Non-Desarguesian plane

In mathematics, a non-Desarguesian plane, named after Gérard Desargues, is a projective plane that does not satisfy Desargues's theorem, or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is valid in all projective spaces of dimension not 2, that is all the classical projective geometries over a field (or division ring), but Hilbert found that some projective planes do not satisfy it. Understanding of these examples is not complete, in the current state of knowledge.

Examples

Known examples of non-Desarguesian planes include:

• The Moulton plane.
• Every projective plane of order at most 8 is Desarguesian, but there are three non-Desarguesian examples of order 9, each with 91 points and 91 lines.
• Hughes planes.
• Moufang planes over alternative division rings that are not associative, such as the projective plane over the octonions.
• André planes.

References

• Albert, A. Adrian; Sandler, Reuben (1968), An Introduction to Finite Projective Planes, New York: Holt, Rinehart and Winston 
• Dembowski, Peter (1968), Finite Geometries, Berlin: Springer Verlag 
• Hughes, Daniel R.; Piper, Fred C. (1973), Projective Planes, New York: Springer Verlag, ISBN 0-387-90044-6 
• Kárteszi, F. (1976), Introduction to Finite Geometries, Amsterdam: North-Holland, ISBN 0-7204-2832-7 
• Lüneburg, Heinz (1980), Translation Planes, Berlin: Springer Verlag, ISBN 0-387-09614-0 
• Room, T. G.; Kirkpatrick, P. B. (1971), Miniquaternion Geometry, Cambridge: Cambridge University Press, ISBN 0-521-07926-8 
• Sidorov, L.A. (2001), "Non-Desarguesian geometry", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/N/n067000.htm 
• Stevenson, Frederick W. (1972), Projective Planes, San Francisco: W.H. Freeman and Company, ISBN 0-7167-0443-9 
• Weibel, Charles (2007), "Survey of Non-Desarguesian Planes", Notices of the AMS 54 (10): 1294–1303, http://www.ams.org/notices/200710/