Four-dimensional space is a concept of space and time. Simply put, any space with four dimensions can be called a "four-dimensional space". However, most of the "four-dimensional space" mentioned in daily life refers to the concept of "four-dimensional space-time" mentioned by Einstein in his "General Theory of Relativity" and "Special Theory of Relativity". According to Einstein's concept, our universe is composed of time and space. The relationship between space and time is that in the structure of space, there is an additional time axis in addition to the three axes of length, width, and height in ordinary three-dimensional space, and this time axis is an imaginary axis.
According to Einstein’s theory of relativity: the three-dimensional space we face in life plus time constitute the so-called four-dimensional space. Since the time we feel on the earth is very slow, we will not obviously feel the existence of the four-dimensional space. However, once we board a spacecraft or arrive in the universe, the speed of our own frame of reference begins to become faster or begins to approach. At the speed of light, we can comparatively find the change in time. If you were traveling in a spacecraft traveling at close to the speed of light, your life would be much longer than that of a person on Earth. There is a potential field here, and the energy of the material changes as the speed changes. Therefore, the changes and comparisons of time are based on the speed of matter as the reference system. This is why time is one of the elements of four-dimensional space.
[Edit this paragraph] Analyze four-dimensional space
What is four-dimensional? The current theory is that three-dimensional space plus the dimension of time constitute the so-called four-dimensional space. However, this argument is immediately shattered. Why?
We can think about it in two dimensions. A two-dimensional creature (if there is one), they think of the so-called three-dimensional space absolutely differently from our three-dimensional space - they will think of time as the third dimension, because they cannot feel the existence of this dimension. Similarly, we have now fallen into this misunderstanding and regard time as the fourth dimension. Maybe the fourth-dimensional creatures also sigh for us when they see us promoting this kind of thinking. So does time count as one dimension? In my opinion, time should be one dimension, that is, adding one dimension to the dimensions of multi-dimensional creatures themselves to form a new N+1 dimensional space. Moreover, this will also help us solve some problems and allow us to compare A deeper understanding of the three-dimensional higher space.
There is a newer idea, that is, all dimensions are composed of time. Without time, there would be no space, including the most basic one-dimensional space. This should be easy to understand, because without time, the existence of space itself has no meaning, because space and time itself is an indivisible whole. So, why can one kind of time form different dimensional spaces? Here, we can think of time as a decomposable constant. Time can be broken down, which may be a little difficult to understand. However, it is very simple as long as you figure out the reason. To understand this truth, you must first understand two points. The first is the inseparability of time and space. I guess everyone knows this. It is meaningless to talk about time without space, or to talk about space without time. The second point is the diversity of time, which may be a little troublesome to understand. In daily life, what we come into contact with is the synthesis of time, that is, the organic combination of various sub-times to form a total time system. You may think that I am quibbling, but I am not. As long as you think about it from another angle, one result may be caused by several different reasons. Taking motion as an example, what we observe is generally a combination of one motion produced by several different motions, that is, combined motion. We can also think about time in this way. The time combination we see can be composed of the time of object movement, historical time (that is, experience time) and other times. As for movement time, we can think of it as the time of up and down movement, the time of left and right movement and the time of forward and backward movement. Of course, the division methods are diverse, which constitutes the diversity of time. As for how to divide it, it depends on different situations. A portion of time corresponds to a portion of space. In this incomplete space, time plays a decisive role.
The reason why we are three-dimensional creatures is that there is only three-dimensional time in this dimension of space. The incompleteness of time determines the incompleteness of space. We cannot understand spaces in other dimensions because we do not have the time to move in that space. The diversity of time determines the diversity of space.
At the same time, because of the different decomposition methods of time, our three-dimensional space is destined to be relative. It can be named one-dimensional, two-dimensional, or even any dimension - completely depending on the different decomposition methods. Time is the key to determining dimensions. At the same time, it is also the key to determining the way low-dimensional objects exist in high dimensions.
