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Axiomatizations of Hyperbolic and Absolute Geometries

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Non-Euclidean Geometries

Part of the book series: Mathematics and Its Applications ((MAIA,volume 581))

Abstract

A survey of finite first-order axiomatizations for hyperbolic and absolute geometries.

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Bibliography

  1. Ahrens, J. (1959). Begründung der absoluten Geometrie des Raumes aus dem Spiegelungsbegriff. Math. Z., 71: 154–185.

    Article  MATH  MathSciNet  Google Scholar 

  2. Artin, E. (1957). Geometric algebra. New York: Interscience.

    MATH  Google Scholar 

  3. Artzy, R. (1966). Non-Euclidean incidence planes. Israel J. Math., 4: 43–53.

    MATH  MathSciNet  Google Scholar 

  4. Bachmann, F. (1973 (1959)). Aufbau der Geometrie aus dem Spiegelungsbegriff, 2. Auflage. Berlin: Springer-Verlag.

    MATH  Google Scholar 

  5. Bachmann, F. (1964). Zur Parallelenfrage. Abh. Math. Sem. Univ. Hamburg, 27: 173–192.

    MATH  MathSciNet  Google Scholar 

  6. Bachmann, F. (1967). Der Höhensatz in der Geometrie involutorischer Gruppenelemente. Canad. J. Math., 19: 895–903.

    MATH  MathSciNet  Google Scholar 

  7. Bachmann, F. (1989). Ebene Spiegelungsgeometrie. Mannheim: B.I.-Wissenschaftsverlag.

    MATH  Google Scholar 

  8. Bachmann, O. (1976). Zur spiegelungsgeometrischen Begründung von Geometrien. Geom. Dedicata, 5: 497–516.

    Article  MATH  MathSciNet  Google Scholar 

  9. Baer, R. (1948). The infinity of generalized hyperbolic planes. Studies and Essays Presented to R. Courant on his 60 th Birthday, January 8, 1948, pages 21–27. New York: Interscience Publishers.

    Google Scholar 

  10. Barbilian, D. (1936). Exkurs über die Dreiecke. Bull. Math. Soc. Roum. Sci., 38: 3–62.

    MATH  Google Scholar 

  11. Bryant, V. (1974). Independent axioms for convexity. J. Geom., 5: 95–99.

    Article  MATH  MathSciNet  Google Scholar 

  12. Cayley, A. (1859). A sixth memoir upon quantics. Collected mathematical papers, vol. 2, pages 561–606. Cambridge, 1889.

    Google Scholar 

  13. Coppel, W. A. (1998). Foundations of convex geometry. Cambridge University Press.

    Google Scholar 

  14. Coxeter, H. S. M. (1969). Introduction to geometry. New York: J. Wiley.

    MATH  Google Scholar 

  15. Csorba, F., Molnár, E. (1982/86) Steiner constructions on the projective-metric plane (Hungarian). Mat. Lapok, 33: 99–122.

    MathSciNet  Google Scholar 

  16. Dehn, M. (1900). Die Legendre’schen Sätze über die Winkelsumme im Dreieck. Math. Ann., 53: 404–439.

    Article  MATH  MathSciNet  Google Scholar 

  17. Diller, J. (1970). Eine algebraische Beschreibung der metrischen Ebenen mit ineinander beweglichen Geraden. Abh. Math. Sem. Univ. Hamburg, 34: 184–202.

    MATH  MathSciNet  Google Scholar 

  18. Doraczyńska, E. (1977). On constructing a field in hyperbolic geometry. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 25: 1109–1114.

    Google Scholar 

  19. Ellers, E., Sperner, E. (1962). Einbettung eines desarguesschen Ebenenkeims in einer projektiven Ebene. Abh. Math. Sem. Univ. Hamburg, 25: 206–230.

    MathSciNet  Google Scholar 

  20. Ewald, G. (1974). Spiegelungsgeometrische Kennzeichnung euklidischer und nichteuklidischer Räume beliebiger Dimension. Abh. Math. Sem. Univ. Hamburg, 41: 224–251.

    Article  MATH  MathSciNet  Google Scholar 

  21. Ferreirós, J. (2001). The road to modern logic — an interpretation. Bull. Symbol. Logic, 7: 441–484.

    Article  MATH  Google Scholar 

  22. Gabrieli, E. (1999). Loops with reflection germ: a characterization of absolute planes. Discrete Math., 208/209: 285–298.

