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Early explicit examples of non-desarguesian plane geometries

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Abstract

We trace the history of geometries where Desargues’ theorem is not valid. Roughly we cover the time from the middle of the nineteenth century until the first decade of the twentieth century, discussing work by Beltrami, Klein, Wiener, Peano, Moulton, and of course Hilbert.

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References

  1. Anisov, S.: The collineation group of Hilbert’s example of a projective plane (Russian). Uspekhi Mat. Nauk 47(3), 147–148 (1992) (English translation in: Russian Math. Surveys 47, 163–164 (1992))

  2. Arana, A., Mancosu, P.: On the relationship between plane and solid geometry. Rev. Symb. Logic (to appear)

  3. Beltrami E.: Risoluzione del problema: Riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette. Ann. Mat. Pura Appl. 7(1), 185–204 (1865)

    Article  Google Scholar 

  4. Cerroni C.: Non-Desarguian geometries and the foundations of geometry from David Hilbert to Ruth Moufang. Hist. Math. 31, 320–336 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dickson, L.E.: On finite algebras. Gött. Nachr. 358–393 (1905)

  6. Hall M.: Projective planes. Trans. Am. Math. Soc. 54, 229–277 (1943)

    Article  MATH  Google Scholar 

  7. Hallett, M., Majer, U. (eds.): David Hilbert’s lectures on the foundations of geometry, 1891–1902. Volume 1 of David Hilbert’s Foundational Lectures. Springer, Berlin (2004)

  8. Hilbert, D.: Elemente der Euklidischen Geometrie. Göttingen, Wintersemester 1898/99. Mimeographed lecture notes by H. von Schaper, vol. 552 of Cod. Ms. D. Hilbert, Göttingen (1899)

  9. Hilbert, D.: Grundlagen der Geometrie (Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen). Teubner, Leipzig (1899)

  10. Hilbert D.: The foundations of geometry. Authorized translation by E.J. Townsend. Open Court Publishing Company, Chicago (1902)

    MATH  Google Scholar 

  11. Hilbert D.: Grundlagen der Geometrie, 7th edn. Teubner, Leipzig (1930)

    MATH  Google Scholar 

  12. Hilbert D., Wiechert E.: Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen. Nabu Press, Charleston (2010)

    Google Scholar 

  13. Hughes D.R., Piper F.C.: Projective planes. Springer, New York (1973)

    MATH  Google Scholar 

  14. Klein F.: Ueber die sogenannte Nicht-Euklidische Geometrie. Zweiter Aufsatz. Math. Ann. 6, 112–145 (1873)

    Article  Google Scholar 

  15. Klein F.: Gesammelte mathematische Abhandlungen. Springer, Berlin (1973)

    Book  Google Scholar 

  16. Löwen R.: Substituting compact disks in stable planes. Adv. Geom. 11, 337–346 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mohrmann H.: Hilbertsche und Beltramische Liniensysteme. Math. Ann. 85, 177–183 (1922)

    Article  MathSciNet  MATH  Google Scholar 

  18. Moufang R.: Alternativkörper und der Satz vom vollständigen Vierseit D9). Abh. Math. Sem. Univ. Hamburg 9, 207–222 (1933)

    Article  Google Scholar 

  19. Moulton F.R.: A simple non-desarguesian plane geometry. Trans. Am. Math. Soc. 3, 192–195 (1902)

    Article  MathSciNet  MATH  Google Scholar 

  20. Peano G.: Sui fondamenti della geometria. Riv. Mat. 4, 51–90 (1894)

    Google Scholar 

  21. Peano, G.: Opere scelte. A cura dell’Unione Matematica Italiana e col contributo del Consiglio Nazionale delle Ricerche, vol. III. Geometria e fondamenti, Meccanica razionale, Varie. Edizioni Cremonese, Roma (1959)

  22. Pickert G.: Projektive Ebenen. Springer, Berlin (1955)

    MATH  Google Scholar 

  23. Polster B., Steinke G.F.: Cut and paste in 2-dimensional projective planes and circle planes. Can. Math. Bull. 38, 469–480 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. Salzmann H.: Kompakte Ebenen mit einfacher Kollineationsgruppe. Arch. Math. 13, 98–109 (1962)

    Article  MathSciNet  Google Scholar 

  25. Salzmann H., Betten D., Grundhöfer T., Hähl H., Löwen R., Stroppel M.: Compact projective planes. De Gruyter, Berlin (1995)

    Book  Google Scholar 

  26. Schneider T., Stroppel M.: Automorphisms of Hilbert’s non-desarguesian affine plane and its projective closure. Adv. Geom. 7, 541–552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Steinke G.F.: Topological affine planes composed of two desarguesian half planes and projective planes with trivial collineation group. Arch. Math. (Basel) 44, 472–480 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  28. Stroppel M.: A note on Hilbert and Beltrami systems. Results Math. 24, 342–347 (1993)

    MathSciNet  MATH  Google Scholar 

  29. Stroppel M.: Bemerkungen zur ersten nicht Desarguesschen ebenen Geometrie bei Hilbert. J. Geom. 63, 183–195 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Toepell M.-M.: On the origins of David Hilbert’s “Grundlagen der Geometrie”. Arch. Hist. Exact Sci. 35, 329–344 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  31. Toepell M.-M.: \"Uber die Entstehung von David Hilberts “Grundlagen der Geometrie”. Vandenhoeck & Ruprecht, Göttingen (1986)

    Google Scholar 

  32. Tschetweruchin N.F.: Eine Bemerkung zu den Nicht-Desarguesschen Liniensystemen. Jber. Deutsch. Math. Verein. 36, 134–136 (1927)

    MATH  Google Scholar 

  33. Vahlen K.T.: Abstrakte Geometrie. Untersuchungen über die Grundlagen der euklidischen und nichteuklidischen Geometrie. Teubner, Leipzig (1905)

    MATH  Google Scholar 

  34. Veblen O., Maclagan-Wedderburn J.H.: Non-desarguesian and non-pascalian geometries. Trans. Am. Math. Soc. 8, 379–388 (1907)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wiener H.: Ueber Grundlagen und Aufbau der Geometrie. Jber. Deutsch. Math. Verein. 1, 45–47 (1892)

    MATH  Google Scholar 

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  1. Fachbereich Mathematik, Universität Stuttgart, 70550, Stuttgart, Germany

    Markus Stroppel

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  1. Markus Stroppel
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Correspondence to Markus Stroppel.

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Dedicated to the memory of Benno Artmann, who passed away on October 14, 2010.

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Stroppel, M. Early explicit examples of non-desarguesian plane geometries. J. Geom. 102, 179–188 (2011). https://doi.org/10.1007/s00022-011-0102-2

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