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; 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. Examples. 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. They are: Numerous other constructions of both finite and infinite non-Desarguesian planes are known, see for example . All known constructions of finite non-Desarguesian planes produce planes whose order is a proper prime power, that is, an integer of the form p, where p is a prime and e is an integer greater than 1. Classification. Hanfried Lenz  gave a classification scheme for projective planes in 1954 and this was refined by Adriano Barlotti in 1957. 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 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." 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. Conics and Ovals. In a Desarguesian projective plane a conic can be defined in several different ways that can be proved to be equivalent. In non-Desarguesian planes these proofs are no longer valid and the different definitions can give  rise to non-equivalent objects. Theodore G. Ostrom had suggested the name "conicoid" for these conic-like figures but did not provide a formal definition and the term does not seem to be widely used. There are several ways that conics can be defined in Desarguesian planes: Furthermore, in a finite Desarguesian plane: Artzy has given an example of a Steiner conic in a Moufang plane which is not a von Staudt conic. Garner gives an example of a von Staudt conic that is not an Ostrom conic in a finite semifield plane.