On the one hand, geometry is understood to mean two- and three-dimensional Euclidean geometry , elementary geometry , which is also taught in mathematics lessons - formerly under the term spatial theory - and which deals with points , straight lines , planes , distances , angles , etc., as well as those concepts and Methods that were developed in the course of a systematic and mathematical treatment of this topic.
On the other hand, the term geometry encompasses a number of large sub-areas of mathematics whose relation to elementary geometry is difficult to recognize for laypeople. This is especially true of the modern concept of geometry, which generally denotes the study of invariant quantities.
History of German-language geometry literature
The oldest surviving geometry treatise in German dates from the beginning of the 15th century. It is the so-called Geometria Culmensis , which was written on behalf of the Grand Master of the Teutonic Order Konrad von Jungingen in the Culm area and contains the Latin text, which is essentially based on the Practica geometriae of Dominicus de Calvasio, and its German translation. Albrecht Dürer's Underweysung of measurement with the zirckel and Richtscheyt in planing lines and whole corporen from 1525 is considered the first printed and independent geometry book in German .
The use of the plural indicates that the term geometry is used in a very specific sense, namely geometry as a mathematical structure, the elements of which are traditionally called points, lines, planes ... and the relationships between them are regulated by axioms . This point of view goes back to Euclid , who tried to reduce the theorems of Euclidean plane geometry to a few postulates (i.e. axioms). The following list is intended to provide an overview of the various types of geometries that fit into this scheme:
- Projective Geometry and Affine Geometry : Such geometries mostly consist of points and straight lines, and the axioms concern straight lines connecting points and the intersection points of straight lines. Affine and projective geometries usually come in pairs: The addition of distance elements turns an affine geometry into a projective one, and the removal of a straight line or a plane with its points turns a two- or three-dimensional projective geometry into an affine one. In important cases, the points can be arranged on a straight line in affine geometry in such a way that half-lines and lines can be defined. In these cases the affine geometry and its projective closure are called 'arranged'.
- Euclidean geometry : This usually means the geometry derived from the axioms and postulates of Euclid. Because the structure of the theory handed down since Euclid still contained gaps, David Hilbert set up a system of axioms in his Fundamentals of Geometry (1899 and many other editions) from which he could clearly build Euclidean geometry down to isomorphism. This can then be clearly described as the three-dimensional real vector space, in which the points are represented by the vectors and the straight lines by the secondary classes of the one-dimensional subspaces. Stretching, standing vertically, angles, etc. are explained as in the analytical geometry customary since Descartes .
- Non-Euclidean Geometry : Geometries whose properties are in many ways analogous to Euclidean geometry, but in which the postulate of parallels (also known as the axiom of parallels) does not apply. A distinction is made between elliptical and hyperbolic geometries.
- Absolute geometry : is the common substructure of the Euclidean and the non-Euclidean geometries, i.e. H. the set of all theorems that are proved without the parallel postulate.
In every geometry, one is interested in those transformations that do not destroy certain properties (i.e. their automorphisms): For example, neither a parallel shift nor a rotation or reflection in a two-dimensional Euclidean geometry changes the distances between points. Conversely, any transformation that does not change the distance between points is a composition of parallel displacements, rotations and reflections. It is said that these mappings form the transformation group that belongs to a plane Euclidean geometry and that the distance between two points represents a Euclidean invariant. In his Erlangen program, Felix Klein defined geometry in general as the theory of transformation groups and their invariants (cf. mapping geometry ); however, that is by no means the only possible definition. Geometries and prominent invariants are listed below:
- Projective geometry : Invariants are the collinearity of points and the double ratio (ratio of partial ratios ) of four points of a straight line (in the complex number plane of any four points; if these lie on a circle, it is real)
- Affine geometry : The parallelism of straight lines, the division ratio of three points of a straight line, area ratios.
- Similarity geometry, in addition to affine geometry, route relationships and angles are invariant.
- Euclidean Geometry ; additional invariants are the distances between points and the angles.
- Non-Euclidean geometry : the collinearity of points, the distances between points and the angles are invariant. However, the two non-Euclidean geometries do not fit into the above hierarchy.
Areas of mathematics that are part of geometry
The following list encompasses 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 , the differential topology , the Riemann geometry and the theory of Lie groups are of differential geometry, among other subregions.
- Algebraic Geometry . It could also be viewed as a field of algebra. Since Bernhard Riemann she has also been using knowledge from function theory. The sub-areas of algebraic geometry include, for example, the theory of algebraic groups , the theory of Abelian varieties or also toric and tropical geometry .
- Convex geometry , which was essentially founded by Hermann Minkowski.
- Synthetic geometry continues the classic approach of "pure" geometry by using abstract geometric objects (points, straight lines) and their relationships (intersection, parallelism, orthogonality ...) instead of algebraic objects (coordinates, morphisms ...). The incidence structure is one of the most general approaches here today. Examples of non-linear incidence structures are the Benz planes .
- Algorithmic geometry ( computational geometry ).
- Discrete geometry , which contains combinatorial geometry as a further, oldest sub-area and deals with polyhedra, tilings , packings of plane and space, matroids , in the sub-area of finite geometry with incidence structures , block plans and the like.
Geometry in schools and classes
Devices such as compasses , rulers and set square , but also computers (see also: dynamic geometry ) are usually used in geometry lessons . The beginnings of geometry lessons deal with geometric transformations or the measurement of geometric quantities such as length , angle , area , volume , ratios , etc. More complex objects such as special curves or conic sections also occur. Descriptive geometry is the graphic representation of three-dimensional Euclidean geometry in the (two-dimensional) plane.
The statements are formulated in sentences .
- HSM Coxeter : Introduction to Geometry .
- HSM Coxeter, L. Greitzer: Geometry Revisited .
- Euclid : The elements .
- Georg Glaeser : Geometry and its applications in art, nature and technology . 2nd Edition. Elsevier, Spektrum Akademischer Verlag, Heidelberg (2007), ISBN 3-8274-1797-X
- David Hilbert : Fundamentals of Geometry
- Max Koecher , Aloys Krieg : level geometry . 3. Edition. Springer, Berlin (2007), ISBN 978-3-540-49327-3
- Hans Schupp: Elementarge Geometry , UTB Schoeningh, Paderborn (1977), ISBN 3-506-99189-2
- Georg Ulrich, Paul Hoffmann: Geometry for self-teaching . 5 volumes. 26th edition. C. Bange Verlag, Hollfeld (1977), ISBN 3-8044-0576-2 (Volume 1)
- M. Wagner: The ABC of geometry . 2nd Edition. CC Buchners Verlag, Bamberg (1920)
- Literature on geometry in the catalog of the German National Library
- Geometric evidence for the secondary level 1 state education server Baden-Württemberg
- Interview (67 MB; AVI ) on the subject of geometry with Hans-Joachim Vollrath
- John B. Conway , Peter Doyle, Jane Gilman , Bill Thurston : Geometry and the Imagination
- Hubert LL Busard : The Practica Geometriae of Dominicus de Clavasio. In: Archive History Exact Sciences. Volume 2, 1965, pp. 520-575.
- Geometry Culmensis. In: Burghart Wachinger u. a. (Ed.): The German literature of the Middle Ages. Author Lexicon . 2nd, completely revised edition, ISBN 3-11-022248-5 , Volume 2: Comitis, Gerhard - Gerstenberg, Wigand. Berlin / New York 1980, col. 1194 f.
- Underweysung with the circle and Richtscheydt . Wikisource