Beltrami Klein model

In geometry , the Beltrami-Klein model is understood to be a model of the hyperbolic plane . It is one of the standard examples of non-Euclidean geometry and goes back to the Italian mathematician Eugenio Beltrami (1835–1900) and the German mathematician Felix Klein (1849–1925). In the German-speaking world, the model is often referred to simply as the Klein model ; sometimes also as a model by Cayley and Klein , the latter designation taking into account the fact that the development of the model by Felix Klein, in addition to the studies of Eugenio Beltrami, also takes into account the results of Arthur Cayley (1821–1895). The Beltrami Klein model is also popularly called beer mat geometry by individual authors . Beltrami's definition is a realization of the hyperbolic plane as a Riemannian manifold , while Cayley and Klein viewed the model as a subset of the projective plane . At the end of the 19th century, David Hilbert established a system of axioms for geometry, for which the Cayley-Klein plane is also a model.

Axiomatic approach according to Hilbert

Hilbert's system of axioms of Euclidean geometry introduces the concepts “point”, “straight line”, “incident”, “between” and “congruent” as undefined terms and formulates 20 axioms for these, including the axiom of parallels. (Hilbert's system of axioms was based on Euclid's original postulates and preliminary work by Hermann Graßmann , Moritz Pasch , Giuseppe Peano and others.)

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

• The underlying set of points is the set of inner points on a circular disk of the Euclidean plane .
• The underlying set of lines consists of all within the open disk located tendons .
• The incidence relation between points and straight lines and the geometric arrangement ( intermediate relation ) are taken from the Euclidean plane.
• The congruence relation is determined with the help of the hyperbolic distance

${\displaystyle d(p,q)={\frac {1}{2}}\log {\frac {|qa||bp|}{|pa||bq|}}}$

defined, whereby the absolute lines stand for Euclidean distances and the points of intersection of the straight line through with the edge of the circular disk are. (This metric is a specific example of a Hilbert metric .) ${\displaystyle a,b}$ ${\displaystyle p,q}$

Beltrami's model as a Riemannian manifold

In his paper published in 1868, Beltrami first looked at the hemispherical model of the hyperbolic plane (which is rarely used today) - that's the set

${\displaystyle \left\{(x,y,z)\in \mathbb {R} ^{3}:x^{2}+y^{2}+z^{2}=1,z>0\right\}}$

with the through

${\displaystyle ds^{2}={\frac {dx^{2}+dy^{2}+dz^{2}}{z}}}$

defined Riemannian metric - and then found that one can by orthogonal projection onto the circular disk

${\displaystyle \left\{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}<1\right\}}$

another model of the hyperbolic plane is obtained in which the straight lines are straight line segments of the Euclidean plane. The open circular disk - with the Riemannian metric, which turns the projection from the hemisphere into an isometry - is the Riemannian manifold known today as the Beltrami-Klein model .

The Beltrami-Klein model in the Erlangen range

The Beltrami-Klein model appeared in a work by Cayley on projective geometry as early as 1859 , but without establishing a connection with hyperbolic geometry. Beltrami as well as Klein recognized that with this model the hyperbolic geometry can be understood as part of the projective geometry : if one considers the Beltrami-Klein model as a subset of the, then the isometries of the Beltrami-Klein model are limitations of projective mappings , which map the circular disc on yourself. ${\displaystyle \mathbb {R} ^{2}\subset \mathbb {R} P^{2}=\mathbb {R} ^{2}\cup \mathbb {R} P^{1}}$

On the importance of the Beltrami Klein model

In the Beltrami-Klein model the Euclidean axiom of parallels is not fulfilled, but all other axioms of the Euclidean plane are . Since the Beltrami-Klein model was developed without contradiction by means of structural elements of the Euclidean level , the following summarizing statement can be made in the words of the mathematician Richard Baldus (geometer and 1933-34 president of the DMV ):

“One can prove from Euclidean geometry that it is not possible to derive the statement of the Euclidean axiom of parallels as a proposition from the other axioms of Euclidean geometry.
This gives ... the solution to the ancient riddle of the Euclidean axiom of parallels. It justifies Euclid, who in a brilliant way felt the necessity of his V postulate. "

The logician and science theorist Godehard Link comments on the following:

“ Non-Euclidean geometries are exactly of this kind: they are models of the core axioms that at the same time make the axiom of parallels wrong. Such geometries were found at the beginning of the 19th century. They are based on a radical reinterpretation of the descriptive meaning of geometric terms. Nevertheless, one can also visualize these geometries by representing their axioms with the help of the terms reinterpreted in this way in classical plane geometry. Again speaking in modern terms, one interprets the non-Euclidean geometry in the Euclidean geometry . ... In the case of geometry, the method of interpretation can be illustrated using the so-called Klein model of hyperbolic geometry within a Euclidean plane. "

With regard to the statement made by Godehard Link that “non-Euclidean geometry can be reinterpreted in Euclidean geometry”, it should be emphasized that the concept of congruence in the Beltrami-Klein model does not agree with the concept of congruence in the Euclidean plane . Line segments that are congruent in the Beltrami-Klein model are generally not congruent in Euclidean geometry (because of the different definition of the concept of distance) . In contrast, the incidence relation and the intermediate relation of the Beltrami-Klein model agree with those of the Euclidean level. It is also correct that the hyperbolic distance can be calculated from Euclidean distances, namely with the formula

