Euclidean ry, Roche ry, and the principle of Cartesian ry [Z]

Ry credits: euclidean and non-European ry. Non-European ry includes the Roche ry and the Cartesian ry.
The traditional description of Euclidean ry (Euclidean ry) is an ordinary system of justice. It proves all true propositions through finite theorems ".
The five theorems of Euclidean ry are:
1. Any two points can be connected in a straight line.
2. Any line segment can be infinitely extended into a straight line.
3. Given any line segment, you can use an endpoint as the center and the line segment as a circle with the radius.
4. All right angles are equal.
5. If the two lines are intersecting with the third line, and the sum of the inner angles of the same side is less than two right angles, the two lines must be intersecting on this side.
The fifth kilometer is called a parallel principle and the following proposition can be derived:
Through a point that is not in a straight line, there is only one line that does not intersection this line.
For a long time, mathematicians have found that the fifth public design is lengthy and not so obvious compared with the first four public ones.
Some mathematicians also noticed that Euclidean was used in the book ry, which was not used until 29th propositions, and was no longer used. That is to say, in geometric original, the first 18 propositions can be launched without relying on the fifth public design.
Therefore, some mathematicians have suggested that the fifth public can be used as a public but as a theorem? Can we rely on the first four public settings to prove the fifth Public configuration? This is the most famous in the history of geometric development. It has argued over more than two thousand years of discussions about the parallel line theory.
Since the problem of proving the fifth public establishment has never been solved, people gradually suspect that the proof is going on, right? Can the fifth Public configuration prove it?
In 1820s, robacski, a professor at Kazan University in Russia, took another path in proving the fifth public institution. He proposed a proposition that contradicts the European-style parallel principle, used it to replace the fifth public design, and then combined it with the first four public settings of the European-style ry into an ordinary system to carry out a series of reasoning. He believes that if there is a conflict in the system-based reasoning, it will prove the fifth public design. We know that this is actually the inverse proof method in mathematics.
However, in his profound and meticulous reasoning, he came up with an intuitive, but logically non-contradictory proposition. Finally, robachski drew two important conclusions:
First, the fifth Public IP cannot be proved.
Second, a series of reasoning in the new system of justice, obtained a series of new theorems without contradictions in logic, and formed a new theory. This theory is perfect and rigorous, just like the European ry.
This type of ry is called the robachwski ry, or the Rochelle ry ". This is the first non-European geometry to be proposed.
From the non-European ry created by robacski, we can draw an extremely important and universal conclusion: logically, a set of non-conflicting assumptions may provide a geometric structure.
Almost along with the creation of non-European ry by robachski, the Hungarian mathematician Bao ye Yano also discovered the existence of the fifth public building and non-European ry. During his study of non-European geometry, Bao ye was also treated with indifference from the family and society. His father, the mathematician Bao ye falkash, believes that the study of the fifth public institution is a waste of effort and effort, and he is advised to give up this research. But Bao ye Yano insisted on working hard to develop new ry. Finally, in 1832, his father published his findings in an appendix.
Gaussian, hailed as the Prince of mathematics in that era, also found that the fifth public institution could not prove and studied non-European ry. However, Gaussian feared that this theory would be cracked down and persecuted by the church forces at that time. He did not dare to publish his own research results, but expressed his views to his friends in his book, nor dare to stand out and publicly support their new theories.
Loose ry
In contrast to the European-style geometric system, the geometric system of the Luo type only replaces the European-style geometric parallel principle with "a point out of a straight line, at least two straight lines can be parallel to this straight line, the other principles are basically the same. Due to the difference in the parallel principle, deductive reasoning leads to a series of new geometric propositions different from the European geometric content.
We know that in addition to a parallel principle, the Luo ry adopts all the principles of the European ry. Therefore, any geometric proposition that does not involve the parallelism principle is also true in the Euclidean ry if it is correct. In Euclidean ry, all the propositions involving the principles of parallelism are not true in the repeat ry, and they all contain new meanings accordingly. The following are examples:
Ry
The vertical lines and diagonal lines of the same line intersect.
Two straight lines or parallel directions perpendicular to the same line.
Similar polygon exists.
Three points that are not in the same straight line can be done and can only be a circle.
Loose ry
The vertical lines and diagonal lines of the same line do not necessarily overlap.
The two straight lines perpendicular to the same line are discrete to infinity when the two ends are extended.
Similar polygon does not exist.
The circle may not be a circle after three points in a line.
From the propositions listed above, we can see that these propositions are in conflict with the intuitive image we are used. Therefore, some geometric facts in the geometric structure are not as easy to accept as those in the Euclidean structure. However, after research, mathematicians have proposed that it is correct to use the fact in the European ry we are used to as an intuitive "model" to explain the lustrory ry.
In 1868, Italian mathematician Beth Lamy published a famous paper "an attempt to explain non-European ry", proving that non-European ry can be achieved on a euclidean space surface (such as a quasi-ball surface. That is to say, the non-European geometric proposition can be translated into the corresponding Euclidean geometric proposition. If there is no contradiction in the Euclidean geometric, the non-European geometric will naturally have no contradiction.
Since we acknowledge that there is no conflict in the European Union, we naturally acknowledge that there is no conflict in the non-European ry. At this time, the long-time unattended non-European ry began to receive widespread attention and in-depth research from the academic community. Therefore, the original research of robacski was highly appraised and praised by the academic community, he himself has been hailed as "cobaini in ry ".
Riman ry
In the Euclidean ry and the Roche ry, the Union principle, sequence principle, continuous principle, and contract principle are all the same, but the parallel principle is different. In the European ry, "a point outside a straight line has only one straight line parallel to a known straight line ". Roche ry says, "at least two straight lines are parallel to known lines at a point outside a straight line ". Is there such a ry that "beyond a straight line, a straight line cannot be parallel to a known straight line "? This is the answer to the question given by the Cartesian geometry.
He was founded by the German mathematician riman. In his essay in 1851, on the assumption of the basis of ry, he explicitly proposed the existence of another ry, which created a new broad field of ry.
A basic rule in the Li-man ry is that any two straight lines in the same plane have common points (intersections ). The existence of parallel lines is not recognized in the column-related geometry. Another theory of this theory is that straight lines can be sung infinitely, but the total length is limited. The model of the Cartesian geometry is a sphere that has been properly "Improved.
In modern times, we have gained an important application in the general theory of relativity. In the general theory of relativity of the physicist Einstein, the space ry is the Cartesian ry. In the general theory of relativity, Einstein gave up his idea about the uniformity of time and space. He believed that time and space was only an approximate and even space in a small space, but the whole time and space was uneven. In physics, this interpretation is exactly the same as the concept of the Cartesian ry.
In addition, the Cartesian geometry is also an important tool in mathematics. It is not only the basis of differential ry, but also applied to differential equations, variational methods, and complex variable functions.
Three geometric relationships
Euclidean, Roche, and Li-man are the three types of ry. All the propositions of these three geometries constitute a strict system of justice, and each of them satisfies the harmony, completeness, and independence. Therefore, these three types of ry are correct.
In our small, not-too-close space, that is, in our daily life, Euclidean ry is applicable. In the space of the universe or in the world of nuclear atoms, roche ry is more in line with objective reality. In the research on the Earth's surface, the actual problems such as sailing and aviation are more accurate.