whole vision. That means that humans can only see two-dimensional images, but they can understand that the world is three-dimensional.

Because the three elements that make up three dimensions are length, width and height, that is, three straight lines perpendicular to each other. In the world, the top and bottom are high and the left and right are wide, so the world you see is two-dimensional.

Some people will say that people have two eyes to judge distance. Does this constitute three-dimensional vision? No, the two retinas see two two-dimensional planes, which add up to just a larger two-dimensional plane, and the brain uses the subtle difference between the two images to make up the distance information.

When we look at the world with our eyes, the resulting images are different in distance, usually right, far, and left, but not absolute. It is this sense of distance that makes up for the length of the three-dimensional needs, so we can understand this The world is three-dimensional.

The meaning of Klein bottle

The so-called "Klein bottle" is always a great mathematician The "fiction" in Mr. Klein's mind could not be created at all. Mathematicians in many countries always want to create it as a gift to the International Congress of Mathematicians. However, what awaits them is one failure after another.

In topology, the Klein bottle is defined as a square area [0,1]×[0,1] modulating the equivalence relation (0,y)~(1,y), 0≤y≤1 And (x,0)~(1-x,1), 0≤x≤1. Similar to Mobius Band, ClayBecause the bottle cannot be oriented. But the Mobius belt can be embedded in R^3, while the Klein bottle can only be embedded in a four-dimensional (or higher dimensional) space, which explains in a mathematical sense why it is impossible to make a Klein bottle in reality.

Klein bottle promotes the development of topology and even has a certain impact on physics, such as space-time grams The thermodynamics on the Rhein bottle is of great significance for humans to explore higher dimensional space.