Geometry originated from the Nile. In production practice, the ancient Egyptians made initial achievements in geometry in order to survey land, demarcate field boundaries, build water conservancy projects and build buildings. In the 3rd century BC, Euclid, an ancient Greek mathematician, collected, sorted out and systematized geometry knowledge by using strict logical reasoning methods adopted by eudoxus and Otto Kush, and compiled the world-famous Geometry Elements, thus creating Euclid geometry.
Euclidean geometry abstracts the undefined and primitive concepts of points, lines and surfaces from objective objects. Geometric propositions with no doubt about the truth summed up by human beings in long-term social life have become so-called axioms (or postulates) in Euclidean geometry, such as "two points determine a straight line" and "the shortest line segment between two points". 1899, Hilbert put forward a set of the most satisfactory axiomatic system at that time in his masterpiece Basic Principles of Geometry. Euclidean geometry starts from 23 definitions, 5 postulates and 5 theorems, arranges propositions in logical order and proves them by strict deductive methods. Poincare believes that it is beautiful to deduce the most mathematical structure from the least premise. Einstein appreciated the "beauty" of Euclidean geometry very much and said with emotion: "If Euclidean geometry failed to arouse your creative enthusiasm in your childhood, then you are not a theorist."
However, in the process of scientific development, the defects of Euclidean geometry become more and more obvious. There is a "fifth postulate" in the Elements of Geometry: when two straight lines are cut by a third straight line, if the sum of two internal angles on one side is less than two right angles, the two straight lines extend to that side and then intersect. The length and ambiguity of this postulate aroused people's doubts, but it proved that its pursuit failed one after another. D'Alembert called it "dirty clothes in the principle of geometry".
1826, Lobachevsky announced his research results in a paper, which marked the establishment of non-Euclidean geometry. Roche made an assertion contrary to Euclid's parallel postulate: at least two straight lines are parallel to the known straight line through a point that is not on the known straight line. Taking this as an axiom and combining with other propositions of Euclid's geometry, he has never been contradictory. So he came to two conclusions: (1) The fifth postulate cannot be proved by other axioms and theorems; (2) On the basis of denying postulate, we can develop a series of inferences-theorems that do not contain contradictions and form a set of logically possible theories. In this new geometry, the sum of the internal angles of the triangle will be less than 180.
Thirty years later, Riemann replaced the parallel postulate with another assertion, that is, it is impossible to draw a straight line that does not intersect with the straight line outside the straight line. Therefore, he introduced a new non-Euclidean geometry-Riemann geometry. In Riemannian geometry, the sum of the internal angles of a triangle will be greater than 180. So what geometry is closer to reality? The actual measurement shows that Euclidean geometry is more in line with the objective reality, but the theory of relativity holds that Euclidean geometry is not the most accurate method to describe the material space. Which is better or worse can only be tested by practice.
With the development of projective geometry, by the end of 19, Euclidean geometry and non-Euclidean geometry were unified in the system of projective geometry. Klein called Euclid geometry "parabolic geometry", Roche geometry "hyperbolic geometry" and Riemann geometry "elliptic geometry". Projective geometry was born in the Renaissance, which originated from the theory of projection of objects on a plane. 1822, Poncelet separated some special properties from geometric figures as research objects, which were called projective properties, and the geometry to study the projective properties of figures was projective geometry.
At the same time that projective geometry came into being, Fermat and Descartes successfully studied geometric problems with algebraic methods and created analytic geometry. The center of analytic geometry is to connect algebraic equations with curves and surfaces, so that geometric figures and algebraic languages can be transformed into each other and the number and shape can be unified. It turns a problem that can't be solved by pure geometric method into algebraic operation, which is relatively easy to solve. Analytic geometry is the result of symbolic algebra. At the same time, the concept of function has been established and developed, which is the basis of calculus.
/kloc-calculus developed rapidly in the 0 ~ (th) century. 173 1 year, differential geometry came into being. This geometry takes mathematical analysis and differential topology as research tools, and mainly discusses the properties of smooth curves and surfaces. Differential geometry is widely used in theoretical physics, such as gravitational theory and gauge field.
Cantor's point set theory expands the scope of shape, and Poincare's topology makes the continuity of shape the object of geometry research, which endows geometry with new contents. With the explosive development of modern mathematics, the branches of geometry have emerged in an endless stream and taken on a new look. No one can predict how tomorrow's geometry will be presented to people.