Hello everyone, today we come to talk about four-dimensional, five-dimensional, six-dimensional, what does such a high-dimensional space look like?
When it comes to this topic, some people may say that the so-called four-dimensional space is a concept composed of a three-dimensional space plus a one-dimensional time.
In fact, if it is understood in this way, this is a complete misunderstanding.
The so-called high-dimensional space, which does not have the participation of time, is just a concept within space. The topic we are talking about today is only representative of space theory.
So how should we define the dimensions of space?
The so-called three-dimensional space, four-dimensional space, five-dimensional space, how did they come from? How should we understand it? Here is a particularly simple way to give you an example, that is, a point can be made into several vertical lines, then this space is a few dimensional space.
For example, zero-dimensional spaces. A point in a zero-dimensional space cannot make a straight line, so it is defined as a zero-dimensional space.
One-dimensional space is a straight line, through this line to select any point, then only one line can be made, so it is defined as one-dimensional space.
A two-dimensional space is a face on which a point can be arbitrarily selected. Only two straight lines perpendicular to each other can be made.
Three-dimensional space is a three-dimensional structure, in this space inside the arbitrary selection of a point, it can form three vertical lines perpendicular to each other. So it is also defined as three-dimensional space, and we can label these three vertical lines as x-axis, y-axis, and z-axis, respectively.
By analogy, let's think about what a four-dimensional space looks like.
Four-dimensional space must be, choose a point, you can make four straight lines perpendicular to each other. Imagine this four-dimensional space according to our existing perception of physical space. It is completely impossible to simulate such a picture, as if there are no four vertical lines in this world.
But we can draw the projection of four-dimensional space in three-dimensional space. A three-dimensional structure of a three-dimensional space, we can easily draw it.
If you project a four-dimensional space into a projection of three-dimensional space, the image drawn should be like this.
Draw a large cube first, then draw a small cube inside the large cube, and finally connect each point of the large cube and the small cube.
Select any point on the large cube, and we can see that there are four straight lines perpendicular to each other. These four straight lines are not vertical if viewed according to the cognition of three-dimensional space.
But suppose there is a space that can form four straight lines perpendicular to each other. Some people here may say, obviously this is not a vertical four lines, why do you have to say that it is vertical?
Remember that this is just a hypothetical projection of a four-dimensional space in our three-dimensional space. According to this line of thinking, we continue to reason, then the five-dimensional space, the six-dimensional space, the seven-dimensional space, they are composed of five vertical lines, six vertical lines and seven vertical lines.
So speaking of which, you may find that the high-dimensional space from the beginning of the four-dimensional space has exceeded our current laws of physics, so the four-dimensional, five-dimensional, and six-dimensional high-dimensional spaces we mention today are not a physical concept, but a high-dimensional theory derived from a mathematical model.
It can't be seen in terms of existing physical cognition, so it doesn't contradict our current cosmic space. Inside all these dimensional spaces, we now introduce a new physical concept, which is spatial distance.
In zero-dimensional space, the spatial distance between any two points. Always zero.
There is a line within the one-dimensional space, and a point is selected as the origin. Building a horizontal coordinate, and arbitrarily choosing another point to the spatial distance from this dot, is the absolute value of this coordinate.
In two-dimensional space, select a point as the origin, establish the x-axis and y-axis respectively, then select the spatial distance from any point to the origin, and we can get the value of this distance according to the Pythagorean theorem as the x square plus the y square under the root number.
Similarly, in three-dimensional space, we can also choose a point as the origin and build the x-axis, y-axis and z-axis. Then the spatial distance from any point to the origin in this three-dimensional space can also be concluded that the x square plus the y square plus the z square under the root number.
And so on to establish axes in four-dimensional space. The four straight lines perpendicular to each other are the x-axis, y-axis, z-axis and m-axis. The coordinates from any point to the origin are also x squares under the root number, plus y square plus z square plus m square.
Why do we mention the concept of spatial distance here? That's because of the spatial distance, no matter how we change the frame of reference, the last period of spatial distance, it will always remain the same.
To give a simple example, in a one-dimensional space, we can put the origin here, and we can also move the origin to any other place, but no matter how much we move the position of the origin 0, the distance of oa will never change.
In two-dimensional space, we can establish a coordinate system in this way, and we can also establish a coordinate system obliquely, but no matter how to establish a coordinate system, even if the coordinates of a change, then the distance of oa will always remain unchanged. This is the invariance of spatial distances in any one dimension.
Of all the dimensional spaces, there is also the most critical commonality. All high-dimensional spaces they can be projected into low-dimensional spaces.
For example, a one-dimensional space is a line, and if we cut a knife at random on this line, we can get a zero-dimensional point.
Two-dimensional space is a surface, and on this surface we cut a knife at will, then we can get a one-dimensional straight line.
Three-dimensional space is a three-dimensional structure, in this three-dimensional structure at will to cut a knife, we can get a two-dimensional surface. This face can be cut out in a rectangle or a triangle.
By analogy, we can derive the so-called four-dimensional space, if cut a knife, it may get a three-dimensional three-dimensional space.
Therefore, so so many high-dimensional spaces have exceeded our existing physical cognition, and we cannot use the laws of physics to look at high-dimensional spaces, but can only use mathematical models to abstractly deduce.
It is also possible that we are human beings in high-dimensional space after being hit by dimensionality reduction and entering three-dimensional space, so our existing three-dimensional cognition simply cannot understand what high-dimensional space looks like?