Dialogs in different dimensions: bring you into the four-dimensional world

When talking about dimensions last time, someone mentioned how to understand four-dimensional space. This is a very interesting topic, but I have never been careful about it. Some time ago, a film called dimensions: a walk through mathematics was published on the Internet. It is said that it details the four-dimensional space. I thought I could recommend this film to write less smelly and long logs. I didn't expect to see it below and find it really bad, people who do not know about the four-dimensional space may not know about the four-dimensional space after half a day. I am not busy with my recent holiday. I plan to take a look at it. If you have never thought about the four-dimensional space before, I believe that today you will have an extraordinary feeling.
Now, if I am a person in a two-dimensional world, I cannot understand what is "height", what is "body", and what is "space ". You want to describe cubes in the 3D world to me. What should you do? You may start with the cube chart: Figure (a) is a cube chart. If we cut a cart of this shape, we can fold it into a cube. I started to wonder.

-How do you do this?
-Fold the square above and stick the corresponding edges together ......
-Wait a moment. These squares are stable shapes. How can they get together?
-Silly! They are not active in the two-dimensional world, but they can be bent towards the third dimension! Draw a picture for you (B). This is how the above squares are bonded. This is a cube without a cap and a gap ......
-You're playing the game! After you bend it like this, the square is not a square, and it is changed to a trapezoid!
-No, they are still square. The six areas in figure (B) are actually square, but they seem to be "skewed" due to perspective.
-Well, you can continue.
-Now we get a covered box. The above five squares (four of which have changed shape due to being in the third dimension) have formed a "space" and can be placed inside. To create a closed cube, you just need to close the remaining square. The final result is as shown in figure (c.
-Token? In figure (c), where is the last square to be merged?
-It is the largest square.
-Nonsense! The big square is made up of five small squares! This big square is also in figure (B) just now!
-No. The big square in figure (B) is indeed the outline of the five small squares, but the big square in figure (c) actually exists, it is the final part. This big positive square is not overlapped with the five small squares. They are different layers in the third dimension. Figure (c) is the cube you dream of. It consists of six squares. The small square you see in figure (c) is a big positive Square. The four trapezoid shapes are actually square, and they are both big. These six squares form the "space" in the middle ".
-I still don't understand. The big square is also in the third dimension. Why is it not deformed?
-This is because the direction of the square is not the third dimension, so it looks the same as before.
-Why are there two more squares in the same direction?
-Alas, it's really troublesome. This is because, although they have the same orientation, they have different positions in the third dimension. The small square is a little farther away from us in the third dimension.
-Oh! I understand. Do you mean that the four "squares" next to a circle span the third dimension, so in the third dimension, we are separated from each other, and some of them are far away from us, so it looks like a transformation from big to small.
-Yes! You understand it very well! To be honest, I have never thought about this in 3D space.
-I seem to understand it. If you say something wrong, don't laugh at me. The "space" refers to the "track" produced by moving the four deformed squares on the third dimension to a small square in the distance ".
-Exactly!
-I fully understand. No wonder we can say that the n-dimensional cube has 2 ^ N points. In fact, the truth is very simple. In fact, you only need to copy the n-1 dimension cube and then connect the corresponding vertex. This is the result of the n-1 dimension Cube's displacement in the n-dimensional dimension. The newly added 2 ^ (n-1) edge is the trajectory of the point.
-Oh, you are so fucking cool. It's a pity to read Chinese. I also want to show you something interesting, so that you can see how three-dimensional cubes rotate. Open your eyes and carefully check where every square changes.

-I am confused again. Why can a small square in the distance cross the left boundary from the third image to the fourth image, so that the half of the square can reach the left of the boundary?
-This is really hard to understand. A small square does not pass through the vertical side. The side is closer to us in the third dimension, and its projection in our direction overlaps with the small square. As you can see, the topological relationships between them remain unchanged.
-Oh, so the small square in the distance will go to the side, and then to the position close to us, replacing the original big square ......
-It's okay to go back. Think about it. We look forward to a three-dimensional dream when you go to bed.
-Okay. Thank you.

 
Well, now I want to tell you a secret. Actually, I am a person from four-dimensional space. Many people ask me what the four-dimensional cube looks like. I am so bored, so I wrote this log today.

I will tell you that a four-dimensional cube is composed of eight three-dimensional cubes of the same size, and its expanded diagram (). Figure (B) is a four-dimensional box bonded, and a lid is not covered yet. These things that look like a pyramid are actually cube with different roots, just because they are in different positions in the four-dimensional space, a perspective occurs.

After the lid is covered, we can see the legendary four-dimensional cube, which I believe many netizens are familiar. There are two standard cubes in one dimension and one small one. These cubes have different positions in the fourth dimension but are facing our two "3D faces ". The other cubes are actually cubes, but they only look distorted due to perspective. A four-dimensional cube can be seen as a moving track of a three-dimensional cube. Therefore, it is easy to draw a four-dimensional cube: draw two three-dimensional cubes and connect them to the corresponding vertex. Observe the rotation of the four-dimensional cube, and you will see the small cube inside go out through a plane, and then become the outermost big cube. All of this is similar to the promotion from two dimensions to three dimensions. If you think carefully, you will find more similarities.