Let’s look at the scientific saying: low dimensions are defects in space. They do not have the space to move within the high-dimensional world. Regarding this, there is a question, that is how we can detect this flaw. The low-dimensional space we think of does not have a certain length because we cannot determine which length it has. That is, we cannot observe the length difference even with the best equipment. So, what about the future? We can't certify it now, but someone may prove in the future that the low-dimensional object does belong to a high-dimensional object. Therefore, there is no so-called spatial difference between low dimensions and high dimensions. So, how do we distinguish between high and low dimensions? Very simple, use time. Using time to explain any dimensional space, we can also think that the reason why low dimensions are inferior to high dimensions is because they have time defects and they cannot feel the existence of high dimensions within the scope of time. Therefore, if we want to understand low or high dimensions, we must first know the time range in which they exist. Transformation between high dimensions and low dimensions can be achieved. The reason is very simple. Just add or remove a time unit. However, it is easy to say, but very complicated to do. Our concept of time is so vague, and it is even more difficult to achieve time conversion in the space range class.
For the four-dimensional space, most people may only think that it is on the axes of length, width, and height, plus a time axis. However, most people still know very little about its specific conditions. An expert once made an analogy: Let us first assume that some flat people live in a two-dimensional space. They only have the concept of a plane. If you want to lock up a two-dimensional flat person, you only need to use a line to draw a circle around him. In this way, within the scope of the two-dimensional space, he will not be able to get out of the circle no matter what. Now those of us who live in three dimensions "interfere" with it. We only need to take the two-dimensional person out of the circle from the third direction (that is, from the direction of the axis indicating height) and put it back elsewhere in the two-dimensional space. For us three-dimensional beings, the situation in four-dimensional space is very similar to the above explanation. If we can overcome four-dimensional space, then it is not impossible to cross the distance of three-dimensional space in an instant.
From zero-dimensional space to four-dimensional space
——A brief discussion on pure conceptual research in geometry
(Ma Lijin, Department of Mathematics, Longdong University, Qingyang, Gansu 745000)
Abstract
Geometry is not necessarily a description of real phenomena. Geometric space and natural space cannot be treated completely identically. The development of purely conceptual research on geometry is a milestone in the field of mathematics. . The development from zero-dimensional space to three-dimensional space, especially from three-dimensional space to four-dimensional space, is a revolution in geometry.
Keywords
Zero dimension; one dimension; two dimensions; three dimensions; four dimensions; n dimensions; geometric elements; points; straight lines; planes.
Text
The concept of n-dimensional space advanced with the development of analytical mechanics in the 18th century. The concept of a fourth dimension appears insignificantly in the works of d'Alembert, Euler and Lagrange, and d'Alembert's entry on dimensions in the Encyclopedia proposed conceiving of time as the fourth dimension. Geometry beyond three dimensions was still rejected in the 19th century. Mobius (Karl August Mobius 1790-1868) pointed out in his "Calculation of the Center of Gravity" that two figures that are mirror images of each other cannot overlap in three-dimensional space, but they can overlap in four-dimensional space. But later he said: Such a four-dimensional space is difficult to imagine, so superposition is impossible. This situation occurs because people regard geometric space and natural space as completely equal. As late as 1860, Ernst Eduard Kummer (1810-1893) still mocked four-dimensional geometry. However, as mathematicians gradually introduced some concepts that had little or no direct physical meaning, such as imaginary numbers, mathematicians learned to get rid of the concept of "mathematics is the description of real phenomena" and gradually embarked on a purely conceptual research method. Imaginary numbers were once very puzzling because they have no reality in nature.
Treating the imaginary number as a directional distance on a straight line, and treating the complex number as a point or vector on the plane, this interpretation is the later four elements, non-Euclidean geometry, complex elements in geometry, n-dimensional geometry As well as the introduction of various weird functions, transfinite numbers, etc., it was the first to get rid of the concept of directly serving physics and ushered in n-dimensional geometry.
In 1844, Grassmann, inspired by quaternions, made a greater promotion and published "Linear Expansion", which was revised into "The Theory of Expansion" in 1862. He first dealt with the concept of general n-dimensional geometry. He said in an article in 1848:
My expansion calculus established the abstract foundation of space theory, that is, it is divorced from all space The intuition becomes a purely mathematical science, constituting geometry only in its special application to (physical) space.
However, the theorems in the expansion calculus are not just about translating geometric results into abstract language, they are of very general importance, because ordinary geometry is limited by (physical) space. Glassman emphasized that geometry could be applied to physics to develop purely intellectual studies. From then on, geometry severed its connection with physics and developed independently.
After the research of many scholars, n-dimensional geometry was gradually accepted by the mathematical community after 1850.
The above is the tortuous process of the development of n-dimensional geometry. The following are some specific processes of the development of n-dimensional geometry.
