Absolute geometry
As absolute geometry in the narrowest sense, the whole of the geometric will sets a three-dimensional designated space, which can solely on the basis of the axioms of the link ( incidence axioms ) (HI), the assembly (H-II), the congruence (H-III) and Continuity (HV) - i.e. without the axiom of parallels - can be derived. The designations in brackets are axiom groups I, II, III and V in Hilbert's system of axioms of Euclidean geometry . In a broader sense, one also counts two-dimensional models that satisfy the axiom groups HI to H-III in their two-dimensional form, the so-called Hilbert planes , to the absolute geometry, these are (in the main cases) Euclidean or hyperbolic planes over Pythagorean solids .
It is therefore a matter of the set of propositions that are valid both in Euclidean geometry and in non-Euclidean geometries , or, in other words, the “common substructure” of these geometries.
For example, some congruence theorems belong to absolute geometry, but the theorem about the sum of angles in the triangle and the Pythagorean theorem do not. In Euclid's elements , the first 28 sentences are proved without the axiom of parallels and thus belong to absolute geometry in the narrower sense.
history
The term “absolute geometry” goes back to one of the founders of non-Euclidean geometries, the Hungarian mathematician János Bolyai . Around 1830 he dealt with the question of the independence of the axiom of parallels from the other axioms of Euclidean geometry, as formulated in the work Elements of Euclid . In addition to Carl Friedrich Gauß , Bolyai was the first to find a model for a non-Euclidean geometry , more precisely a hyperbolic geometry .
Since Euclid's axiomatics did not meet modern mathematical requirements, the discussion about absolute and non-Euclidean geometry was only put on a sustainable basis by Hilbert's system of axioms of Euclidean geometry in 1899. On this basis Johannes Hjelmslev founded the theory of the Hilbert planes in 1907. In 1926, Max Dehn called this axiomatic justification of absolute geometry by Hjelmslev “the highest point that modern mathematics has reached beyond Euclid in the justification of elementary geometry”. At that time, however, there was no overview of the models for these levels. In 1960 W. Pejas was able to describe all Hilbert levels algebraically and thus bring this classical theory, absolute geometry in the narrower sense, to a certain conclusion. All Hilbert levels are either Euclidean or hyperbolic in the main cases.
In the years 1929–1949, Hjelmslev himself generalized absolute geometry with his "General Congruence Theory" to a geometry of reflections . The basic idea is to use axioms about the movement group instead of axioms about points and straight lines . Friedrich Bachmann builds on this basic idea with his "Structure of Geometry from the Concept of Reflection". For him, this leads to the concept of metric absolute geometry . Finite models of this geometry are always Euclidean, infinite models can in the main cases be Euclidean, hyperbolic or elliptic or, under slightly weakened conditions, Minkowskish . Every plane or spatial metric absolute geometry can be embedded in a projective-metric geometry of the corresponding dimension determined by it .
Axiomatics
There is no generally accepted axiomatic of absolute geometry. The Hilbert axioms mentioned in the introduction without the axiom of parallels are often used as a basis for discussion, with individual axioms then being weakened or omitted entirely. This is historically due to the fact that the whole theory had its starting point in the discussion of the parallel axiom and its independence in Euclid . And the best-known modern axiomatic in Euclid's sense was and is Hilbert's. A verbatim quote from Bachmann:
- “While the axiomatic foundation of a theory that has been researched for a long time was established by the Hilbert axiom system of Euclidean geometry, there is not such a clearly defined theory whose axiomatization is to be achieved by our axiom system. ... "
All the essential differences between a geometry with parallel postulate and without ( non-Euclidean geometry ) occur in just two dimensions comparable to higher dimensions - very different from the discussed also in geometry since the 19th century problem of the set of Desargues , who just only in two-dimensional spaces is independent of the usual axioms. Therefore many axiom systems are limited to the plane case. Then some of the axioms of incidence (HI) become superfluous and one can restrict oneself to I-1 to I-3:
Incidence axioms for one level
- I.1. Two different points P and Q always determine a straight line g.
- I.2. Any two different points on a straight line determine this straight line.
- I.3. On a straight line there are always at least two points, in a plane there are always at least three points that are not on a straight line.
These are the existence and uniqueness of the connecting lines and an axiom of richness - it is clear that this “absolute minimum for an absolute geometry” is still too general.
Arrangement and congruence
Therefore axioms from groups II (arrangement) and III (congruence) are usually added. The full set of axioms II of the arrangement excludes elliptical planes. The problem of congruence can be avoided by the idea that instead of the congruence (of figures of the plane) the group of congruence maps is described as a group created by reflections . This is the basic idea of the geometry of the reflections by Hjelmslev mentioned in the historical section of this article . A more recent system of axioms, which formally bases absolute geometry only on axioms about reflections and the group generated by reflections, is metric absolute geometry .
Axioms of the circle
Hilbert's axioms of continuity (group V with Hilbert) are often not required in absolute geometry (for example with Hjelmslev) or are replaced by weaker axioms of the circle . This means that in absolute geometry the same constructions with compasses and rulers can be carried out as when the axioms of continuity are required. In the special case of Euclidean geometry (with the axiom of parallels) this corresponds to the generalization from ordinary Euclidean space over the real numbers , in which any continuous "lines" intersect, if they should do it vividly, to a space over a Euclidean field in which this is for Conic sections, especially for circles and straight lines.
literature
- Friedrich Bachmann : Structure of geometry from the concept of reflection . 2nd supplemented edition. Springer, Berlin / Heidelberg / New York 1973, ISBN 3-540-06136-3 .
- Richard Baldus , Frank Löbell : Non-Euclidean Geometry (= Göschen Collection . 970 / 970a). 3. Edition. Berlin 1954.
- David Hilbert : Fundamentals of Geometry . 14th edition. Teubner, Stuttgart / Leipzig, ISBN 3-519-00237-X ( archive.org - edition from 1903).
- Jeremy Gray: Ideas of Space: Euclidean, Non-Euclidean, and Relativistic . 2nd Edition. Oxford University Press, Oxford 1989, ISBN 978-0-19-853935-3 .
- Benno Klotzek: Euclidean and non-Euclidean elementary geometries . 1st edition. Harri Deutsch, Frankfurt am Main 2001, ISBN 3-8171-1583-0 .
References and comments
- ↑ ^{a } ^{b } ^{c} The main cases here are levels above bodies, subsidiary cases are sub-levels of these levels that fulfill the axioms, but are only coordinated by coordinates from certain sub-rings of "their" body. Bachmann (1973) §20.13 Hilbert levels.
- ↑ ^{a } ^{b } ^{c} Bachmann (1973) foreword to the second edition.
- ↑ W. Pejas: The models of the Hilbert axiom system of absolute geometry . In: Math. Ann. tape 147 , 1961, pp. 110-119 .
- ↑ Bachmann (1973)
- ↑ For Minkowski geometry, the requirement for the existence of connecting lines , axiom 1, must be weakened.
- ↑ Bachmann (1973) § 6.10 Justification of Metric Geometry
- ↑ But whether a thought in the direction of Hilbert's axiom V.II of linear completeness could even be tangible before the first steps to the foundation of analysis and real numbers, is doubtful.
- ↑ what is meant here is his axiomatic for metric absolute geometry according to Bachmann (1973).
- ↑ Bachmann (1973), p. 25
- ↑ Klotzek (2001) 1.2.4, axioms of the circle and their meaning in construction