    Article  MathSciNet  Google Scholar 

  23. Greenberg, M. J. (1979). On J. Bolyai’s parallel construction. J. Geom., 12: 45–64.

    Article  MATH  MathSciNet  Google Scholar 

  24. Greenberg, M. J. (1988). Aristotle’s axiom in the foundations of geometry. J. Geom., 33: 53–57.

    Article  MATH  MathSciNet  Google Scholar 

  25. Grochowska-Prażmowska, M. (1993). A proof of Pasch’s axiom in the absolute theory of oriented parallelity. J. Geom., 46: 66–81.

    Article  MathSciNet  MATH  Google Scholar 

  26. Hartshorne, R. (2000). Geometry: Euclid and Beyond. New York: Springer-Verlag.

    MATH  Google Scholar 

  27. Heimbeck, G. (1980). Zum Aufbau der absoluten Geometrie nach Ewald. Abh. Math. Sem. Univ. Hamburg, 50: 70–88.

    MATH  MathSciNet  Google Scholar 

  28. Hellwig, G. (1991). Modelle der absoluten Geometrie. Dissertation, Universität Kiel.

    Google Scholar 

  29. Hessenberg, G., Diller, J. (1967). Grundlagen der Geometrie. Berlin: W. de Gruyter.

    MATH  Google Scholar 

  30. Hilbert, D. (1902/1903). Über den Satz von der Gleichheit der Basiswinkel im gleichschenkligen Dreieck. Proc. London Math. Soc. 35: 50–68.

    Google Scholar 

  31. Hilbert, D. (1899). Grundlagen der Geometrie. Stuttgart: B. G. Teubner, 12. Auflage, 1977.

    Google Scholar 

  32. Hilbert, D. (1903). Neue Begründung der Bolyai-Lobatschefskyschen Geometrie. Math. Ann., 57: 137–150.

    Article  MATH  MathSciNet  Google Scholar 

  33. Hjelmslev, J. (1907). Neue Begründung der ebenen Geometrie. Math. Ann., 64: 449–474.

    Article  MATH  MathSciNet  Google Scholar 

  34. Hjelmslev, J. (1929, 1929, 1942, 1945, 1949). Einleitung in die allgemeine Kongruenzlehre. Danske Vid. Selsk., mat.-fys. Medd., 8, Nr. 11; 10, Nr. 1; 19, Nr. 12; 22, Nr. 6; Nr. 13; 25, Nr. 10.

    Google Scholar 

  35. G. Hübner, (1969). Klassifikation n-dimensionaler absoluter Geometrien, Abh. Math. Sem. Univ. Hamburg, 33: 165–182.

    MATH  MathSciNet  Google Scholar 

  36. Karzel, H. (1999). Recent developments on absolute geometries and algebraization by K-loops. Discrete Math., 208/209: 387–409.

    Article  MathSciNet  Google Scholar 

  37. Karzel, H., König, M. (1981). Affine Einbettung absoluter Räume beliebiger Dimension. In Butzer, P. L. and Fehér, F., editors, E. B. Christoffel. The influence of his work on mathematics and the physical sciences, pages 657–670. Basel: Birkhäuser Verlag.

    Google Scholar 

  38. Karzel, H., Konrad, A. (1994). Eigenschaften angeordneter Räume mit hyperbolischer Inzidenzstruktur. Beiträge Geom. Algebra TU München, 28: 27–36.

    Google Scholar 

  39. Karzel, H., Konrad, A. (1995). Reflection groups and K-loops. J. Geom., 52: 120–129.

    Article  MathSciNet  MATH  Google Scholar 

  40. Karzel, H., Kroll, H.-J. (1988). Geschichte der Geometrie seit Hilbert. Darmstadt: Wissenschaftliche Buchgesellscahft.

    MATH  Google Scholar 

  41. Karzel, H., Wefelscheid, H. (1995). A geometric construction of the K-loop of a hyperbolic space. Geom. Dedicata 58: 227–236.

    Article  MathSciNet  MATH  Google Scholar 

  42. de Kerékjártó, B. (1955). Les fondements de la géométrie I. Budapest: Akademiai Kiádo.

    Google Scholar 

  43. Klingenberg, W. (1954). Eine Begründung der hyperbolischen Geometrie. Math. Ann., 127: 340–356.

    Article  MATH  MathSciNet  Google Scholar 

  44. Knüppel, F. (1991). On pre-Hjelmslevgroups and related topics. In Barlotti, A., Ellers, E. W., Plaumann, P. and Strambach, K., editors, Generators and relations in groups and geometries, pages 125–177. Kluwer Academic Publishers.

    Google Scholar 

  45. Konrad, A. (1995). Nichteuklidische Geometrie und K-loops. Dissertation, Technische Universität München.

    Google Scholar 

  46. Kordos, M., Szczerba, L. W. (1969). On the ΠΣ-axiom systems for hyperbolic and some related geometries. Bull. Acad. Polon. Sci. Sér, Sci. Math. Astronom. Phys., 17: 175–180.

