Non-Euclidean geometry

The non-Euclidean geometries are specializations of the absolute geometry. They differ from Euclidean geometry, which can, by the fact that the parallel axiom does not apply to them also be formulated as a specialization of the absolute geometry.

History

Early on, it had given consideration to the mathematical structure of space. Beginning of the 19th century provided the mathematician János Bolyai, Nikolai Lobachevsky and Carl Friedrich Gauss states that need not necessarily be a Euclidean geometry of the room, and began to develop a non-Euclidean geometry. However, this work has long remained unnoticed. Carl Friedrich Gauss did not publish his results in this respect.

Between 1818 and 1826 led Gauss to Hannovarian Surveying and developed this method with significantly increased accuracy. In this context, the idea that he had empirically been looking for a curvature of space by having measured the sum of the angles in a triangle, which is formed from the Brocken in the Harz Mountains, the island in the Thuringian Forest and the Hohen Hagen was born in Göttingen. Today, it is mainly viewed as a legend, even if the possibility of Gauss was looking for deviations from the normal value of the angle sum of 180 °, can not be excluded with absolute consistency. The accuracy of his instruments would not have been sufficient for the detection of tiny curvature of space in the gravitational field of the earth by far. It is not yet possible today.

Gauss's student Bernhard Riemann was the one who developed the differential geometry of curved spaces and 1854 introduced. At this time, no one expected a physical relevance of this topic. Tullio Levi -Civita, Gregorio Ricci - Curbastro Elwin Bruno Christoffel and built the differential geometry further. Einstein found in her work a variety of mathematical tools for his general theory of relativity.

Basics

Non-Euclidean geometries were developed not with the aim to clarify our experience of space, but as axiomatic theories in dealing with the parallel problem. The existence of models for non-Euclidean geometries (eg, by Felix Klein and Henri Poincaré ) proves that the parallel axiom of Euclid can not be deduced from the other axioms and is independent of them.

Non-Euclidean geometries are obtained by omitting the axiom of parallels from the axiom system or modifies it. The basic options for change are:

• There is no parallel to a straight line and a point outside the line. Two distinct lines in a plane intersect each other so always. This results in an elliptical geometry. An illustrative model of a two-dimensional elliptical geometry is the geometry of a spherical surface. Here, the sum of the angles of a triangle is larger than 180 °, of the circumference of a circle is less than and less than the surface area. In elliptic geometry, however, the axioms of order are no longer the same.
• There are at least two parallels to a line and a point outside the line. Here, all other axioms are met. This gives a hyperbolic geometry. It can be described by the models of small and Poincaré in two ways. On a small scale (or local) can be illustrated on a saddle surface of constant Gaussian curvature ( the so-called pseudo- sphere ). The sum of the angles of a triangle is now less than 180 °, of the circumference of a circle is greater than its surface area and over.

Meanwhile, the non-Euclidean geometry plays an important role in theoretical physics and cosmology. According to the general theory of relativity differs from the geometry of the universe from Euclidean because gravitational fields " bend " the room. Whether the geometry of the Universe " in the large" spherical ( elliptical), flat ( ie Euclidean ) or is hyperbolic, one of the major current issues in physics.

Mathematician who made ​​important contributions

• Giovanni Girolamo Saccheri
• Franz taurinus
• Ferdinand Karl Schweikart