What is a four-dimensional space?

first, we need to know what a four-dimensional space is. Four-dimensional space refers to the space including time a and three-dimensional space composed of length x width y height z.

If we walk in a long and narrow tunnel, there are only two directions in which we can get out of the tunnel-front and back; And when we walk in the open field, we will have four directions-front, back, left and right; When our astronaut performs a spacewalk in space, he will have six directions, front, back, left, right, up and down. So where can we find the seventh and eighth directions, that is, the fourth pair of directions? Of course, that can only be found in four-dimensional space.

However, there are these two directions in the space we live in, which are the front and back of time. Think about what happened in the past and what will happen in the future, and we will find that there is coherence in all this.

there is a phenomenon that I have been interested in for a long time. That's why a square has four sides, but a cube has twelve edges. Later, after studying mathematics, I understand that this is the difference of spatial dimensions. A square can exist in two-dimensional space, but a cube can't. It has to stay in three-dimensional space at least. So is there anything that can't stay in three-dimensional space but can only stay in four-dimensional space and above?

In Chinese, there is an orthography in the square. When you think about it carefully, it means that both the square and the cube are square. Can they represent anything in space?

after years of research, I think that all squares are a kind of primitive body of spatial dimension, which can be explained from the perspective of geometry. For example, their angles are perpendicular to each other, their opposite sides are parallel to each other and they represent the most extreme directions of a certain dimensional space.

in this case, will there be a connection between them? Why is one dimension two endpoints of a straight line, two dimensions four sides and four vertices, and three dimensions twelve sides and eight vertices? Are these numbers isolated and unrelated? Did God arrange these numbers as the characteristics of space?

some intuition tells me that they must be related, and I must find out.

if you think about it in detail, the simplest thing that can directly match the dimensions of space is the number of coordinate axes in the N-dimensional geometric coordinate system. Therefore, I took this as a breakthrough to conduct a deeper exploration.

I added a new coordinate axis-axis A, which is different from the other three coordinate axes. It is assumed that axis A is perpendicular to the other three coordinate axes in this coordinate system. Based on this, a new four-dimensional coordinate system is formed, which seems absurd.

let's imagine that the space we can feel now is only two dimensions. In other words, if we only live in a two-dimensional space with only four directions now, suppose we can't feel up and down. Then, when someone puts forward that there is such a third direction, will it be surprising? We only have four directions, and all directions are perpendicular to each other, so where can we find the third pair of directions? Just like an ordinary little ant crawling on the flat ground, it has never flown or thought about flying, so is flying absurd for this poor ant?

Later, I took a point on the A axis and assumed that the distance from this point to the origin was equal to the side length (1mi) of the cube. This is the first side of the later four-dimensional cube. Then make some parallel lines in the same direction for each vertex of the three-dimensional cube. After many failures, several candidate figures of four-dimensional hypercube geometric model were finally established. Among these figures to be selected, a real four-dimensional cube model has just come out.

this is a graph which is composed of four groups of parallel lines with eight numbers in each group. Like the previous dimensions, this graph is also a spatially symmetric graph. What is even more surprising is that the centers of six planes pointed in each direction in a three-dimensional cube become eight cubes in a four-dimensional space. This is the same as four directions in two-dimensional space pointing to four sides respectively. This makes me more aware that there is an inseparable connection between adjacent dimensions, which is the "split theory" that evolved later.

2.1 There is a "split-aggregate relationship" in the one-,two-and three-dimensional space

After studying this model, I think that the transformation of adjacent dimensions of space is formed by the vertical split of space. In the early days of space formation, the universe is a point, which we can call zero-dimensional space; Then this point splits into two points, and the space between these two points is called linear space, which is one-dimensional space; These two points later split in the third direction (+Y) respectively, and the space between these four points appears as a plane, which is a two-dimensional space; Four points in a two-dimensional space continue to split in the fifth direction (+Z), and eight poles appear. The space between these eight poles is called a three-dimensional space (cube). From this, we can continue to infer that the high-dimensional space is formed because the poles of the low-dimensional space are split separately. Then, the four-dimensional space must be formed by splitting eight poles of the three-dimensional space respectively; That is, it has 2×8 poles and 2×12+8 edges.

observing this four-dimensional model, you will find that two three-dimensional cubes with the same shape are connected at both ends of any group of parallel lines, just as there are a pair of squares with the same shape at both ends of each group of parallel lines of the cube, that is, each group of parallel lines is equivalent and interchangeable.

2.2 Basic properties of four-dimensional cube and formulas for calculating vertices and edges of n-dimensional cube

Further observation shows that in this model, each pole is connected with four line segments with different directions, and none of the directions are coincident. No one pole connects four directions that are consistent with the four directions on other poles. This shows that each pole has its own independence and is indispensable. Anyone who has studied binary system knows that there are only sixteen binary digits with four weights. In other words, sixteen poles are saturated for four-dimensional space, and it is impossible to have one or two more poles. Conversely, there are eight poles in three-dimensional space, is it also indispensable? We have the right and obligation to doubt the known or confirmed views. There are only eight binary numbers with three weights. If this is doubtful, then we should doubt mathematics, because binary can be said to be the foundation of mathematics, and mathematics is the foundation of the universe, then the universe will lose its meaning. So I think this is a real fact. Similarly, two-dimensional space has four poles, which is enough to illustrate this fact. One-dimensional space has two poles. In this way, we will find that the number of poles in space is the number of digits with corresponding weights in binary. That is, F=2N, where n is the dimension of the corresponding space and f is the number of poles (divination limits) of the corresponding space. We can calculate the number of poles in the corresponding space, and we can also calculate the number of edges in the corresponding space by splitting. When the upper one-dimensional space is transformed into the next one through splitting, firstly, the number of edges in the upper one-dimensional space is doubled, and then the number of new parallel lines (the same as the number of poles in the previous one-dimensional space) is added, that is, G=B×2+2N-1, where n is the dimension of the corresponding space, g is the number of edges in the corresponding space, and b is the number of edges in the previous one-dimensional (N-1-dimensional) space of the corresponding space. Further observation shows that in the N-dimensional space, * * * has n groups of parallel segments with 2N-1 in each group. So G =2N-1N again, that is, G= B×2+2N-1=2N-1N. This method can also be used to calculate the number of sides in a certain dimension and the number of side bodies and N-dimensional bodies (n >: 3,N< The dimension of the space). That is to say, we can use some formulas to calculate the number of poles, edges, sides, and bounding volumes in a certain space.