Absolute geometry

As an absolute geometry in the narrowest sense, the whole of the geometric theorems about a three-dimensional space is called, the one solely on the basis of the axioms of logic (incidence axioms ) ( HI), the arrangement (H -II), the congruence ( H-III ) and the continuity (HV ), ie without the parallel postulate - can be derived. The names listed in brackets are here Axiom Group I, II, III and V in Hilbert's system of axioms of Euclidean geometry. In another sense, it also includes two-dimensional models which satisfy the axiom groups HI to H -III in their two-dimensional form, the so-called Hilbert - levels, the absolute geometry, these are ( in the main cases ) Euclidean or hyperbolic planes over Pythagorean bodies.

It is the set of sentences that are valid both in the Euclidean geometry as well as in the non-Euclidean geometries, or in other words to the " common base " of these geometries.

For example, do not apply to some congruence to the absolute geometry, the theorem on the sum of the angles in a triangle and the Pythagorean theorem. In Euclid's Elements, the first 28 sets are proved without the axiom of parallels and are therefore the absolute geometry in a narrow sense.

History

The term "absolute geometry " comes from one of the founders of non-Euclidean geometries, the Hungarian mathematician János Bolyai. This dealt in 1830 with the issue of the independence of the parallel postulate from the other axioms of Euclidean geometry, as formulated in the work elements of Euclid. In addition to Carl Friedrich Gauss, Bolyai was first to a model of non-Euclidean geometry, more precisely, a hyperbolic geometry.

Since the axioms of Euclid was not enough for modern mathematical claims, discussion of absolute and non-Euclidean geometry was first detected by Hilbert's system of axioms of Euclidean geometry in 1899 on a sustainable basis. On this basis, founded Johannes Hjelmslev 1907, the theory of Hilbert planes. Max Dehn in 1926 called this axiomatic justification of absolute geometry by Hjelmslev " the highest point reaches the modern mathematics Euclid also going in the grounds of elementary geometry ". But at that time we still had no overview of the models for these levels. In 1960, W. Pejas could describe all levels Hilbert algebra and able to bring this classical theory, the absolute geometry in a narrower sense, to a certain degree. All Hilbert levels are either Euclidean or hyperbolic in the main cases.

Hjelmslev himself generalized in the years 1929-1949, the absolute geometry with its "General Kongruenzlehre " to a geometry of the reflections. The basic idea is to put in place of axioms about points and lines axioms about the movement group based. On this basic idea Friedrich Bachmann builds with his "Structure of the geometry of the specular term" on. This leads him to the concept of absolute metric geometry. Finite models of this geometry are always Euclidean, infinite Euclidean models in the main cases, hyperbolic or elliptic or even under slightly weaker conditions be minkowskisch. Any flat or spatial metric absolute geometry can be embedded in a projective- metric determined by them geometry of the corresponding dimension.

Axiomatics

There is no universally accepted axioms of absolute geometry. Referred to in the introduction to Hilbert 's axioms without axiom of parallels are often used as a basis for discussion, which then weakened individual axioms or be omitted entirely. Historically this is the fact that the whole theory had its starting point in the discussion of the parallel postulate and its independence in Euclid. And the most famous modern axiomatic in the sense of Euclid was and is the Hilbert. A direct quote from Bachmann about this:

All significant differences between a geometry with parallel axiom and one without ( non-Euclidean geometry) already occur in two dimensions comparable to higher dimensions - quite unlike the also discussed in geometry since the 19th century problem of the set of Desargues, who just only in two-dimensional space is independent of the usual axioms. Therefore, many systems of axioms restricted to the planar case. Then, by the incidence axioms (HI) some unnecessary and you may be limited to I-1 to I-3:

Incidence axioms for plane

• I.1. Two distinct points P and Q determine always a straight line g
• I.2. Any two distinct points determine a straight line this straight.
• I.3. On a straight line, there is always at least two points in a plane, there are always at least three points not located on a straight line.

These are the existence and uniqueness of the connecting line and a Reichhaltigkeitsaxiom - it is clear that this " absolute minimum for an absolute geometry " is still too general.

Arrangement and congruence

Therefore, in general nor axioms from the groups will be added taken II ( arrangement ) and III ( congruence ). The full axiom group II, the arrangement precludes elliptical planes. The problem of congruence can be circumvented by the idea that one instead of the congruence (of figures in the plane) describes the group of Kongruenzabbildungen as generated by reflections group. This is just the basic idea of that in the historical section of this article geometry of the reflections of Hjelmslev. A recent axiom system that formally defines only axioms about reflections and the group generated by reflections based on the absolute geometry, the metric absolute geometry.

Axioms of the Circle

Hilbert's axioms of continuity ( group V with Hilbert ) are in absolute geometry often not demanded replaced ( for example, in Hjelmslev ) or by weaker axioms of the circle. Thus, the same constructions with ruler and compass can be performed as if the axioms of continuity required in the absolute geometry. In the special case of Euclidean geometry ( parallel axiom ) this corresponds to the generalization of the ordinary Euclidean space over the real numbers in which any continuous "lines" intersect when they should do it graphically, to a room on a Euclidean body in this for conic sections, ie especially true for circles and lines.