Journal of Geometry

, 110:9 | Cite as

The Thomsen–Bachmann correspondence in metric geometry I

  • Rolf StruveEmail author
  • Horst Struve


We study in this two part paper 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’. We show that (1) the correspondence can be precisely stated in a framework of first-order logic, (2) 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 (3) these geometries and the theory of their group of motions are not only mutually interpretable but also bi-interpretable. 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. In the first part of the paper we introduce the fundamental logical and geometric notions and show that the Thomsen–Bachmann correspondence can be rephrased in first-order logic by ‘The theory of the group of motions is a conservative extension of the underlying geometric theory’.


Thomsen–Bachmann correspondence calculus of reflections reflection group Bachmann group absolute geometry Cayley–Klein geometries sentential equivalence bi-interpretability symmetric space 

Mathematics Subject Classification

03B30 51F05 51F15 



  1. 1.
    A’Campo, N., Papadopoulos, A.: On Klein’s so-called non-Euclidean geometry. In: Ji, L., Papadopoulos, A. (eds). Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics. EMS IRMA Lectures in Mathematics and Theoretical Physics, vol. 23. EMS Publishing House (2015)Google Scholar
  2. 2.
    Bachmann, F.: Zur Begründung der Geometrie aus dem Spiegelungsbegriff. Math. Ann. 123, 341–344 (1951)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bachmann, F.: Hjelmslev planes. Atti del Convegno di Geometria Combinatoria e sue Applcazioni, Perugia, pp. 43–56 (1970)Google Scholar
  4. 4.
    Bachmann, F.: Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd edn. Springer, Heidelberg (1973)CrossRefGoogle Scholar
  5. 5.
    Bachmann, F.: Ebene Spiegelungsgeometrie. BI-Verlag, Mannheim (1989)zbMATHGoogle Scholar
  6. 6.
    Barrett, T.W., Halvorson, H.: Glymour and Quine on theoretical equivalence. J. Philos. Log. 45, 467–483 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Barrett, T.W., Halvorson, H.: Quine’s conjecture on many-sorted logic. Synthese 194, 3563–3582 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Behnke, H., Bachmann, F., et al.: Fundamentals of Mathematics, vol. II. Geometry. MIT Press, London (1974)Google Scholar
  9. 9.
    Benz, W.: Vorlesungen über Geometrie der Algebren. Springer, Berlin (1973)CrossRefGoogle Scholar
  10. 10.
    Button, T., Walsh, S.: Philosophy and Model Theory. Oxford University Press, Oxford (2018)CrossRefGoogle Scholar
  11. 11.
    Chang, C.C., Keisler, H.J.: Model Theory. North Holland, Amsterdam (1990)zbMATHGoogle Scholar
  12. 12.
    Ferreirós, J.: The road to modern logic–an interpretation. Bull. Symb. Log. 7, 441–484 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hilbert, D.: Neue Begründung der Bolyai-Lobatschefskyschen Geometrie. Math. Ann. 57, 137–150 (1903)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hjelmslev, J.: Neue Begründung der ebenen Geometrie. Math. Ann. 64, 449–474 (1907)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hjelmslev, J.: Einleitung in die allgemeine Kongruenzlehre. Danske Vid. Selsk. mat-fys. Medd. 8(11) (1929); 10(1) (1929); 19(12) (1942); 22(6, 13) (1945); 25(10) (1949)Google Scholar
  16. 16.
    Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993)CrossRefGoogle Scholar
  17. 17.
    Klein, F.: Collected Works of Klein. Springer, Berlin (1921)Google Scholar
  18. 18.
    Lingenberg, R.: Metric Planes and Metric Vector Spaces. Wiley, New York (1979)zbMATHGoogle Scholar
  19. 19.
    Monk, J.D.: Mathematical Logic. Springer, New York (1976)CrossRefGoogle Scholar
  20. 20.
    Oberschelp, A., Glubrecht, J.-M., Todt, G.: Klassenlogik. BI-Verlag, Mannheim (1983)Google Scholar
  21. 21.
    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)CrossRefGoogle Scholar
  22. 22.
    Pambuccian, V., Struve, H., Struve, R.: Metric geometries in an axiomatic perspective. In: Ji, L., Papadopoulos, A., Yamada, S. (eds.) From Riemann to Differential Geometry and Relativity, pp. 413–455. Springer, Berlin (2017)CrossRefGoogle Scholar
  23. 23.
    Pinter, C.: Properties preserved under definitional equivalence and interpretations. Zeitschr. f. math. Logik und Grundlagen d. Mathematik 24, 481–488 (1978)Google Scholar
  24. 24.
    Praźmowski, K.: On the group of similarities in classical geometrical planes. Demonstr. Math. 18, 933–943 (1985)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Rothmaler, P.: Introduction to Model Theory. Taylor & Francis, London (2000)zbMATHGoogle Scholar
  26. 26.
    Schnabel, R.: Kennzeichnungen euklidischer Ebenen als affine Ebenen mit Spiegelungsoperator. Mitt. Math. Ges. Hambg. XI, 183–195 (1983)Google Scholar
  27. 27.
    Schwan, W.: Elementare Geometrie. Akademische Verlagsgesellschaft, Leipzig (1929)zbMATHGoogle Scholar
  28. 28.
    Struve, H.: Ein gruppentheoretischer Aufbau der äquiformen Geometrie. Beitr. Algebra Geom. 19, 193–204 (1985)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Struve, H., Struve, R.: Non-euclidean geometries: the Cayley-Klein approach. J. Geom. 98, 151–170 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Struve, R.: An axiomatic foundation of Cayley-Klein geometries. J. Geom. 107, 225–248 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Struve, R., Struve, H.: The Thomsen–Bachmann correspondence in metric geometry II. J. Geom. (to appear) Google Scholar
  32. 32.
    Tarski, A.: On the calculus of relations. J. Symb. Log. 6, 73–89 (1941)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Tarski, A.: What is elementary geometry? In: Henkin, L., Suppes, P., Tarski, A. (eds.) The Axiomatic Method. North Holland, Amsterdam (1959)Google Scholar
  34. 34.
    Thomsen, G.: Grundlagen der Elementargeometrie in gruppenalgebraischer Behandlung. Hamburger Math. Einzeilschr., vol. 15. Teubner, Leipzig (1933)Google Scholar
  35. 35.
    Visser, A.: On Q. Soft. Comput. 21, 39–56 (2016)CrossRefGoogle Scholar
  36. 36.
    Visser, A.: Categories of theories and interpretations. In: Enayat, A., Kalantari, I., Moniri, M. (eds.) Logic in Tehran, pp. 284–341. Cambridge University Press, Cambridge (2017)Google Scholar
  37. 37.
    Yaglom, I.M.: A Simple Non-Euclidean Geometry and Its Physical Basis. Springer, Heidelberg (1979)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.BochumGermany
  2. 2.Seminar für Mathematik und ihre DidaktikUniversität zu KölnCologneGermany

Personalised recommendations