What are non-Euclidean dates

non-Euclidean geometry

In a broader sense, generic term for geometries that are different from Euclidean geometry, d. H. thus both for elliptical geometry and for hyperbolic geometry. While in hyperbolic geometry there are at least two parallel straight lines for every straight line through every point not lying on it, in elliptical geometry there are no parallel straight lines at all.

However, the term “non-Euclidean geometry” is often used (in a narrower sense) for the also as Lobachevsky geometry or hyperbolic geometry The theory referred to is used, which is based on the axioms of absolute geometry and the negation of the parallels axiom of Euclid. This emerged from the centuries-long efforts of many mathematicians to prove the axiom of parallels on the basis of the other axioms of geometry (parallel problem). In the 1820s, the three mathematicians Janos Bolyai, Carl Friedrich Gauß and Nikolai Iwanowitsch Lobatschewski came to the conclusion, largely independently of one another, that a geometry exists in which the axioms of absolute geometry and the negation of the Euclidean axiom of parallels apply So there are not in all cases uniquely determined parallels to given straight lines through given points, so there are points through which two or more different parallels run to given given straight lines. This knowledge, which is difficult to perceive, had enormous philosophical consequences that prevented Gauss from publishing his findings in this regard during his lifetime. In 1830 he wrote in a letter to Bessel:

We must humbly admit that if the number is merely a product of our spirit, space also has a reality outside of our spirit, for which we cannot completely prescribe its laws a priori to work out very extensive investigations about it for public announcement and perhaps this will never happen in my lifetime, since I shy away from the screams of the Boiotians if I wanted to express my opinion fully.

Lobachevsky was the first to publish his work on non-Euclidean geometry in 1829, which explains the name "Lobachevsky geometry" mentioned earlier. However, the response to his work was initially low. The first leading mathematician to recognize the importance of the new geometry was Bernhard Riemann. He also dealt with spherical geometry from a completely new perspective and developed it into an independent non-Euclidean geometry (elliptical geometry or Riemann geometry). Above all, however, with his general theory of manifolds (1854) he created a theoretical building that led to a systematization of the already existing geometries and produced further geometrical systems.

The starting point for this is the concept of the internal geometry of a surface already introduced by Gauss. Riemann examined the geometries on surfaces of constant curvature, which can be classified into three categories:

  1. Riemannian or elliptical geometry as geometry on a surface of constant positive curvature,
  2. Euclidean geometry as geometry on a surface of curvature zero,
  3. Lobachevskian or hyperbolic geometry as geometry on a surface of constant negative curvature.
See also elliptic geometry, Euclidean geometry, hyperbolic geometry.