Non-Desarguesian plane

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In mathematics, a non-Desarguesian plane, named after Girard Desargues, is a projective plane that does not satisfy Desargues' 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,[1] 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.


Several examples are also finite. For a finite projective plane, the order is one less than the number of points on a line (a constant for every line). Some of the known examples of non-Desarguesian planes include:


According to Weibel (2007, pg. 1296), H. Lenz gave a classification scheme for projective planes in 1954[3] and this was refined by A. Barlotti in 1957.[4] This classification scheme is based on the types of point–line transtitivity permitted by the collineation group of the plane and is known as the Lenz–Barlotti classification of projective planes. The list of 53 types is given in Dembowski (1968, pp.124–5) and a table of the then known existence results (for both collineation groups and planes having such a collineation group) in both the finite and infinite cases appears on page 126. According to Weibel "36 of them exist as finite groups. Between 7 and 12 exist as finite projective planes, and either 14 or 15 exist as infinite projective planes."

Other classification schemes exist. One of the simplest is based on the type of planar ternary ring (PTR) which can be used to coordinatize the projective plane. The types are fields, skewfields, alternative division rings, semifields, nearfields, right nearfields, quasifields and right quasifields.[5]


  1. ^ Desargues' theorem is vacuously true in dimension 1, it is only problematic in dimension 2
  2. ^ see Room & Kirkpatrick 1971 for descriptions of all four planes of order 9.
  3. ^ Lenz, H. (1954). "Kleiner desarguesscher Satz und Dualitat in projektiven Ebenen". Jahresbericht der Deutschen Mathematiker-Vereinigung 57: 20–31. 
  4. ^ Barlotti, A. (1957). "Le possibili configurazioni del sistema delle coppie punto-retta (A,a) per cui un piano grafico risulta (A,a)-transitivo". Boll. Un. Mat. Ital. 12: 212–226. 
  5. ^ Colbourn & Dinitz 2007, pg. 723 article on Finite Geometry by Leo Storme.