Supplementary mathematics/Non-Euclidean geometry

Geometry is an area of mathematics that considers the regularities of position, size and shape of sets of points (e.g. on and between lines and surfaces), including their change and mapping. Depending on whether metric relationships (length, angular sizes, areas, volumes) are examined or whether only the mutual position of the objects is considered, one speaks of metric or projective geometry.

Metric geometries are Euclidean geometry, which is based on the parallel axiom, and the non-Euclidean geometries, such as Bolyai-Lobachevskian (hyperbolic) geometry, which retains all the axioms of Euclidean geometry but does not use the parallel axiom, and Riemannian (elliptical) Geometry that is also based on the assumption that not every straight line is infinitely long.

geometries in the overview

Geometry is an area of mathematics that considers the regularities of position, size and shape of sets of points (e.g. on and between lines and surfaces), including their change and mapping. Depending on whether metric relationships (length, angular sizes, areas, volumes) are examined or whether only the mutual position of the objects is considered, one speaks of metric or projective geometry. Metric geometries are Euclidean geometry, which is built on the parallel axiom, and the non-Euclidean geometries, such as Bolay-Lobachevskian (hyperbolic) geometry, which retains all the axioms of Euclidean geometry but does not use the parallel axiom, and Riemannian (elliptical) Geometry that is also based on the assumption that not every straight line is infinitely long. The projective geometry can develop these three geometries as special forms of a general dimension geometry.

on the emergence of non-Euclidean geometries

For around 2000 years, the general validity of Euclidean geometry was believed to exist, e.g. to describe the real physical space. But then the increasing criticism of this view led to two important discoveries:

The criticism of EUCLID's separation of geometry and arithmetic led to the creation of the concept of the real number, with the help of which not only commensurate but also incommensurable quantities could be characterized. The starting point for the emergence of a mathematics of constantly changing quantities was laid. RENÉ DESCARTES and CARL FRIEDRICH GAUSS worked in this field. Criticism of individual postulates, especially the fifth (parallel postulate), led to the development of other geometries that did not contradict reality - the non-Euclidean geometries (by LOBATSCHEWSKI, BOLYAI, GAUSS or RIEMANN), which led to the transition from a mathematics of constant relations to one of the mutable relationships meant. So the parallel axiom has been replaced by its opposite statement: "In a plane, through any point outside a given line, one can draw more than one line that does not intersect the given line". This geometry turned out to be just as consistent as Euclid-Hilbert's geometry. If many geometries are possible, there can be no general definition of the basic terms. The effort (of EUKLID and others) to define basic terms is therefore impossible in principle. Basic terms therefore only refer to the system under consideration.

But which geometry is valid in reality?

The statement that Euclidean geometry is a model of the real space of our intuition does not answer this question. In experiments on small areas of the earth's surface, the assumption that the surface is a plane may apply. If, on the other hand, you are experimenting in large areas, you have to imagine the surface as curved or as a sphere. Correspondingly, the non-Euclidean geometries in small areas hardly differ from the Euclidean geometry. The difference only becomes apparent in large room areas. This question about the geometric structures of the real world led to new discoveries and developments in the natural sciences, such as B. the theory of relativity by EINSTEIN, which broke radically with usual geometric ideas.

expansion

Neo-Euclidean geometry was later developed by Gauss and Riemann in the form of a more general geometry. This is the more general geometry that was used in Einstein's theory of general relativity. In non-Euclidean geometry, the set of interior angles is not like 180 degrees. For example, if the sides of the triangle are hyperbolic, the set of internal angles never reaches 180 degrees and is less. Also, if the geometry is elliptical, it will never be 180 degrees; Rather, it is more.