Abstract
We continue the investigations of the Thomsen–Bachmann correspondence between metric geometries and groups, which is often summarized by the phrase ‘Geometry can be formulated in the group of motions’. In the first part (H. Struve and R. Struve in J Geom, 2019. https://doi.org/10.1007/s00022-018-0465-8) of this paper it was shown that the Thomsen–Bachmann correspondence can be precisely stated in a framework of first-order logic. We now prove that the correspondence, which was established by Thomsen and Bachmann for Euclidean and for plane absolute geometry, holds also for Hjelmslev geometries, Cayley–Klein geometries, isotropic and equiform geometries, and that these geometries and the theory of their group of motions are mutually faithfully interpretable (and bi-interpretable, but not definitionally equivalent). Hence a reflection-geometric axiomatization of a class of motion groups corresponds to an elementary axiomatization of the underlying geometry and provides with the calculus of reflections a powerful proof method.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bachmann, F.: Zur Begründung der Geometrie aus dem Spiegelungsbegriff. Math. Ann. 123, 341–344 (1951)
Bachmann, F.: Axiomatischer Aufbau der ebenen absoluten Geometrie. In: Henkin, L., Suppes, P., Tarski, A. (eds.) The Axiomatic Method, pp. 114–126. North-Holland, Amsterdam (1959)
Bachmann, F.: Hjelmslev planes. Atti del Convegno di Geometria Combinatoria e sue Applcazioni, Perugia, 43–56 (1970)
Bachmann, F.: Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd edn. Springer, Heidelberg (1973)
Bachmann, F., Knüppel, F.: Starrheit in der Geometrie involutorischer Gruppenelemente. Arch. Math. 35, 155–163 (1980)
Bachmann, F.: Ebene Spiegelungsgeometrie. BI-Verlag, Mannheim (1989)
Behnke, H., Bachmann, F., et al.: Fundamentals of Mathematics, vol. II, Geometry. MIT Press, London (1974)
de Bouvère, K.L.: Logical synonymity. Indag. Math. 27, 622–629 (1965)
Button, T., Hodge, W., Walsh, S.: Philosophy and Model Theory. Oxford University Press, Oxford (2018)
Cayley, A.: A sixth memoir upon quantics. Philos. Trans. R. Soc. Lond. 149, 61–90 (1859). (cp. Collected Math. Papers, Vol. 2, Cambridge (1889))
Giering, O.: Vorlesungen über höhere Geometrie. Vieweg, Braunschweig (1982)
Hartshorne, R.: Geometry: Euclid and Beyond. Springer, Heidelberg (2000)
Hilbert, D.: Grundlagen der Geometrie. Leipzig, Teubner (1899). Translated by Unger, L., under the title: Foundations of Geometry. Open Court, La Salle, IL (1971)
Hjelmslev, J.: Die natürliche geometrie. Hamb. Math. Einzelschriften. vol. 1 (1923)
Hjelmslev, J.: Einleitung in die allgemeine Kongruenzlehre. Danske Vid. Selsk. mat-fys. Medd. 8, Nr. 11 (1929); 10, Nr. 1 (1929); 19, Nr. 12 (1942); 22, Nr. 6, Nr. 13 (1945); 25, Nr.10 (1949)
Karzel, H., Kroll, H.-J.: Geschichte der Geometrie seit Hilbert. Wissenschaft-liche Buchgesellschaft, Darmstadt (1988)
Karzel, H.: Gruppentheoretische Begründung der absoluten Geometrie mit abgeschwächtem Dreispiegelungssatz. Habilitation, Hamburg (1956)
Klein, F.: Vorlesungen über nicht-euklidische Geometrie. Springer, Berlin (1928)
Klingenberg, W.: Euklidische Ebenen mit Nachbarelementen. Math. Z. 61, 1–25 (1954)
Lingenberg, R.: Metric Planes and Metric Vector Spaces. Wiley, New York (1979)
Müller, H.: Zur Begründung der ebenen absoluten Geometrie aus Bewegungs-axiomen. Dissertation, TU München, München (1966)
Pambuccian, V.: Fragments of Euclidean and hyperbolic geometry. Sci. Math. Jpn. 53, 361–400 (2001)
Pambuccian, V.: Constructive axiomatization of non-elliptic metric planes. Bull. Pol. Acad. Sci. Math. 51, 49–57 (2003)
Pambuccian, V.: Groups and plane geometry. Studia Log. 81, 387–398 (2005)
Pambuccian, V.: Axiomatizations of hyperbolic and absolute geometries. In: Prékopa, A., Molnár, E. (eds.) Non-Euclidean Geometries: Janos Bolyai Memorial Volume, pp. 119–153. Springer, New York (2006)
Pambuccian, V.: Orthogonality as single primitive notion for metric planes. With an appendix by H. and R. Struve. Beitr. Algebra Geom. 48, 399–409 (2007)
Prusińska, A., Szczerba, L.: Geometry as an extension of the group theory. Log. Log. Philos. 10, 131–135 (2002)
Schmidt, A.: Die Dualität von Inzidenz und Senkrechtstehen in der absoluten Geometrie. Math. Ann. 118, 609–625 (1943)
Schütte, K.: Gruppentheoretisches Axiomensystem einer verallgemeinerten euklidischen Geometrie. Math. Ann. 132, 43–62 (1956)
Snapper, E., Troyer, R.J.: Metric Affine Geometry. Dover Publications, New York (1971)
Sörensen, K.: Ebenen mit Kongruenz. J. Geom. 22, 15–30 (1984)
Sperner, E.: Ein gruppentheoretischer Beweis des Satzes von Desargues in der absoluten Axiomatik. Arch. Math. 5, 458–468 (1954)
Struve, H.: Singulär projektiv-metrische und Hjelmslevsche Geometrie. Diss. Univ. Kiel (1979)
Struve, H., Struve, R.: Non-euclidean geometries: the Cayley–Klein approach. J. Geom. 98, 151–170 (2010)
Struve, R.: An axiomatic foundation of Cayley–Klein geometries. J. Geom. 107, 225–248 (2016)
Struve, R., Struve, H.: The Thomsen–Bachmann correspondence in metric geometry I. J. Geom. (2019). https://doi.org/10.1007/s00022-018-0465-8
Tarski, A.: Einige methodologische Untersuchungen über die Definierbarkeit der Begriffe. Erkenntnis 5, 80–100 (1935)
Thomsen, G.: Grundlagen der Elementargeometrie in gruppenalgebraischer Behandlung. Hamburger Math. Einzeilschr. 15, Teubner, Leipzig (1933)
Yaglom, I.M.: A Simple Non-Euclidean Geometry and Its Physical Basis. Springer, Heidelberg (1979)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Struve, R., Struve, H. The Thomsen–Bachmann correspondence in metric geometry II. J. Geom. 110, 14 (2019). https://doi.org/10.1007/s00022-019-0467-1
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00022-019-0467-1