${\displaystyle d(p,q)={\frac {1}{2}}\log {\frac {|qa||bp|}{|pa||bq|}}}$  ,

and insofar as the consistency of hyperbolic geometry follows from the consistency of Euclidean geometry.

literature

• Norbert A'Campo , Athanase Papadopoulos: On Klein's so-called non-euclidean geometry. In: Sophus Lie, Felix Klein: The Erlangen program and its impact in mathematics and physics Ed .: L. Ji, A. Papadopoulos, European Mathematical Society Publishing House, 2014, arxiv : 1406.7309v1 .
• Richard Baldus: Non-Euclidean Geometry. Hyperbolic geometry of the plane. Edited and supplemented by Frank Löbell (=  Göschen Collection . 970 / 970a). 4th edition. Walter de Gruyter publishing house, Berlin 1964.
• Eugenio Beltrami: Saggio di interpetrazione della geometria non-euclidea . In: Giornale di Matematiche . tape 6 , 1868, p. 284-312 ( gallica.bnf.fr ).
• Andreas Filler: Euclidean and non-Euclidean geometry (=  mathematical texts . Volume 7 ). BI Wissenschaftsverlag, Mannheim [u. a.] 1993, ISBN 3-411-16371-2 .
• David Hilbert and Stephan Cohn-Vossen : Descriptive Geometry . 2nd Edition. Springer-Verlag, Berlin [a. a.] 1996, ISBN 3-540-59069-2 .
• Helmut Karzel ; Kay Soerensen; Dirk Windelberg: Introduction to Geometry (=  Uni-Taschenbücher . Volume 184 ). Vandenhoeck & Ruprecht, Göttingen 1973, ISBN 3-525-03406-7 .
• Felix Klein: About the so-called non-Euclidean geometry . In: Math. Ann. tape 4 , 1871, p. 573-625 .
• Horst Knörrer : Geometry . 2nd updated edition. Vieweg-Verlag, Wiesbaden 2006, ISBN 978-3-8348-0210-1 .
• Godehard Link: Collegium Logicum . Mentis, Paderborn 2009, DNB  996736883 .
• Georg Nöbeling : Introduction to the non-Euclidean geometries of the plane . Verlag Walter de Gruyter, Berlin 1976, ISBN 3-11-002001-7 .
• Harald Scheid [editor]: Duden arithmetic and mathematics . 4th, completely revised edition. Bibliographisches Institut, Mannheim [u. a.] 1985, ISBN 3-411-02423-2 .

References and comments

1. ↑ The finite geometry knows also hyperbolic levels - cf. about Heinz-Richard Halder, Werner Heise: Introduction to combinatorics . Carl Hanser Verlag, Munich [u. a.] 1976, ISBN 3-446-12140-4 , pp. 235-236 .  - which, however, are not meant in this article.
2. ↑ Beltrami: Giornale di Matematiche . 1868, p. 284 ff .
3. ↑ Klein: Math. Ann. tape 4 , 1871, p. 573 ff .
4. ↑ Knörrer: pp. 148–153, 364.
5. ↑ Duden arithmetic and mathematics . S. 435 .
6. ↑ ^{a b} Filler: p. 194.
7. ↑ See introduction to the original work by Felix Klein ( Math. Ann. Volume 4 , p. 573 ff . ) and Baldus: p. 146.
8. ↑ Duden arithmetic and mathematics . S. 435, 703 .
9. ^ Godehard Link: Collegium Logicum . Mentis, Paderborn 2009, DNB  996736883 , p. 7-8 .
10. ↑ David Hilbert: Fundamentals of Geometry . Leipzig 1899, with numerous new editions, most recently 14th edition by Teubner, Stuttgart 1999, ISBN 3-519-00237-X ; archive.org (1903 edition).
11. ↑ Hilbert / Cohn-Vossen: p. 214.
12. ↑ Karzel et al .: pp. 184-187.
13. ↑ Knörrer: p. 149.
14. ↑ Nöbeling: p. 19.
15. ↑ Thus, of each secant through the circular disk, the segment located inside is considered, excluding the two secant points located on the circular line .
16. ↑ Eugenio Beltrami: Saggio di interpretazione della geometria non-euclidea. Giornale Matemat. 6: 284-312 (1868)
17. ^ Milnor, John: Hyperbolic geometry: the first 150 years. Bull. Amer. Math. Soc. (NS) 6 (1982) no. 1, 9-24.
18. ↑ John Stillwell: Sources of hyperbolic geometry. In: History of Mathematics. 10. American Mathematical Society, Providence RI; London Mathematical Society, London 1996, ISBN 0-8218-0529-0 , x + 153 pp.
19. ↑ Richard Baldus: Non-Euclidean Geometry. Hyperbolic geometry of the plane. Walter de Gruyter, Berlin 1964, p. 66.
20. ↑ In fact, it wasn't until 1868 that Beltrami found hyperbolic geometries. From the end of the 1820s, Lobachevsky and others had worked out far-reaching implications for the axioms of hyperbolic geometry, but they had not found a model and thus also failed to prove the consistency of hyperbolic geometry.
21. ^ Godehard Link: Collegium Logicum . Mentis, Paderborn 2009.