# What is the definition of postulation for geometry

## Euclidean geometry

II. For a long time the question whether the parallel postulate can be proven on the basis of the other axioms and postulates has been controversial. This so-called parallel problem occupied generations of mathematicians and resulted in countless attempts at proof< agenum="" _86="" hervor,="" die="" jedoch="" alle="" daran="" scheiterten,="" daß="" unbemerkt="" aussagen="" verwendet="" wurden,="" die="" nicht="" aus="" den="" übrigen="" axiomen="" und="" postulaten="" ableitbar="" sind="" und="" vielmehr="" zum="" 5.="" postulat="" äquivalente="" aussagen="">

It was only when Gauss, Bolyai and Lobachevsky were able to show in the first half of the 19th century that the theory, which consists of the other axioms and postulates as well as the negation of the 5th postulate, is a consistent theory (namely a non-Euclidean geometry), the problem was solved, albeit in a completely unexpected way.

Since then, Euclidean geometry in the sense of an axiomatic structure of geometry has been the geometry in which all axioms of geometry, i.e. both those of absolute geometry and the 5th postulate of Euclid or the Euclidean axiom of parallels apply.

III. The Erlangen program, which Felix Klein presented in his inaugural lecture in 1872, represents a completely different way of describing different geometries than the axiomatic way. According to this, a geometry can be understood as an invariant theory with respect to a transformation group on a set. The (point) set on which the Euclidean geometry is based is the Euclidean point space: an affine point space with an associated vector space on which a positively definite symmetrical bilinear form (scalar product) is defined. The transformation group that characterizes Euclidean geometry is the group of movements (Euclidean movement group), the associated transformation group of the Euclidean vector space is the group of orthogonal mappings, i.e. H. those mappings that leave the scalar product of two vectors unchanged. Thus, movements are equidistant images of the Euclidean point space on themselves.

Euclidean geometry in the sense of the Erlangen program is thus the theory of invariants with regard to the movements of a Euclidean point space. These invariants include, among others. Line lengths, line ratios, areas or volumes and the dimensions of angles, whereby the invariance of line lengths distinguishes Euclidean geometry from other (more general) geometries.

Compare this with Felix Klein's Erlangen program.

- Are Nutrela pieces vegetarian
- Has the Gulf Stream changed course?
- What is digital marketing and the future
- Life What is the 2468 Diaet
- Ernest Hemingway was a double agent
- Why is discarded airline food not recycled
- Where can I buy a star
- Who is your favorite designer 1
- What is a Semi-Voice Process in BPO
- Women treat handsome men differently
- What is the supermoon phenomenon
- Think arranged marriages are out of date
- Is ghee milk free
- How do I fix this WiFi problem
- Which country is Lithuania's best friend today?
- What is the introduction of JavaScript
- Should hypocritical politicians be ousted
- How can I suppress my morning fear
- How do you test executable Haskell files
- What is a Julian Calendar
- What does a current meter do with registers
- How accurate are drug sniffers
- Royal families are still important
- What shocked you