The principle of duality in Euclidean and in absolute geometry
type: Item
instance of: scholarly article
title: The principle of duality in Euclidean and in absolute geometry
zbMATH Open document ID: 1372.51005
author: Rolf Struve
published in: Journal of Geometry
publication date: 2017-04-20T00:00:00Z
review text: This paper provides a complete answer to the following two questions the first of which was raised by \textit{F. Bachmann} [Aufbau der Geometrie aus dem Spiegelungsbegriff. Berlin: Springer (1973; Zbl 0254.50001)]: (1) For which statements \(\sigma\) holding in the theory \({\mathcal B}\) of Bachmann's metric planes does the dual \(\sigma*\), obtained by interchanging the words ``point'' and ``line'', also hold in \({\mathcal B}\)? and (2) the same question with \({\mathcal B}\) replaced by the theory of all metric planes satisfying Playfair's form of the Euclidean parallel postulate. Bachmann [loc. cit.] wrote (p.\ 49): ``Ein allgemeines Theorem, welches den vollen Umfang der erlaubten Analogisierungen beschreibt, ist allerdings nicht bekannt.'' The answer is found by carefully stating first-order axiom systems for both an absolute and a Euclidean theory in which the dual of each axiom is also an axiom. These theories, \({\mathcal A}\) and \({\mathcal E}\), which have the property that the models of \({\mathcal A}\) are precisely the groups of metric planes (in the sense of Bachmann) and the associated dual groups, whereas the models of \({\mathcal E}\) are precisely the Euclidean and the co-Euclidean Cayley-Klein groups, that were axiomatized in reflection-geometric terms by the author in [J. Geom. 107 225--248 (2016; Zbl 1360.51006)]. The answer to questions (1) and (2) is that \(\sigma\) and its dual \(\sigma*\) hold in metric planes (or in metric planes with the Euclidean parallel postulate) if and only if \(\sigma\) can be derived from \({\mathcal A}\) (or from \({\mathcal E}\) respectively).
reviewed by: Victor V. Pambuccian
Mathematics Subject Classification ID: 51F05
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 6706783
zbMATH Keywords: Bachmann's metric planes
zbMATH Keywords: principle of duality
zbMATH Keywords: absolute geometry
zbMATH Keywords: Euclidean geometry
zbMATH Keywords: co-Euclidean geometry
zbMATH Keywords: Gödel's completeness theorem
MaRDI profile type: Publication
full work available at URL: s00022-016-0314-6
OpenAlex ID: W2297322919
cites work: Q3254794
cites work: Q3992511
cites work: Worlds Out of Nothing
cites work: Q5683869
cites work: Q5495525
cites work: Die Dualität von Inzidenz und Senkrechtstehen in der absoluten Geometrie
cites work: Q3678266
cites work: Non-euclidean geometries: the Cayley-Klein approach
cites work: An axiomatic foundation of Cayley-Klein geometries
cites work: Q4176080
DOI: 10.1007/S00022-016-0314-6
Recommended article: Point-reflections in metric plane
Recommended article: Ternary Operations as Primitive Notions for Constructive Plane Geometry V
Recommended article: Orthogonality as single primitive notion for metric planes
Recommended article: Q3830117
Recommended article: Metric Geometries in an Axiomatic Perspective
Recommended article: Curvature of images in the spherical and paraboloidal mirrors.
Recommended article: On the equivalence of Lagrange's axiom to the Lotschnittaxiom
Recommended article: On the principal of duality.
Recommended article: A theory of duality in Euclidean geometry
Recommended article: Geometria proiettiva non euclidea.
description: scientific article; zbMATH DE number 6706783