First, we regard the point as a zero-dimensional space, the straight line as a one-dimensional space, and the plane as a two-dimensional space, and observe the following postulates:
Two objects belonging to a straight line Point determines this straight line. 1.1
Two planes belonging to a straight line determine this straight line. (Compare this postulate with Postulate 1.1). 1.2
Two straight lines belonging to the same point also belong to the same plane. (Corollary to Postulate 1.2) 1.3
Two straight lines belonging to the same plane also belong to the same point. 1.4
It can be inferred:
1. Two spaces with the same dimension, under certain conditions, determine the other space with one higher dimension. For example: two points (we think of them as two zero-dimensional spaces) determine a straight line (one-dimensional space). Two straight lines (two one-dimensional spaces) belonging to the same point (specified conditions) also belong to the same plane (two-dimensional space).
2. Two spaces with the same dimension can also determine a lower one-dimensional space under certain conditions. For example: two planes (two two-dimensional spaces) determine a straight line belonging to them (one-dimensional space). Two straight lines (two one-dimensional spaces) belonging to the same plane (limited conditions) determine a point (zero-dimensional space).
3. Conclusion 2 does not include the fact that two planes can determine a space with one higher dimension. It only assumes that they define a straight line, which is one dimension of space lower than the plane. This leaves a gap to extend our thoughts to higher-dimensional space. This gap can be eliminated in Corollary 1.3 "Two straight lines belonging to the same point also belong to the same plane", by replacing the geometric elements point, straight line and plane with the geometric elements straight line, plane and three-dimensional space in sequence.
The following inference is the result of substitution. Two planes belonging to the same straight line also belong to the same three-dimensional space (Figure 1).
With this new corollary, we also include three-dimensional space, a geometric element that directly corresponds to other geometric elements.
The next step is to apply the duality principle to this reasoning and draw some inherent conclusions from these newly extended inferences. The principle of duality will be applied through the exchange of positions of geometric elements - plane and space. At this time we get the following inference:
Two three-dimensional spaces belonging to the same straight line also belong to the same plane (Figure 2). 1.5
From Corollary 1.5 we can get the following postulate:
The two existing three-dimensional spaces belonging to a plane determine this plane. 1.6
Based on the above 1.5 and 1.6, the following opinions can be put forward:
1. The geometric conditions of the four-dimensional space are obvious, because two with the same dimensions have Knowing space can only exist in a space one dimension higher than them.
For example: two different storage lines (one dimension) are located in a plane (two dimensions); two different storage planes (two dimensions) (storage along a line) are located in a three-dimensional space Here; two different stored three-dimensional spaces (stored along a plane) are located in a four-dimensional space.
2. Geometry is regarded as two planes that do not belong to the same straight line but intersect at a point, and belong to different three-dimensional spaces (Figure 3).
The concept of four-dimensional space can also be studied through analytical geometry. There we can represent geometric concepts using algebraic equations. In order to use this means of observation leading to the understanding of four-dimensional space, we will study the equations of the three geometric elements of the three-dimensional space system - points, lines and planes. Using the Cartesian system representation, we can write:
The equation of the point: ax + b = 0 (coordinate system: a point on the straight line).
The equation of a straight line: ax + by + c = 0 (coordinate system: two orthogonal straight lines on the plane).
The equation of the plane: ax + by + cz + d = 0 (coordinate system: three mutually perpendicular planes in three-dimensional space).
We can see from the above research:
The number of variables in the equation of each geometric element (or space) represented is equal to the dimension of the space plus 1.
The geometric elements in the coordinate system have the same dimensions as the geometric elements in the represented geometric space.
In this coordinate system, the number of geometric elements is equal to the dimension of the space being represented plus one. In a coordinate system, this number of geometric elements is the minimum requirement.
The coordinate system used to represent geometric elements is located in a space one dimension higher than the geometric elements it contains.
Based on the above observations, we can write the following equations for three-dimensional space. It should be noted that this equation has four variables (x, y, z, u).
ax + by + cz + du + e = 0
Now we can conclude:
1. The geometric elements of this coordinate system have three dimensions, that is, they is a three-dimensional space.
2. There are four three-dimensional spaces in this coordinate system.
3. This coordinate system is located in a four-dimensional space.
Our research on four-dimensional space and even higher space is not summarized through experiments. In reality, it is difficult for us to discover and deduce their general laws. For these problems, we can adopt a new method. research methods. That is: purely conceptual research. In this way, we can easily derive these new contents that are important but difficult to imagine in reality.
