nLab
synthetic projective geometry

Contents

Idea

Synthetic projective geometry is an axiomatic approach to projective geometry (usually of projective spaces) without use of (algebraic or analytic) coordinate calculations (unlike the wider, modern study of projective and quasiprojective algebraic varieties). In particular, the method does not require that that projective space be defined over an algebraically closed ground field, or even a field at all; for example the noncommutative ground division rings are possible as well as combinatorial descriptions of finite projective geometries. In dimension different from 22 only the projective spaces over division rings are possible, while in dimension 22 other cases are possible, in particular non-Desarguesean planes (e.g. planes coming from Hall quasifields).

References

  • U. Schoenwaelder, Literatur zur synthetischen projektiven Geometrie
  • Karl Georg Christian von Staudt, Geometrie der Lage, 1857 (projective geometry over reals)
  • O. Veblen, J. W. Young, Projective geometry, 2 Vols. Blaisdell Publishing Company, 1966
  • H. S. M. Coxeter, Reelle projektive Geometrie der Ebene, Mathematische Einzelschriften Band 3. Oldenbourg, 1955; The Real projective plane, 1949, 1992; Projective Geometry, Springer–Verlag, 1994
  • D. R. Hughes, F. C. Piper, Projective planes, GTM 6. Springer–Verlag, 1973
  • H. Lenz, Vorlesungen uber projektive Geometrie, Akademische Verlagsgesellschaft, Leipzig, 1965
  • G. Pickert, Projektive Ebenen, Die Grundlehren der math. Wissenschaften 80. Springer–Verlag, 1955, 1975
  • A. Seidenberg, Lectures in projective geometry, The University Series in Undergraduate Mathematics. D. van Nostrand Company, 1962
  • A. Seidenberg, Pappus implies Desargues, Amer. Math. Monthly, 83(3):190–192, 1976
  • O. Veblen, J.H.M. Wedderburn, Non-Desarguesian and non-Pascalian geometries, Trans AMS 8, 379–388 (1907) pdf
  • Marshall Hall Jr., Projective Planes, Transactions of the American Mathematical Society 54: 229–277 (1943) pdf
  • Charles Weibel, Survey of Non-Desarguesian Planes, Notices of the American Mathematical Society 54 (10): 1294–1303 (2007) pdf
  • wikipedia en: category: projective geometry, Hall plane, de: Projektive Geometrie, Satz von Pappos
  • Wolfram MathWorld Projective geometry (many references!)

The following treatment is not axiomatic but still concentrates mainly on the classical subject

  • Lars Kadison, Matthias T. Kromann, Projective geometry and modern algebra, Birkhäuser Boston Inc., Boston, MA, 1996
  • R. Hartshorne, Foundations of projective geometry, Lecture notes Harvard University, W. A. Benjamin, New York 1967

category: geometry

Last revised on April 24, 2016 at 10:38:20. See the history of this page for a list of all contributions to it.