    MathSciNet  MATH  Google Scholar 

  47. Kreuzer, A. (1996). Locally projective spaces which satisfy the bundle theorem. J. Geom., 56: 87–98.

    Article  MATH  MathSciNet  Google Scholar 

  48. Kroll, H.-J., Sörensen, K. (1998). Hyperbolische Räume. J. Geom., 61: 141–149.

    Article  MathSciNet  MATH  Google Scholar 

  49. Liebmann, H. (1905). Elementargeometrischer Beweis der Parallelenkonstruktion und neue Begründung der trigonometrischen Formeln der hyperbolischen Geometrie. Math. Ann., 61: 185–199.

    Article  MATH  MathSciNet  Google Scholar 

  50. Lingenberg, R. (1979). Metric planes and metric vector spaces. New York: J. Wiley.

    MATH  Google Scholar 

  51. List, K. (2001). Harmonic mappings and hyperbolic Plücker transformations. J. Geom., 70: 108–116.

    Article  MATH  MathSciNet  Google Scholar 

  52. Menger, K. (1971). The new foundation of hyperbolic geometry. In Butcher, J. C., editor, Spectrum of Mathematics, pages 86–97. Auckland University Press.

    Google Scholar 

  53. Moler N., Suppes, P. (1968). Quantifier-free axioms for constructive plane geometry. Compositio Math., 20: 143–152.

    MathSciNet  MATH  Google Scholar 

  54. Moszyńska, M. (1977). Theory of equidistance and betweenness relations in regular metric spaces. Fund. Math., 96: 17–29.

    MathSciNet  MATH  Google Scholar 

  55. Nolte, W. (1966). Zur Begründung der absoluten Geometrie des Raumes. Math. Z., 94: 32–60.

    Article  MATH  MathSciNet  Google Scholar 

  56. Nolte, W. (1974). Metrische Räume mit dreiseitverbindbaren Teilräumen I,II. J. Geom., 4: 53–90, 91–117.

    Article  MATH  MathSciNet  Google Scholar 

  57. Pambuccian, V. (1994). Zum Stufenaufbau des Parallelenaxioms. J. Geom., 51: 79–88.

    Article  MATH  MathSciNet  Google Scholar 

  58. Pambuccian, V. (2001). Constructive axiomatization of plane absolute, Euclidean and hyperbolic geometry, Math. Logic Quart., 47: 129–135.

    Article  MATH  MathSciNet  Google Scholar 

  59. Pambuccian, V. (2001). Fragments of Euclidean and hyperbolic geometry. Sci. Math. Jpn., 53: 361–400.

    MATH  MathSciNet  Google Scholar 

  60. Pambuccian, V. (2003). Constructive axiomatization of non-elliptic metric planes. Bull. Pol. Acad. Math., 51: 49–57.

    MATH  MathSciNet  Google Scholar 

  61. Pambuccian, V. (2003). Sphere tangency as single primitive notion for hyperbolic and Euclidean geometry. Forum Math., 15: 943–947.

    Article  MATH  MathSciNet  Google Scholar 

  62. Pambuccian, V. (2004). Hyperbolic geometry in terms of point-reflections or of line-orthogonality, Math. Pannon., to appear.

    Google Scholar 

  63. Pambuccian, V. (2004). An axiomatics of hyperbolic projective-metric planes in terms of lines and orthogonality, Bull. Pol. Acad. Sci. Math., 52: 297–302.

    Article  MATH  MathSciNet  Google Scholar 

  64. Pasch, M., Dehn, M. (1926 (1882)). Vorlesungen über neuere Geometrie. 2. Auflage. Berlin: Springer-Verlag.

    MATH  Google Scholar 

  65. Peckhaus, V. (2002). Regressive Analysis. Log. Anal. Hist. Philos., 5: 97–110.

    Google Scholar 

  66. Pejas, W. (1961). Die Modelle des Hilbertschen Axiomensystems der absoluten Geometrie. Math. Ann., 143: 212–235.

    Article  MATH  MathSciNet  Google Scholar 

  67. Pejas, W. (1964). Eine algebraische Beschreibung der angeordneten Ebenen mit nichteuklidischer Metrik. Math. Z., 83: 434–457.

    Article  MATH  MathSciNet  Google Scholar 

  68. Prażmowski, K. (1989). An axiomatic description of the Strambach plane. Geom. Dedicata, 32: 125–156.

    MathSciNet  MATH  Google Scholar 

  69. Sakowicz, K. (1988). Axiomatizability of the dual external hyperbolic plane (Polish), MS Thesis, University of Warsaw, Białystok Section.

