As we all know, human beings as creatures of the three-dimensional world cannot perceive the higher dimensional space, and we can only take the trouble to envision it. Previously, famous mathematicians have designed a bottle, which has an undirected plane, named Klein bottle.
In the eyes of many people, this bottle is a high-dimensional space artifact, and, it is said that all the water on Earth into it, are not enough to fill the "Klein bottle", which is why? Let's take a look at it!
What is a Klein bottle?
First of all, let's understand what exactly is a Klein bottle. It was discovered by German mathematician Felix Klein in 1882, and the German name was originally "Klein plane", but due to translation problems, it was mistakenly turned into Klein bottle. However, the "Klein plane" does look similar to the bottle, so the misnomer has been used.
By nature, Klein bottles have no boundaries and their structure appears to be relatively simple. There is a hole at the bottom of the bottle, and the neck of the bottle, when stretched, is twisted into the bottle and eventually reattached to the hole at the bottom. In this case, there is no distinction between the inside and the outside.
Speaking of which, you should think of another thing that has a similar structure, which is the Möbius strip. The Möbius strip is simpler to make than the Klein bottle, and it shows three dimensions through two dimensions, so it seems acceptable to us.
And if you glue two Möbius strips together, you get a closed surface with no edges, which is somewhat similar to a Klein bottle.
Scientists say they call such surfaces with local Euclidean geometry topological manifolds. Möbius bands and the usual paper rings are edged manifolds, while their bonded Klein bottles and torus surfaces are closed manifolds with no edges.
But the Klein bottle tries to show us four dimensions, which is why it looks rather "weird". It is worth mentioning that, in fact, all the Klein bottles on the market are not really, because we can not cross the dimensions to create such a high-dimensional space artifact, just as if you are in a two-dimensional world, you can not create a three-dimensional sphere, this is the limit.
However, why is it said that even if all the water on earth is poured in, it will not be able to fill this bottle? Does it mean that this bottle also has the nature of a bottomless pit?
Klein bottle with water
As we all know, from the current observations, the Earth is indeed the most water-rich existence in the solar system, except for the name does not carry water. So on the surface, it is still easy to fill such a small bottle. In that case, why would it be said that it is not filled?
This is because there is a hole at the bottom of the bottle, and the Klein bottle itself has a "loop" inside, so in fact, in the four-dimensional modeling, the water does not flow over the surface of the bottle from the beginning to the end. Since there is no "end" at the surface, the bottle is never full.
It should be noted that there is a limit to this never-full, which refers to the three-dimensional space alone. If you can enter four-dimensional space, and use the Klein bottle, then it can still be easily filled. It can be seen that the Klein bottle can not be filled up the fundamental reason is that the principle of this thing does not belong to three-dimensional space, of course, is not limited to the three-dimensional space "capacity volume" restriction rules.
It is worth mentioning that there had been a different story about the matter of the Klein bottle holding water. That is, because this bottle has no inside and outside, causing it to be different from the sphere and the ring surface, so it can not be used as a container to hold water, although it is a closed surface.
Anyway, no matter which explanation is given, it has something to do with the characteristics of the Klein bottle itself, and it is difficult for us to say which one is correct.
In addition, although we often say that the Klein bottle belongs to four dimensions, it may also belong to a higher dimensional space by nature. After all, as physics research continues to advance, humans have long discovered that not only time can become a new dimension, but also some other factors can become a "ruler" in higher dimensions.
The "descending blow" in high-dimensional space
I believe that anyone who has seen "Three Bodies" is impressed by the "two-way foil" that the singer threw down, which mankind thought beforehand was well prepared to take a chance in the confrontation with higher civilizations. But to our surprise, the whole process was over before we even saw anyone else's face.
Just about anything from higher-dimensional beings may be able to deal a devastating blow to lower dimensions, and it is amazing that the entire solar system has gone from three-dimensional to flat. And then look back at the Klein bottle that seems to belong to the higher dimensional space thought, it will be found that it also has a threat to the three-dimensional world from a certain point of view.
Let's use it in three-dimensional space to fill the characteristics of water, if high-dimensional creatures can put an oversized caliber Klein bottle on the Earth, then it can make it easy to suck the entire planet dry. And the human world will suffer unprecedented havoc because of this situation, so this is also considered a "downgrade".
So, if it is really possible for humans to enter the higher dimensions in the future, what kind of world will they see? Take four-dimensional space as an example, which is currently the most interesting and relatively easy to understand. In four-dimensional space, time is no longer a torrent that never returns, but becomes a scale.
This means that your life within the four dimensions may be infinitely long, and even have the opportunity to be able to travel back and forth within each time period, completely realize the so-called "time travel", regardless of whether it is back to 18 years old or back to 8 years old, as long as you want can be achieved.
However, we are still faced with multiple limitations, after all, the real four-dimensional space should not be as simple as imagined, but more complex. There time becomes a scale, but we are three-dimensional creatures, so it is difficult to understand or jump out of the way, so the final end may still be trapped.
So, if there are really many creatures within the higher dimensions, or even more radiant civilizations, what should they be like? Can we see these higher-dimensional creatures?
What higher-dimensional creatures look like
From the current findings, the creatures of higher dimensions should have long since transcended the "limits of the physical body" to another form of existence. But now, even out of the Earth are subject to a variety of environmental constraints, so before we also transcend the "physical body", I'm afraid it is impossible to enter the fourth dimensional space.
And to say whether you can see or perceive their presence, by and large, is also very difficult. We will only find some bizarre coincidences in life, but can not find the originator of these events. After all, under the augmentation of the dimension, these creatures not only travel through space, time is not limited, there are some other superpowers.
This is actually like humans are to ants, when you pass a tall ant mound on the side of the road, you may deliberately kick the dirt bag. For you, this behavior is irrelevant, and it doesn't even take much effort. But for the ants inside the mound, they may have experienced a huge earthquake.
What's more, by the time the ant scouts come out to check, you're probably already far away, so the disaster will eventually be classified as "natural causes" in the ant community. Of course, the ants probably have no way to understand what it is to be so many times larger than they are, and this is the limit of dimensionality.
It is important to note that ants are not really "two-dimensional creatures", but still have a three-dimensional appearance. The reason why scientists like to use the ant analogy is to help people understand the complexity of high-dimensional space, so you should not directly classify ants as two-dimensional creatures.
That said, humans have not only never seen a higher-dimensional creature, but also a two-dimensional one. But maybe we see those "paper people" in the game is the so-called two-dimensional creatures, the difference is that we think they are artificial without their own consciousness, but maybe they are conscious in the two-dimensional world, only in accordance with the three-dimensional world of human design path in walking, but never noticed.
In this case, in the three-dimensional world of human beings, will not also be the eyes of higher-dimensional creatures "paper people"? What we think of as autonomous consciousness may also be set up.