Non-Desarguesian plane

In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective spaces of dimension not 2;[1] in other words, the only projective spaces of dimension not equal to 2 are the classical projective geometries over a field (or division ring). However, David Hilbert found that some projective planes do not satisfy it.[2][3] The current state of knowledge of these examples is not complete.[4]

  1. ^ Desargues' theorem is vacuously true in dimension 1; it is only problematic in dimension 2.
  2. ^ Hilbert, David (1950) [first published 1902], The Foundations of Geometry [Grundlagen der Geometrie] (PDF), English translation by E.J. Townsend (2nd ed.), La Salle, IL: Open Court Publishing, p. 48
  3. ^ Hilbert, David (1990) [1971], Foundations of Geometry [Grundlagen der Geometrie], translated by Leo Unger from the 10th German edition (2nd English ed.), La Salle, IL: Open Court Publishing, p. 74, ISBN 0-87548-164-7. According to the footnote on this page, the original "first" example appearing in earlier editions was replaced by Moulton's simpler example in later editions.
  4. ^ Weibel 2007, p. 1296