Non-Desarguesian geometry

A plane geometry in which the Desargues assumption need not be true. The plane is then called a non-Desarguesian plane. Desargues's theorem cannot be proved in the plane using only the projective axioms of the plane, without recourse to the axioms of congruence (the metric axioms) or to space axioms. For example, it cannot be deduced from an axiom system consisting of all Hilbert's axioms for the plane with the exception of the axiom of congruence for triangles (cf. Hilbert system of axioms). The plane geometry based on such a system is non-Desarguesian; it cannot be considered as part of a three-dimensional geometry in which all axioms of Hilbert's system except the above-mentioned congruence axiom are valid. A non-Desarguesian projective $2$-plane cannot be imbedded in a projective space of a higher dimension (see [1], [4], [5]).

The fact that a non-Desarguesian plane geometry can actually be constructed yields independence proofs for various groups of axioms in Hilbert's system, and also highlights the role of Desargues's theorem as an independent additional axiom of plane projective geometry (see [2]).

Attention has also been given to what are known as non-Desarguesian systems, in which Desargues's theorem is not valid as a configurational proposition (see Configuration). Non-Desarguesian systems exist, in particular, on certain surfaces and in general on certain Riemannian manifolds that are straight spaces. A simple example is the paraboloid $z=xy$, on which the points and the shortest joins constitute a non-Desarguesian system. Another example is provided by the torus: There exist metrizations of the torus without conjugate points in which the geodesics of the universal covering space constitute a non-Desarguesian system (see also [5], [6]).

References

[1]	D. Hilbert, "The foundations of geometry" , Open Court (1950) (Translated from German)
[2]	L.A. Skornyakov, "Projective planes" Uspekhi Mat. Nauk , 6 : 6 (1951) pp. 112–154 (In Russian)
[3]	H. Busemann, "The geometry of geodesics" , Acad. Press (1955)
[4]	H. Mohrmann (ed.) , Festschrift D. Hilbert: zu seinem 60.ten Geburtstag , Springer, reprint (1982)
[5]	L. Bieberbach, "Einleitung in die höhere Geometrie" , Teubner (1933)
[6]	M.L. Narayana Rao, K. Kuppuswamy Rao, "A class of non-desarguesian planes" J. Comb. Theory Ser. A , 19 (1975) pp. 247–255

Comments

Some finite projective planes are non-Desarguesian, see [a2].

References

[a1]	H.S.M. Coxeter, "Twelve geometric esays" , Southern Ill. Univ. Press (1968) pp. 248
[a2]	M. Hall, "The theory of groups" , Macmillan (1959) pp. 394–397
[a3]	G. Pickert, "Projective Ebenen" , Springer (1975)
[a4]	D.R. Hughes, F.C. Piper, "Projective planes" , Springer (1973)