The principle of duality in Euclidean and in absolute geometry (Q523404)

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).