Synthetic geometry

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Synthetic geometry is the branch of geometry that starts from geometric axioms and theorems and often uses synthetic considerations or construction methods - in contrast to analytical geometry , in which algebraic structures such as solids and vector spaces are already used to define geometric structures.

Modern synthetic geometry is based on axiomatically formulated "geometric" principles that implicitly define the geometric objects, points , straight lines , planes , etc. through their relationships to one another, and investigates the logical dependencies between differently formulated axiom systems. The geometric axioms are mostly modeled by algebraic structures ( sets of coordinates in the broadest sense or structure-preserving maps, such as collineations ) and thus incorporated into modern mathematics, which is based on set theory and evidence arguments drawn from the visual space , as they were still self-evident for Euclid , excludes from evidence.

history

The geometry of Euclid was essentially synthetic, even if not all of his works of pure geometry dedicated. His main work " Elements " builds all of mathematics on geometrical foundations. Numbers are also initially established as ratios of lengths and their relationships are geometrically justified.

The reverse approach of analytical geometry, in which geometric objects are first defined by numbers and equations -  coordinates  - and later by more general algebraic structures , became dominant in mathematics in the 17th century through the reception of the works of René Descartes - presumably they go essential ideas on this to other scientists, see the section on mathematics in Descartes . The analytical approach then initiated generalizations of Euclidean geometry and perhaps made them possible in the first place.

The modern synthetic geometry according to Descartes described in the introduction dealt intensively with the question of the logical presuppositions and consequences of the axiom of parallels . This led to non-Euclidean geometries , to elliptical and hyperbolic geometries, and to common generalizations in absolute geometry .

Modern synthetic geometry reached a high point in the 19th century a . a. with the contributions of Jakob Steiner to projective geometry .

Geometrical axioms

Since synthetic geometry explores the axiomatic prerequisites for “geometry” in a very general sense, there are a large number of axioms here that can be classified according to different points of view.

  • Hilbert's system of axioms of Euclidean geometry has 5 groups of axioms with which the prerequisites for theorems of “classical” geometry (in the real plane and in three-dimensional real space) can be examined.
    • Most of the time "incidence axioms" are assumed, the group (group I) of which is also fundamental for Hilbert and Euclid. On the basis of the incidence geometry one can build absolute as well as projective and affine geometries . In the affine case affine planes are often examined , in the projective case projective planes . The affine and projective geometries are also in a diverse relationship in the further construction of the geometries by projective expansion of an affine or slots of a projective plane.
    • Group II of Hilbert's axioms, the axioms of arrangement , lead in certain affine planes to the introduction of interrelationships for points on a straight line and to side divisions and half-planes which are defined by side division functions. A weak division of the sides is possible in a Pappus plane if its coordinate body allows a nontrivial quadratic character . A “strong” arrangement exactly when the coordinate body allows a body arrangement .
    • Group III, the axioms of congruence, are treated in the more recent literature as properties of subgroups in the group of collineations of an affine plane and are therefore no longer used in the classical form. Instead, an orthogonality relation can be introduced and examined.
    • The axiom of parallels, which Hilbert's own group IV forms, is counted among the axioms of incidence in the more recent literature. In absolute geometry it is completely omitted, in projective geometry it is replaced by incidence axioms which exclude its validity.
    • The axioms of continuity (group V in Hilbert's) are replaced in the more recent literature on synthetic geometry by the weaker axioms of a Euclidean plane in which the possibilities of classical constructions with compasses and rulers can be examined.
  • Closing theorems of Euclidean geometry are axioms in synthetic geometry: Desargues 'theorem and his special cases and Pappos' theorem correspond to reversibly uniquely different generalizations of the usual concept of coordinates for affine and also for projective planes. For an overview, see the article on ternary bodies , a concept of geometric algebra that can adequately algebraize certain classes of affine and projective geometries. For a complete isomorphic algebraic description of model classes of synthetic geometries, including the cited closure theorems, reference is made to the main article Geometric Relation Algebra .
  • Desargues' theorem can be proved in at least three-dimensional spaces from very weak incidence axioms for both affine and projective spaces . This is one of the reasons why synthetic geometry studies particularly flat structures. (see also the axiom of Veblen-Young ).

Computational synthetic geometry

Although the study of problems of analytic geometry, the focus of the particular computer -based computational geometry is, in this context, synthetic geometry (is computational synthetic geometry ) operated. For example, it is investigated for which orders (number of elements of a straight line) finite incidence levels can exist (see block diagram ).

literature

  • Euclid: The Elements - Books I – XIII. Edited and translated by Clemens Thaer . 4th edition. Harri Deutsch, Frankfurt am Main 2003, ISBN 3-8171-3413-4 (first published 1933–1937).
  • Benno Artmann: Euclid - The Creation of Mathematics . Springer, Berlin / Heidelberg 1999, ISBN 0-387-98423-2 . - English-language introduction to the structure and proof technique of the elements
  • Jürgen Bokowski, Bernd Sturmfels : Computational synthetic geometry . Lecture Notes in Mathematics 1355. Springer, New York 1988, ISBN 0-387-50478-8 .
  • Gino Fano : Contrast of synthetic and analytical geometry in its historical development in the XIX. Century. In: Encyclopedia of Mathematical Sciences including its applications. Third volume in three parts: Geometry . Teubner, Leipzig 1910 ( full text at the Göttingen Digitization Center [PDF]).
  • David Hilbert : Fundamentals of Geometry . Teubner, Stuttgart 1999, ISBN 3-519-00237-X (first edition: Leipzig 1899).
  • Jeremy Gray: Worlds out of nothing: a course of the history of geometry of the 19th century . Springer, 2007, ISBN 978-0-85729-059-5 .

Individual evidence

  1. ↑ Jürgen Bokowski, Bernd Sturmfels : Computational synthetic geometry . Lecture Notes in Mathematics 1355. Springer, New York 1988, ISBN 0-387-50478-8 .