Euclidean geometry refers to geometry constructed according to Euclid's Elements of Geometry.
Euclidean geometry sometimes refers to the geometry on a plane, that is, plane geometry. Euclidean geometry in three-dimensional space is usually called solid geometry. See Euclidean space for high-dimensional cases.
Mathematically, Euclidean geometry is a common geometry in plane and three-dimensional space, based on the assumptions of points, lines and surfaces. Mathematicians also use this term to represent high-dimensional geometry with similar properties.
The five axioms of Euclidean geometry are:
Any two points can be connected by a straight line.
Any line segment can be infinitely extended into a straight line.
Given an arbitrary line segment, you can make a circle with one of its endpoints as the center and the line segment as the radius.
All right angles are congruent.
If both lines intersect the third line and the sum of the internal angles on the same side is less than two right angles, the two lines must intersect at that side.
The fifth axiom is called the parallel axiom, and the following propositions can be derived:
Through points that are not on a straight line, there is one and only one straight line that does not intersect the straight line.
Parallel axioms are not so obvious as other axioms. Many geometricians tried to prove this axiom with other axioms, but all failed. 19th century, by constructing non-Euclidean geometry, it was proved that the parallel axiom could not be proved. If the parallel axiom is removed from the above axiom system, we can get a more general geometry, that is, absolute geometry. )
On the other hand, the five axioms of Euclidean geometry are incomplete. For example, there is a theorem in this geometry: any line segment is part of a triangle. He constructed it in the usual way: taking the line segment as the radius, taking the two endpoints of the line segment as the center respectively, and taking the points of the two circles as the third vertex of the triangle. However, his axiom does not guarantee that these two circles must intersect. Therefore, many revised versions of axiomatic systems have been proposed, including Hilbert axiomatic system. Euclid also put forward five "general concepts", which can also be used as axioms. Of course, he also took advantage of other properties of quantity afterwards.
Things that are equal to the same thing are equal.
Equivalent plus equivalent or equivalent.
What is equal minus what is equal is still equal.
If one thing coincides with another, they are equal.
The whole is greater than the part.
Non-Euclidean geometry is a branch of mathematics. Generally speaking, it has three different meanings: broad, narrow and ordinary. The so-called generalized geometry refers to all geometries different from Euclidean geometry, and the narrow non-Euclidean geometry only refers to Roche geometry. As for non-Euclidean geometry in the usual sense, it refers to Roche geometry and Riemann geometry.
Euclid's Elements of Geometry put forward five postulates. For a long time, mathematicians found that the fifth postulate was tedious and not as obvious as the first four postulates.
Some mathematicians also noticed that Euclid didn't use it until the 29th proposition in the book Elements of Geometry, and never used it again. In other words, the first 28 propositions can be derived from the geometric primitive without relying on the fifth postulate.
So some mathematicians ask whether the fifth postulate can be regarded as a theorem, not a postulate. Can we rely on the first four postulates to prove the fifth postulate? This is the most famous discussion on the theory of parallel lines in the history of geometric development, which has been debated for more than two thousand years.
Because the proof of the fifth postulate has not been solved, people gradually doubt whether the path of proof is correct. Can the fifth postulate be proved?
1In the 1920s, Lobachevsky, a professor at Kazan University in Russia, took another road in the process of proving the fifth postulate. He put forward a proposition that contradicted the European parallel axiom, replaced the fifth postulate with it, and then combined with the first four postulates of European geometry to form an axiom system and launched a series of reasoning. He believes that if there are contradictions in reasoning based on this system, it is equivalent to proving the fifth postulate. We know that this is actually the reduction to absurdity in mathematics.
However, in his meticulous and in-depth reasoning process, he put forward one proposition after another that is incredible in intuition but not contradictory in logic. Finally, Lobachevsky drew two important conclusions:
First of all, the fifth postulate cannot be proved.
Secondly, a series of reasoning in the new axiom system has produced a series of logically non-contradictory new theorems and formed new theories. This theory is as perfect and rigorous as Euclidean geometry.
This geometry is called Luo Barczewski geometry, or Roche geometry for short. This is the first non-Euclidean geometry.
From the non-Euclidean geometry founded by Lobachevsky, we can draw an extremely important and universal conclusion: a set of logically contradictory assumptions may provide a geometry.
Almost at the same time that Lobachevsky founded non-Euclidean geometry, Hungarian mathematician Bao Ye Janos also discovered the existence of unprovable fifth postulate and non-Euclidean geometry. In the process of learning non-Euclidean geometry, Baoye was also given a cold shoulder by his family and society. His father, Bao Ye Facas, a mathematician, thought it was a foolish thing to study the Fifth Postulate, and advised him to give up this kind of research. But Bao Ye Janos insisted on developing new geometry. Finally, in 1832, in a book by his father, the research results were published in the form of an appendix.
Gauss, then known as the "prince of mathematics", also found that the fifth postulate could not be proved and studied non-Euclidean geometry. However, Gauss was afraid that this theory would be attacked and persecuted by the church forces at that time, and he dared not publish his research results publicly. He just expressed his views to his friends in his letters, but he didn't dare to stand up and publicly support the new theories of Lobachevsky and Bao Ye.
Roche geometry
The axiom system of Roche geometry differs from Euclid geometry only in that the parallel axiom of Euclid geometry is replaced by "from a point outside a straight line, at least two straight lines can be parallel to this straight line", and other axioms are basically the same. Due to the difference of parallel axioms, a series of new geometric propositions different from Euclidean geometry are derived through deductive reasoning.
As we know, Roche geometry adopts all axioms of European geometry except one parallel axiom. Therefore, any geometric proposition that does not involve parallel axioms, if correct in Euclidean geometry, is also correct in Roche geometry. In European geometry, all propositions involving parallel axioms are not valid in Roche geometry, and they all have new meanings accordingly. Here are a few examples to illustrate:
Euclidean geometry
The perpendicular and diagonal of the same line intersect.
Two straight lines perpendicular to the same straight line or parallel.
There are similar polygons.
Crossing three points that are not in a straight line can be done, and only a circle can be made.
Roche geometry
The perpendicular and diagonal of the same line do not necessarily intersect.
Two straight lines perpendicular to the same straight line spread to infinity when both ends are extended.
There are no similar polygons.
Passing three points that are not on the same straight line may not necessarily make a circle.
From some propositions of Roche geometry listed above, we can see that these propositions are contradictory to the intuitive image we are used to. Therefore, some geometric facts in Roche geometry are not as easily accepted as European geometry. However, mathematicians put forward that we can use the facts in European geometry we are used to as an intuitive "model" to explain Roche geometry, which is correct.
1868, Italian mathematician Bertrami published a famous paper "An Attempt to Explain Non-Euclidean Geometry", which proved that non-Euclidean geometry can be realized on the surface of Euclidean space (such as quasi-sphere). In other words, non-Euclidean geometry propositions can be translated into corresponding Euclidean geometry propositions. If there is no contradiction in Euclidean geometry, there is no contradiction in non-Euclidean geometry.
Since people admit that Euclid has no contradiction, it is natural to admit that non-Euclid geometry has no contradiction. Until then, non-Euclidean geometry, which has been neglected for a long time, began to get extensive attention and in-depth research in academic circles, while Lobachevsky's original research was highly praised and praised by academic circles, and he himself was also known as "Copernicus in geometry".
Euclidean geometry, Roche geometry and Riemann geometry are three different geometries. All the propositions of these three kinds of geometry constitute a strict axiom system, which meets the requirements of harmony, completeness and independence. So these three geometries are all right.