Geometry

The geometry ( ancient Greek γεωμετρία geometria, earth-measure ',' measurement of land ') is a branch of mathematics.

On the one hand we mean by geometry, the two - and three-dimensional Euclidean geometry, elementary geometry, which is also taught in the classroom and dealing with points, lines, planes, distances, angles, etc., as well as those conceptions and methods in the course of a systematic and mathematical treatment of this issue have been developed.

On the other hand, the term geometry includes a number of major areas of mathematics, their relation to elementary geometry for lay people is just more difficult to see.

Topics

Geometries

The use of the plural indicates that the term geometry is used in a very specific sense, namely geometry as a mathematical structure whose elements traditionally points, lines, planes, .... hot, and their relationships with each other are governed by axioms. This view goes back to Euclid, who has been trying to return the records of the plane Euclidean elementary geometry on a few postulates (ie axioms ). The following list is intended to give an overview of different types of geometries that fit into this scheme:

• Projective geometry and affine geometry: Such geometries usually consist of points and lines, and the axioms concerning connecting straight points and the intersection points of lines. Affine and projective geometries usually come in pairs: Adding remote elements makes an affine geometry of a projective, and the removal of a straight line or a plane with their points makes a two - or three-dimensional projective, affine geometry. In important cases, the points can be arranged on a straight line in the affine geometry that can half-lines and paths define. In these cases, it is called the affine geometry and its projective completion ' arranged '.

• Euclidean geometry: Below is customarily derived from the axioms and postulates of Euclid's geometry. Because since Euclid traditional structure of the theory still contained gaps accuracy, David Hilbert in his Foundations of Geometry (1899 and many further editions ) established a system of axioms from which he was able to establish clearly the Euclidean geometry up to isomorphism. Thereafter it can be clearly described as the three-dimensional real vector space in which the points are represented by the vectors, and the straight line passing through the cosets of the one-dimensional sub-spaces. Distances, perpendicularity, angle, etc. are explained in the usual since Descartes analytical geometry.

• Non-Euclidean geometry: geometric shapes, their properties are analogous to Euclidean geometry in many ways, but in which the parallel postulate (also called parallel postulate ) does not apply. A distinction is elliptic and hyperbolic geometries.

• Absolute geometry is the common foundation of Euclidean and non-Euclidean geometries, ie the set of all sets that are proved without the parallel postulate.

In each geometry one is interested in those transformations that do not destroy certain properties (ie, their automorphisms ): For example, to change either a parallel translation or a rotation or reflection in a two-dimensional Euclidean geometry, the distances of points. Conversely, every transformation that does not change the spacing of spots, a composition of parallel translations, rotations and reflections. It is said that these figures constitute the transformation group, which belongs to a plane Euclidean geometry, and that the distance between two points is a Euclidean invariant. Felix Klein in his Erlangen Program geometry generally known as the theory of transformation groups and their invariants defined ( see Figure geometry ); However, this is by no means the only possible definition. The following geometries and prominent invariants are enumerated:

• Projective Geometry: invariants are the collinearity of points and the cross-ratio (ratio of partial ratios ) of four points on a line ( in the complex plane of any four points, and when they lie on a circle, it's real)

• Affine geometry: The parallelism of lines, the division ratio of three points on a line, surface area ratios.

• Similarity geometry, in addition to the affine geometry are distance ratios and angles invariant.

• Euclidean geometry; Additional invariants are the distances of points and angles.

• Non-Euclidean geometry are invariant collinearity of points, the distances between points, and the angle. However, the two non-Euclidean geometries do not fit into the above hierarchy.

Areas of mathematics, which belong to the geometry

The following list includes very large and far-reaching areas of mathematical research.

• Elementary geometry
• The differential geometry is the branch of geometry in which particular methods of analysis and topology are used. The elementary differential geometry, differential topology, Riemannian geometry and the theory of Lie groups include branches of differential geometry.
• Algebraic Geometry. You could also consider a field of algebra. Since Bernhard Riemann It also uses knowledge of the theory of functions.
• Convex geometry, which was established mainly by Hermann Minkowski.
• Synthetic geometry continues the classic approach of pure geometry, by be based instead of algebraic objects (coordinates, morphisms ... ) abstract geometric objects ( points, lines ) and their relationships (cut, parallelism, orthogonality ... ). The incidence geometry heard here today among the most common approaches.
• Computational Geometry (computational geometry )
• Discrete geometry containing the combinatorial geometry as another, the oldest sub-area, and is concerned with polyhedra, tilings, packings of the plane and space, matroids, in the subfield of finite geometry with incidence structures, block plans and the like.

Geometry in school and teaching

(: Dynamic geometry also see ) used Traditionally, the teaching of geometry devices such as compass, ruler and protractor, but also the computer. The rudiments of geometry lessons about dealing with geometric transformations, or the measurement of geometrical quantities such as length, angle, area, volume, ratios, etc. More complex objects such as curves or special conics occur. Descriptive Geometry is the graphical representation of the three-dimensional Euclidean geometry in the ( two-dimensional ) plane.