Synthetic geometry

Synthetic geometry is the branch of geometry which emanates from geometric axioms and theorems and often synthetic considerations and construction methods used - in contrast to analytic geometry, are already used in the algebraic structures such as body and vector spaces for the definition of geometric structures.

The modern synthetic geometry goes from axiomatically formulated " geometric " principles, which define the geometric objects, points, lines, planes, etc. implicitly through their relationships, and investigates the logical dependencies between differently formulated axiom systems. Here, the geometrical axioms mostly by algebraic structures (coordinates amounts in the broadest sense or structure -preserving mappings, as collineations ) modeled and thus incorporated into modern mathematics based on set theory and drawn from the space of intuition, Evidence arguments, as they were, of course, for Euclid, exclude from evidence.

History

The " geometry of Euclid " was synthesized essentially, although not devoted all his works of pure geometry. His major work "Elements" builds all of mathematics on geometric principles. Even numbers are first established as ratios of lengths and justify its geometric relationships.

The reverse approach of analytic geometry, in which geometric objects only by numbers and equations - coordinates - are defined and later by more general algebraic structures, has become prevalent in the 17th century by the reception of the works of René Descartes in mathematics - probably go the back to essential ideas to other scientists, see the section on mathematics in Descartes. The analytical approach has subsequently initiated generalizations of Euclidean geometry and perhaps only made possible.

The invention described in the introduction to modern synthetic geometry by Descartes dealt extensively with the question of the logical presuppositions and implications of the parallel axiom. This led to non-Euclidean geometries, for elliptic and hyperbolic geometry and common generalizations in absolute geometry.

Reached a high point of modern synthetic geometry in the 19th century from with the contributions of Jakob Steiner on projective geometry.

Geometric axioms

Since the synthetic geometry explores the axiomatic requirements for " geometry " in a very general sense, there are a number of axioms that can be classified according to different criteria.

• Hilbert's system of axioms of Euclidean geometry has five groups of axioms, which sets the conditions for the "classical" geometry can be investigated ( in the real plane and in three-dimensional real space ).

• Mostly are first " incidence axioms " assuming this group (Group I) is also fundamental in Hilbert and Euclid. On the basis of incidence geometry can build both absolute and projective and affine geometries. In the affine case frequently affine planes in projective projective planes are investigated. The affine and projective geometries are also in the further development of the geometries by projective extension of an affine or a projective plane slots in a variety of relationship.
• The Group II of Hilbert 's axioms, the axioms of the arrangement, result in certain affine planes for the introduction of intermediate relations for points on a line and page dividers and half-planes defined by pagination functions. A weak pagination is in a pappusschen level if and only possible if the coordinates of the body allows a nontrivial quadratic character. A "strong" arrangement if and only if the coordinates of the body allows a state array.
• The group III, the axioms of congruence are treated as properties of subgroups in the group of collineations of affine plane in the recent literature, and therefore no longer used as a basis in the classical form. Alternatively, a orthogonality can be introduced and evaluated.
• The parallel axiom that a separate group IV forms in Hilbert, is expected in the more recent literature on the incidence axioms. In absolute geometry, it is omitted completely, in projective geometry it is replaced by incidence axioms, which exclude its validity.
• The axioms of continuity ( group V with Hilbert ) are replaced in the recent literature for synthetic geometry by the weaker axioms of a Euclidean plane in which the possibilities of classical constructions can be studied by ruler and compass.

• Closing sentences are axioms of Euclidean geometry in synthetic geometry: The set of Desargues and its special cases and the set of Pappus correspond reversible clearly different generalizations of the usual concept of coordinates for affine and projective planes. ( → For a review, see Ternärkörper ).
• The set of Desargues may be demonstrated in at least three-dimensional spaces of very low incidence axioms for both affine and projective spaces. That's one of the reasons for which the synthetic geometry studied particularly planar structures. (→ See also Axiom of Veblen - Young).

Computational synthetic geometry

Although the employment problems of analytic geometry is a particular focus of computer-based algorithmic geometry, is operated within this framework, synthetic geometry ( "computational synthetic geometry" ). This evaluation will, for example, at what magnitude ( number of elements in a straight line ) can exist finite incidence levels (see block diagram).