References
1. "Geometry of Four-Dimensional Drawing"
[American] C.E.S. Lindgren, S.M. Slaby (authors)
p>
Xie Shen (translator), Zhou Jiyi (principal)
Tsinghua University Press
2. "The Philosophical Walk of Fractal"
Lin Xia Water (Waiting)
Capital Normal University Press
3. "Analytical Geometry"
(Third Edition) edited by Lu Lingen, Xu Zidao, etc.
p>Higher Education Press
4. "Philosophy of Mathematics"
[U.S.] Paul Benacerraf, [U.S.] Hilary Putnam (Ed. )
The Commercial Press
[Edit this paragraph] Why is space-time four-dimensional
Authentic dimensional research methods are usually inseparable from the principle of human existence. For example, if space is two-dimensional, two-dimensional animals cannot digest normally. If space were more than four dimensions, the world would be much more exciting. If we are animals in four-dimensional space, Poincaré's conjecture about three-dimensional balls should not be the problem of the century. Unfortunately, the extra three-dimensional space makes the gravitational force and electrostatic force change more drastically with distance than in the three-dimensional space. This makes electrons as small as atomic nuclei and as large as planets in the solar system no longer stable, and they soon fly away in vortexes. away from or hit the center.
Many people cannot accept the principle of human existence and think it goes against the scientific tradition. The scientific method is to start from the first principle and deduce everything and even the observer. The principle of human existence is to deduce the universe from the conditions for the existence of observers. They are exactly at the opposite poles.
Hawking believes that the boundary condition of the universe is that it has no boundaries. Using the Karucha-Klein model, space-time is originally high-dimensional, and the reason why we feel it is four-dimensional is because the extra dimensions are rolled into small sizes that we cannot observe, such as Plan Gram scale. Just like the surface of a hair is two-dimensional, but at a glance, only the length of the hair remains. People call the space that they feel is outer space, and the space that is not perceptible is called inner space. Time is a dimension in outer space.
When using quantum cosmology to study the economic origin of space-time dimensions, it is necessary to avoid artificially adjusting the total dimensions of Karucha-Klein to obtain the required outer space dimensions. Because artificial adjustment will fall into a logical loop, this approach can achieve as many dimensions of space as you want. Therefore, the total dimensions of the available Kalucha-Klein models must be derived from first principles. The eleven-dimensional supergravity model is derived from first principles. There may be a so-called supersymmetry in nature.
In 1980, Freund and Rubin discovered a very beautiful universe model of eleven-dimensional supergravity, in which the inner space is a seven-dimensional sphere and the outer space is a four-dimensional sphere. But in the classical framework, one cannot prove that there are no solutions to external spacetimes with other dimensions.
In quantum cosmology, instantons are the seeds of the creation of the universe. Instantons are solutions of Einstar's equations and other field equations in which time and space coordinates are indistinguishable. The instanton that creates the universe of eleven-dimensional supergravity must be the product of the two factor spaces of the four-dimensional sphere and the seven-dimensional sphere space. If time is surrounded by four dimensions, the four-dimensional space-time will then expand and evolve into the macroscopic universe we live in and feel four-dimensional. Otherwise, the outer space-time will be seven-dimensional.
In the scenario of the creation of a charged black hole, the correct representation of the cosmic wave function must be used to calculate the probability of creation. Because regular instantons are very rare, the concept of constrained gravity must be introduced to study the creation of general black holes. Finding the right representation is crucial for the wave function not only of charged but also rotating black holes.
Starting from the same instant, after choosing the correct representation, the probability of time being created in a four-dimensional ball is much greater than the probability of time being created in a seven-dimensional manifold. Therefore, it is proven in quantum cosmology that outer space-time must be four-dimensional.
[Edit this paragraph] The four-dimensional space of the physical world
There are various multi-dimensional spaces in mathematics, but so far, the physical world we know is only four-dimensional, that is, three-dimensional space plus one dimension time. The high-dimensional space mentioned in modern microphysics has another meaning, which has only mathematical meaning.