    Google Scholar 

  70. Schröder, E. M. (1984). Aufbau metrischer Geometrie aus der Hexagrammbedingung. Atti Sem. Mat. Fis. Univ. Modena, 33: 183–217.

    MATH  MathSciNet  Google Scholar 

  71. Schwabhäuser, W., Szmielew, W., Tarski, A. (1983). Metamathematische Methoden in der Geometrie. Berlin: Springer-Verlag.

    MATH  Google Scholar 

  72. Scott, D. (1959). Dimension in elementary Euclidean geometry. In Henkin, L., Suppes P. and Tarski, A., editors, The axiomatic method, pages 53–67. North-Holland, Amsterdam.

    Google Scholar 

  73. Skala, H. L. (1992). Projective-type axioms for the hyperbolic plane. Geom. Dedicata, 44: 255–272.

    Article  MATH  MathSciNet  Google Scholar 

  74. Smid, L. J. (1936). Eine absolute Axiomatik der Geometrie in einer begrenzten Ebene. Math. Ann., 112: 125–138.

    Article  MathSciNet  Google Scholar 

  75. Smith, J. T. (1973). Orthogonal geometries, I, II. Geom. Dedicata, 1: 221–235, 334–339.

    MATH  Google Scholar 

  76. Smith, J. T. (1985). Point-hyperplane axioms for orthogonal geometries, Geom. Dedicata, 17: 279–286.

    MATH  MathSciNet  Google Scholar 

  77. Sörensen, K. (1984). Ebenen mit Kongruenz. J. Geom., 22: 15–30.

    Article  MATH  MathSciNet  Google Scholar 

  78. Sörensen, K. (1999). Eine Bemerkung über absolute Ebenen. J. Geom., 64: 160–166.

    Article  MATH  MathSciNet  Google Scholar 

  79. Sperner, E. (1954). Ein gruppentheoretischer Beweis des Satzes von Desargues in der absoluten Axiomatik. Arch. Math. (Basel), 5: 458–468.

    Article  MATH  MathSciNet  Google Scholar 

  80. Strommer, J. (1961). Ein elementarer Beweis der Kreisaxiome der hyperbolischen Geometrie. Acta. Sci. Math., 22: 190–195.

    MATH  MathSciNet  Google Scholar 

  81. Strommer, J. (1990). Ein neuer elementarer Beweis der Kreisaxiome der hyperbolischen Geometrie. Acta Sci. Math., 54: 269–271.

    MATH  MathSciNet  Google Scholar 

  82. Szász, P. (1958). Unmittelbare Einführung Weierstrassscher homogenen Koordinaten in der hyperbolischen Ebene auf Grund der Hilbertschen Endenrechnung. Acta Math. Acad. Sci. Hungar., 9: 1–28.

    Article  MATH  MathSciNet  Google Scholar 

  83. Szász, P. (1962). Einfache Herstellung der hyperbolischen Trigonometric in der Ebene auf Grund der Hilbertschen Endenrechnung. Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 5: 79–85.

    MATH  Google Scholar 

  84. Szczerba, L. W. (1972). Weak general affine geometry. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 20: 753–761.

    MATH  MathSciNet  Google Scholar 

  85. Szmielew, W. (1961/1962). A new analytic approach to hyperbolic geometry. Fund. Math., 50: 129–158.

    MATH  MathSciNet  Google Scholar 

  86. Tarski, A. (1959). What is elementary geometry? In Henkin, L., Suppes P. and Tarski, A., editors, The axiomatic method, pages 16–29. North-Holland, Amsterdam.

    Google Scholar 

  87. Tarski A., Givant, S. (1999). Tarski’s system of geometry. Bull. Symb. Log., 5: 175–214.

    Article  MathSciNet  MATH  Google Scholar 

  88. Ziegler, M. (1982). Einige unentscheidbare Körpertheorien. Enseign. Math., 28: 269–280.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Integrative Studies, Arizona State University West, Phoenix, AZ, 85069-7100, USA

    Victor Pambuccian

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  1. Victor Pambuccian
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Editors and Affiliations

  1. Rutgers Center for Operations Research, Piscataway, New Jersey, USA

    András Prékopa

  2. Budapest University of Technology and Economics, Hungary

    Emil Molnár

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Pambuccian, V. (2006). Axiomatizations of Hyperbolic and Absolute Geometries. In: Prékopa, A., Molnár, E. (eds) Non-Euclidean Geometries. Mathematics and Its Applications, vol 581. Springer, Boston, MA. https://doi.org/10.1007/0-387-29555-0_7

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