The method of understanding multidimensional space

Multidimensional space means that when observing things, we can think and look at objects from several independent angles.

For example, one-dimensional space is like the same highway, with only front and back; Two-dimensional space is like a square, with front and back, left and right; Three-dimensional space is the space of human life, with front, back, left, right and up; On the basis of four-dimensional space, you can also adjust the time freely ... to understand the dimension in another way: one dimension: two people stand on a railroad track, and one person can just find the other person along the railway. Two-dimensional: two people are on a square. Stereo: Two people are in a building. Siwei: No matter whether you are on the railway track, in the square or in the building, your friend goes in the morning, and you will never find him again in the afternoon, so if you want to find him, the time must match, and time is the fourth dimension.

No matter in several dimensions, a point can be described by a coordinate. One-dimensional straight line is represented by X axis, and two-dimensional plane can be represented by X axis length and Y axis width. Three-dimensional solid adds a high Z axis on the basis of plane, and four-dimensional, five-dimensional and six-dimensional axes add A axis, B axis and C axis respectively. Dimension is an independent parameter invented by human beings, which is only used to describe the continuity of time and space in which we live. In mathematics, the basic meaning of dimension is that object X is based on the premise that A is n dimension. Dimension is a perspective, which is only used to describe the research object conveniently.

One-dimensional world is a straight line, a straight line, a curve or even a closed chord. To improve the dimension, to illustrate the problem, if this dimension is closed or curled up, the front line will become a tube and the string will become a bagel after adding this dimension. Raise the dimension again. For the three-dimensional world, it may be hard to imagine to get another three-dimensional world by discussing the situation of two extended dimensions and one curled dimension, but just think that every point in the original two dimensions can extend to a new dimension. Of course you can't draw it, but you can show it. However, in the real world, there are six curled dimensions besides the three extended dimensions that can be perceived, and their sizes are in the order of Planck's length, so they cannot be perceived. Now, for the case of two extended dimensions, discuss what it looks like after adding six curled dimensions. For the three extended dimensions, it is actually the same, but it is not easy to draw, so we get nine dimensions, plus the time dimension, and finally get the ten-dimensional world. P.S. superstring theory is actually 1 1 dimension after supersymmetry, so I won't discuss it here.

A science student, from middle school to master's or doctor's degree, mostly goes through the following stages in understanding dimensions:

The first stage is like this: the point is zero-dimensional, the line is one-dimensional, the surface is two-dimensional, the body is three-dimensional, and the cognition is concrete. Speaking of which, there must be a corresponding image in my brain.

In the second stage, after studying classical mechanics, we know that the motion of particles is described by six coordinates and the phase space is six dimensions. At this time, the vector space has been abstractly understood. Speaking of multidimensional vectors, the brain is a low-dimensional image, and the pen is a formula derived from the low dimension. These formulas need not be memorized. Know the low dimension and you will know the high dimension. This stage ends with understanding the ensemble theory of statistical mechanics, and the dimensions can be expanded at will.

The third stage is to learn the theory of relativity, which is the first non-European space to be deeply contacted. Only then did I know that dot product can have different definitions. When thinking, the brain is a two-dimensional and three-dimensional Min's space, and the pen is a four-dimensional space of analogy.

Then quantum mechanics, his space, integral changes. I began to understand infinite dimensional space and realize the particularity of infinite dimension. You don't need images when you think, and an analogy formula is at the end of your pen. At the end of this stage is learning function and various induction.

Followed by quantum field theory, Lie group algebra. The concept of dimension here comes from the role of groups. Simply put, if you use a matrix to represent the changes of physical objects, how big the matrix needs to be. At this time, the image can be a three-dimensional space or various objects. As long as the group can play a role, it is easy to imagine.

Finally, study pure mathematics, topology and differential geometry. See all kinds of spaces. At this time, you don't need images to think, because clarity is very important. But until then, low-dimensional images will still be very helpful. There are also special cases, such as later understanding Hawking's space-time structure, because it is a hyperbolic space, images are only used to think about the structure, and most of them prove that deduction depends on definition.

For example, the image in the four-dimensional space-time brain is R {2+ 1}, that is, two light cones (light cones) are opposite. You can imagine all kinds of surface curves here and then extend them to multiple dimensions. But at the end of the pen, there is still only definition.

The following uses the projection method to understand the dimension space:

It is difficult for any creature in three dimensions to try to think of four or more dimensions from the brain. However, human beings can mathematically describe the space of 4 or more dimensions and conceive its projection in the 3-dimensional space. Understanding projection is very important. Four-dimensional objects can only be understood if they are projected into three dimensions. Over time, the brain can construct the change of projection of a four-dimensional object in three dimensions when it rotates around an axis.

For example, four-dimensional space draws a three-dimensional figure on paper, as shown below.

Then spread it on the table, holding 1 stick, perpendicular to the paper, perpendicular to O. So, X, Y and Z sticks, four axes, perpendicular to each other, imagine. The starting point of this thinking is to reduce three dimensions to two dimensions, and then establish one dimension on three dimensions. The world that humans usually see at all times is "the projection of the three-dimensional world on the two-dimensional optic nerve plane". What human beings can't feel is four or more dimensions of space, which all exist in mathematics. What human beings can feel is a three-dimensional space. When you touch something uneven, it is the most real 3-D world.

If we take a "person" as the object of investigation, the lens is gradually reduced, and we can see that: people have space and volume; A person's life is a process, and there are countless possibilities and choices, but one can only choose one. Finally, the timeline depicting a person's life is unique. People live in a universe, which can be regarded as a point in a larger space. There are countless other universes in this space, and each universe has its own birth and death and development process. This space can be regarded as a point in a larger space. This can be summarized as follows:

1-3 dimension is commonly known as length, width and height. ?

The duration is introduced into the 4-dimensional space, which can describe the process of human activities, and a person's life is equivalent to a unique timeline. ?

Five-dimensional introduction possibility. At a certain point, if people make other choices, they will have different timelines. 4-dimensional space is equivalent to one of countless timelines in 5-dimensional space. ?

6-dimension introduces all kinds of choices and changes that people may get before a certain point in time. 5-dimension space can be regarded as a point in 6-dimension space that people get under the influence of various factors and choices. ?

Starting from the seventh dimension, the observed object is magnified to the universe. A universe can be regarded as a point in seven dimensions, and every universe, like people, has different birth and death and development processes. Like people in 4-dimensional space, the universe has a unique development timeline in 7-dimensional space.

8-dimensional space, introducing all possible universe development history after the birth of the universe. ?

9-D introduces all the conditions that may affect the birth of the universe before its birth. ?

7-9 dimensions describe all possible development processes of the universe, just as 4-6 dimensions describe all possible development processes of human beings. ?

Dimensional space contains all possibilities and imaginations, and 9 dimensional space can be regarded as 1 points in 10 dimensional space.

Ten dimensional space is equivalent to a coordinate system, which is used to describe all the possibilities of human space.

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