Four-dimensional space-time is the lowest dimension that constitutes the real world. Our world happens to be four-dimensional. As for high-dimensional real space, at least for now, we cannot perceive it. I mentioned an example in a post. When a ruler is rotated in a three-dimensional space (excluding time), its length remains unchanged. However, when it is rotated, its coordinate values ??change, and there are differences between the coordinates. contact. The meaning of four-dimensional space-time is that time is the fourth-dimensional coordinate, which is related to space coordinates. That is to say, space-time is a unified and indivisible whole. They are in a relationship of "one goes down and the other goes up."
The four-dimensional space-time is not limited to this. From the mass-energy relationship, mass and energy are actually the same thing. Mass (or energy) is not independent, but related to the state of motion. For example, the greater the speed, the greater the speed. , the greater the mass. In four-dimensional space-time, mass (or energy) is actually the fourth-dimensional component of four-dimensional momentum. Momentum is a quantity that describes the motion of matter, so it is natural that mass is related to the state of motion. In the four-dimensional space-time, momentum and energy are unified and are called the four vectors of energy and momentum. In addition, four-dimensional velocity, four-dimensional acceleration, four-dimensional force, four-dimensional form of electromagnetic field equations, etc. are also defined in four-dimensional space-time. It is worth mentioning that the four-dimensional form of the electromagnetic field equations is more perfect, completely unifying electricity and magnetism. Electric fields and magnetic fields are described by a unified electromagnetic field tensor. The physical laws of four-dimensional space-time are much more perfect than those of three-dimensional space, which shows that our world is indeed four-dimensional. It can be said that at least it is much more perfect than Newtonian mechanics. At least due to its perfection, we cannot doubt it.
In the special theory of relativity, time and space constitute an indivisible whole - four-dimensional space-time. Energy and momentum also constitute an indivisible whole - four-dimensional momentum. This shows that there may be profound connections between some seemingly unrelated quantities in nature. When we discuss general relativity in the future, we will also see that there is a profound connection between space and time and the four vectors of energy and momentum.
[Edit this paragraph] Related events
Event 1:
In 1960, a strange thing also happened in the mysterious Bermuda Sea. In front of many onlookers, the American fighter jet was swallowed up by the clouds and disappeared.
One of the witnesses, H. Victor recalled: "I was working at the artificial satellite station at Kindley Air Force Base at the time. The weather was good that day, and the sky was clear except for one cloud.
"Five fighter jets were engaged in training flights. Including me, many base personnel were watching the sky. Five fighter jets rushed into a floating white cloud 800 meters above the coast. They stretched their necks desperately to look at the sky, but it never appeared again.
"The base suddenly became commotion. The commander of the control tower was an eyewitness from beginning to end. He also did not see any objects falling from the clouds to the sea. The radar screen also showed the original five fighter jets. The sudden disappearance of the shadow immediately attracted the attention of officials, and a search team was dispatched.
"The search range was from the coast of the base to the shoal 800 meters away. "I searched again and again, but couldn't find even a fragment of a fighter jet. The white cloud swallowed up a fighter jet and disappeared unknowingly..."
Event 2:
Another strange thing happened on June 1, 1968. On that day, on the outskirts of Buenos Aires, the capital of Argentina, South America, two cars were driving on the highway. One car carried the lawyers Bittle and his wife, and the other car carried their friends, the Gordons. Their destination was the city of Mabu, 150 kilometers away. The Gordons led the way. Soon, the car arrived at the outskirts of Mabu City in the dusk. Looking back, they saw that the Bittles' car was missing. They thought the lawyer's car had broken down. After entering the city, they made phone calls separately. They sent people to the villages and towns along the way to search along the highway.
After two days and finding nothing, the Gordons had no choice but to call the police. On the same day, Gordon received a long-distance call from Mexico, and the speaker was actually Lawyer Bittle himself. It turned out that they encountered an incredible thing:
When the Bittles' car passed through the city of Sheskom, white fog suddenly shrouded the front of the car. Soon, the car was completely surrounded by white fog. Bittel looked at his watch and saw that the time was 12:10 midnight. At this moment, the couple suddenly passed out. I don't know how much time passed before they woke up. It was getting light and the car was still driving on the highway. The strange thing is that the scenery on the road, as well as the clothing of the pedestrians, are all different from those in Argenyan. When I stopped and asked, I was really surprised: it turned out that they were already in Mexico City! Argentina is at least 6,000 kilometers away from Mexico. How could they drive the car from Argentina to Mexico? The lawyer himself couldn't figure it out.
