There are many examples of both finite and infinite non-Desarguesian planes. Some of the known examples of infinite non-Desarguesian planes include:
Regarding finite non-Desarguesian planes, 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.[5] They are:
Numerous other constructions of both finite and infinite non-Desarguesian planes are known, see for example Dembowski (1968). All known constructions of finite non-Desarguesian planes produce planes whose order is a proper prime power, that is, an integer of the form pe, where p is a prime and e is an integer greater than 1.
Classification[edit]
Hanfried Lenz gave a classification scheme for projective planes in 1954,[6] which was refined by Adriano Barlotti in 1957.[7] This classification scheme is based on the types of point–line transitivity 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. 124–125) 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. As of 2007, "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."[4]
Other classification schemes exist. One of the simplest is based on special types of planar ternary ring (PTR) that can be used to coordinatize the projective plane. These types are fields, skewfields, alternative division rings, semifields, nearfields, right nearfields, quasifields and right quasifields.[8]