A theory of duality in Euclidean geometry

Original Paper
  • 42 Downloads

Abstract

The principle of duality is well established in projective geometry but can hardly be found in the literature on Euclidean geometry where it is “more a principle of analogy than a scientific principle with a logical foundation” (cp. Sommerville, The Elements of Non-Euclidean Geometry. The Open Court, London, 1919). We close this gap and develop a theory of duality in Euclidean geometry. Following Hilbert’s Grundlagen der Geometrie we consider the incidence, order and metric structure of a Euclidean plane and show (a) that there is a large class of theorems of Euclidean incidence geometry which allow a dualization (b) that Hilbert’s order structure can be introduced in a Euclidean plane in a self-dual way and (c) that appropriate definitions of metric notions (e.g., of an angle, a segment or a circle) lead to Euclidean theorems with meaningful dual versions. This shows that duality in Euclidean geometry is not a collection of isolated phenomena but corresponds to a rich and coherent theory.

Keywords

Euclidean geometry Principle of duality Self-dual order structure Circle geometry 

Mathematics Subject Classification

51M05 03B30 51F15 51G05 

References

  1. Atiyah, M.F.: Duality in Mathematics and Physics. Conferencies FME. University of Barcelona, Barcelona (2008)Google Scholar
  2. Bachmann, F.: Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd edn. Springer, Heidelberg (1973)CrossRefMATHGoogle Scholar
  3. Bachmann, F.: Ebene Spiegelungsgeometrie. BI-Verlag, Mannheim (1989)MATHGoogle Scholar
  4. Behnke, H., Bachmann, F., et al.: Fundamentals of Mathematics, vol. II. Geometry. MIT Press, London (1974)Google Scholar
  5. Buekenhout, F.: Ensembles quadratiques des espaces projectifs. Math. Z. 110, 306–318 (1969)MathSciNetCrossRefMATHGoogle Scholar
  6. Coxeter, H.S.M.: The Real Projective Plane. Cambridge University Press, London (1955)MATHGoogle Scholar
  7. Coxeter, H.S.M.: Introduction to Geometry. Wiley, London (1961)MATHGoogle Scholar
  8. de Villiers, M.: Some adventures in Euclidean geometry. Dynamic Mathematics Learning (2009)Google Scholar
  9. Dyckhoff, R., Negri, S.: Geometrisation of first-order logic. Bull. Symb. Logic 21, 123–163 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. Ewald, G.: Geometry: An Introduction. ISHI, New York (2013)MATHGoogle Scholar
  11. Hallett, M., Majer, U. (eds.): David Hilbert’s Lectures on the Foundations of Geometry, pp. 1891–1902. Springer, Berlin (2004)Google Scholar
  12. Hartshorne, R.: Geometry: Euclid and Beyond. Springer, Heidelberg (2000)CrossRefMATHGoogle Scholar
  13. Hessenberg, G., Diller, J.: Grundlagen der Geometrie. de Gruyter, Berlin (1967)MATHGoogle Scholar
  14. Hilbert, D.: Grundlagen der Geometrie, 11th edn. Teubner, Stuttgart (1972)MATHGoogle Scholar
  15. Hjelmslev, J.: Neue Begründung der ebenen Geometrie. Math. Ann. 64, 449–474 (1907)MathSciNetCrossRefMATHGoogle Scholar
  16. Hjelmslev, J.: Danske Vid. Selsk. Mat-fys. Medd. 8(11) (1929a); 10(1) (1929b); 19(12) (1942); 22(6, 13) (1945); 25(10) (1949)Google Scholar
  17. Karzel, H., Kroll, H.-J.: Geschichte der Geometrie seit Hilbert. Wissenschaftliche Buchgesellschaft, Darmstadt (1988)MATHGoogle Scholar
  18. Lenz, H.: Vorlesungen über projektive Geometrie. Akademische Verlagsgesellschaft, Leipzig (1965)MATHGoogle Scholar
  19. Lingenberg, R.: Grundlagen der Geometrie, 2nd edn. BI-Verlag, Wien (1976)MATHGoogle Scholar
  20. Pambuccian, V.: The axiomatics of ordered geometry. I. Ordered incidence spaces. Expo. Math. 29, 24–66 (2011)MathSciNetMATHGoogle Scholar
  21. Pambuccian, V.: Negation-free and contradiction-free proof of the Steiner–Lehmus theorem. Notre Dame J. Formal Logic (2017).  https://doi.org/10.1215/00294527-2017-0019
  22. Prieß-Crampe, S.: Angeordnete Strukturen: Gruppen, Körper, projektive Ebenen. Springer, Berlin (1983)CrossRefMATHGoogle Scholar
  23. Sommerville, D.M.Y.: The Elements of Non-Euclidean Geometry. The Open Court, London (1919)MATHGoogle Scholar
  24. Sperner, E.: Die Ordnungsfunktion einer Geometrie. Math. Ann. 121, 107–130 (1949)Google Scholar
  25. Struve, H., Struve, R.: Zum Begriff der projektiv-metrischen Ebene. Z. Math. Logik Grundlagen Math. 34, 79–88 (1988)MathSciNetCrossRefMATHGoogle Scholar
  26. Struve, H., Struve, R.: An axiomatic analysis of the Droz–Farny line theorem. Aequat. Math. 90, 1201–1218 (2016)MathSciNetCrossRefMATHGoogle Scholar
  27. Struve, R.: An axiomatic foundation of Cayley–Klein geometries. J. Geom. 107, 225–248 (2016a)Google Scholar
  28. Struve, R.: The principle of duality in Euclidean and in absolute geometry. J. Geom. 107, 707–717 (2016b)Google Scholar
  29. Thomsen, G.: Grundlagen der Elemenatargeometrie in gruppenalgebraischer Behandlung. Hambg. Math. Einzelschriften 15 (1933)Google Scholar
  30. Wyler, O.: Order in projective and in descriptive geometry. Compos. Math. 11, 60–70 (1953)MathSciNetMATHGoogle Scholar

Copyright information

© The Managing Editors 2017

Authors and Affiliations

  1. 1.BochumGermany

Personalised recommendations