Beltrami–Klein model

In geometry we mean by the Beltrami -Klein model, a model of the hyperbolic plane. It is one of the standard examples of non-Euclidean geometry and passes to the Italian mathematician Eugenio Beltrami (1835-1900) and the German mathematician Felix Klein (1849-1925) back. In German-speaking countries, the model is often referred to simply as small cal model; sometimes also as a model of Cayley and Klein, the latter name the invoice fact contributes to the development of the model by Felix Klein, in addition to the investigations of Eugenio Beltrami particular degree also takes into account results of Arthur Cayley ( 1821-1895 ). Popular the Beltrami -Klein model is called by various authors coasters geometry.

In Beltrami's definition, it is a realization of the hyperbolic plane as a Riemannian manifold, while Cayley and Klein considered the model as a subset of the projective plane. End of the 19th century, David Hilbert presented a system of axioms of geometry, for which the Cayley-Klein plane is also a model.

Axiomatic access by Hilbert

Hilbert's system of axioms of Euclidean geometry results in the concepts of " point ", " line", " incident ", " between " and " congruent " as undefined terms and formulated for these 20 axioms, including the axiom of parallels. ( Hilbert's axiom system built on the original postulates of Euclid, as well as preparatory work of Hermann Grassmann, Moritz Pasch, Giuseppe Peano, and others. )

In Hilbert's system of axioms of hyperbolic geometry the parallel axiom is replaced by the axiom that there is through a point outside a line any number of parallels. A model of this axiom system provides the following from the Beltrami -Klein model construction derived:

• The point set is based the amount of the interior points of a circular disk of the Euclidean plane.
• The Straight Qty underlying consists of all located within the open disk tendons.
• The incidence relation between points and straight lines and geometric arrangement (intermediate relation) are taken from the Euclidean plane.
• The congruence is using the hyperbolic distance

Beltrami's model as a Riemannian manifold

In his 1868 published work Beltrami initially regarded the ( now hardly common ) hemispheric model of the hyperbolic plane - this is the amount

With by

Defined Riemannian metric - and then realized that by orthogonal projection onto the disc

Another model of the hyperbolic plane is replaced, in which the lines are straight line segments of the Euclidean plane. The open disk - with the Riemannian metric which makes the projection of the hemisphere to an isometry - is today known as Beltrami -Klein model Riemannian manifold.

The Beltrami -Klein model in the Erlangen program

The Beltrami -Klein model came in 1859 in a work of Cayley projective geometry before, but without making explicit the link to hyperbolic geometry. Beltrami as well as small realized that the hyperbolic geometry can be considered as part of projective geometry with this model: if one of the views the Beltrami -Klein model as a subset, then the isometries of the Beltrami -Klein model are restrictions of projective images, which map the disc to itself.

On the significance of the Beltrami -Klein model

In the Beltrami -Klein model, the Euclidean parallel axiom is not satisfied, but all other axioms of Euclidean plane. Now that the Beltrami -Klein model was developed consistent means of structural elements of the Euclidean plane, is in the words of mathematician Richard Baldus ( Geometer and 1933-34 president of the DMV) to take the following summary statement:

" One can choose from Euclidean geometry to prove that it is not possible to deduce the statement of the Euclidean parallel postulate from the other axioms of Euclidean geometry as a set.

This is ... the solution to the age-old riddle of the Euclidean parallel axiom given. You computational manufactures Euclid, has felt the need of his V. postulate in an ingenious way. "

The logician and philosopher of science Godehard Link commented the following about it:

" Non - Euclidean geometries are exactly of this kind: they are models of the core axioms that make both the parallel axiom wrong. At the beginning of the 19th century such geometries were found. They are based on a radical reinterpretation of the philosophical significance of geometric concepts. Nevertheless, one can also illustrate these geometries by representing their axioms using such re-interpreted terms in the classical plane geometry. Again, in modern terms, you interpret the non-Euclidean geometry in Euclidean geometry. In the case of geometry ... you can illustrate the method of interpretation about by the so-called Klein model of hyperbolic geometry in a Euclidean plane. "

The formulation, one could interpret " the non-Euclidean geometry in Euclidean geometry " is misleading in that one of the concepts of Euclidean geometry - the concept of " congruence " - in the Beltrami -Klein model does not match the congruence concept of Euclidean level matches. In the Beltrami -Klein model congruent line segments are (because of the different distance defined term ) is generally not congruent in Euclidean geometry. Only the incidence and intermediate relations of the Beltrami -Klein model are consistent with those of the Euclidean plane.

Is true, however, that the hyperbolic distance from Euclidean distances can be calculated, namely the formula, and thus the consistency of hyperbolic geometry from the consistency of Euclidean geometry follows.

Swell

• Richard Baldus: Non-Euclidean geometry. Hyperbolic geometry of the plane. Edited and supplemented by Frank loebell ( = Sammlung Goschen. 970 / 970a ). 4th edition. Walter de Gruyter, Berlin 1964.
• Eugenio Beltrami: Saggio di interpetrazione della geometria non- euclidea. In: Giornale di Matematiche. 6, 1868, pp. 284-312 ( Saggio di interpetrazione della geometria non- euclidea on http://gallica.bnf.fr ).
• Andreas Filler: Euclidean and non-Euclidean geometry ( = Mathematical texts 7. ). BI Publisher Science, Mannheim [ua ] 1993, ISBN 3-411-16371-2.
• David Hilbert and Stephan Cohn- Vossen: Descriptive geometry. 2nd edition. Springer -Verlag, Berlin [ ua] 1996, ISBN 3-540-59069-2.
• Helmut Karzel; Kay Sorensen; Dirk Windelberg: Introduction to geometry ( = university paperbacks 184. ). Cambridge University Press, Göttingen 1973, ISBN 3-525-03406-7.
• Felix Klein: With the so-called non - Euclidean geometry. In: Math Ann .. 4, 1871, pp. 573-625.
• Horst Knörrer: geometry. Second, updated edition. Vieweg -Verlag, Wiesbaden 2006, ISBN 978-3-8348-0210-1.
• Godehard link: Collegium Logicum. Mentis, Paderborn 2009, DNB 996 736 883 ( Online ( PDF) ).
• Georg Nöbeling: Introduction to non-Euclidean geometry of the plane. Walter de Gruyter, Berlin 1976, ISBN 3-11-002001-7.
• Harald Scheid [ Arranger ]: Duden arithmetic and mathematics. Fourth, completely revised edition. Bibliographical Institute, Mannheim [ua ], 1985, ISBN 3-411-02423-2.

References and Notes

• Geometry