The Bittles quickly called the Argentina Consulate in Mexico and asked for help. At this time, the hands of both of their watches stopped at 12:10, but in fact, it was already 12:10. It’s June 3rd. Strange things like this have been discovered many times in the world, so they have attracted the attention of many scientists.
Event 3
In 1934, at the Port of Philadephia in the United States, there was a destroyer full of officers and soldiers setting off to the distant sea. Suddenly, a wave hit, and before the helmsman could stabilize the direction, the ship magically appeared in the harbor of Norfolk, southeast of Fortuna.
The captain, first mate, pilot, helmsman and sailors all opened their eyes and looked at each other in confusion. No one knew what happened. The captain frowned and wondered at Philardi. The distance between the Port of Alphia and the Port of Norfolk is more than 500 kilometers. How could it be possible to sail from one port to another in a short period of time? Moreover, the chief mate, pilot, and helmsman did not neglect their duties and controlled this at all levels. How could such an incredible thing happen to a ship? It’s so baffling! ...
Incident 4
On May 10, 1956, in a city called Otas in Oklahoma, in the western United States, an eight-year-old child named Jimmy was having sex with his little friend. Tom and Ken play the game of "catch the robber" together. Jimmy climbed up the fence of a nearby house and caught Ken who passed under the fence.
While he was having fun, Jimmy suddenly shouted: "Ken, wait a minute!" and jumped down from the wall. At that moment, Jimmy disappeared. Tim and Ken were shocked and hurriedly shouted: " "Hey! Jimmy!" "Jimmy! Where are you hiding? Come out!"
The two children shouted to their friends, but there was no reply. People heard that Jimmy suddenly disappeared in front of two companions, and there was an uproar. Jimmy's mother hurriedly reported to the police station. The police thought that a case of child abduction had occurred and immediately launched a search, but to no avail.
A month passed, and one day, Jimmy's mother unexpectedly disappeared. Since no one was at the scene at the time, it is not known what the circumstances were when she disappeared. However, two consecutive incidents of sudden disappearances made the police nervous, and they conducted a comprehensive investigation again, but still found nothing. No one knows why Jimmy and his son are missing.
[Edit this paragraph] Scientists’ explanation
Scientists believe that there is an elusive channel between the earth and some mysterious world. On both sides of the passage are two worlds at different levels. People who study this phenomenon call the mysterious world hidden on the other side of the passage "four-dimensional space."
The universe is endless, and there are countless secrets hidden in the vast and boundless universe. Scientists' in-depth exploration of the "four-dimensional space" will reveal the mysteries of this "mysterious world". The mystery of the so-called "four-dimensional space" will surely be understood by mankind in the near future.
[Edit this paragraph] Related information
One dimension refers to a straight line with an origin, such as a number axis, which means that after the origin is set, a number can be used to represent the position.
Two-dimensional refers to a plane, which needs to be positioned by perpendicular intersecting axes, and the position is represented by two numbers
The three-dimensional analogy is three numbers, just like three-dimensional space
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Fourth dimension usually refers to adding a time axis to the three-dimensional space, and using three-dimensional numbers at a certain point in time to mark the position status. We should be in four-dimensional space
Fifth dimension is The dynamic space is called "speed"
The sixth dimension is the "temperature" generated by the friction caused by motion
The seventh dimension is the "electricity" generated by the temperature generating heat and explosion
Scientists believe that the three-dimensional space model is already unrealistic. Now cosmologists regard time as the fourth dimension, and the fifth dimension refers to the unlimited energy. According to the hypothesis of scientists, the universe is flat, and this makes time travel possible.
This entry is helpful to me
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Reference materials:
1.1. "Four-dimensional Drawing Geometry"
p>2.[US] C.E.S. Lindgren, S.M. Slaby (author)
3. Xie Shen (translator), Zhou Jiyi (editor)
4 .Tsinghua University Press
5.2. "The Philosophical Stroll of Fractal"
6. Lin Xiashui (waiting)
7. Capital Normal University Press
8.3. "Analytical Geometry"
9. (Third Edition) edited by Lu Lingen, Xu Zidao, et al.
10. Higher Education Press
11.4. "Philosophy of Mathematics"
12.[US] Paul Benacerraf, [US] Hilary Putnam (editor)
13. Business Press