Desarguesian plane

A Desarguesian plane is characterized by the validity of Desargues’ theorem for every suitable pair of triangles. If two triangles are perspective from a point, meaning that the three lines joining corresponding vertices meet at one center, then the intersections of corresponding sides lie on one axis; the converse formulation is equivalent in a projective plane.

The name comes from Girard Desargues, whose seventeenth-century work helped establish the projective treatment of incidence and perspective. The condition is stronger than the basic axioms of a projective plane: non-Desarguesian planes satisfy those axioms but fail Desargues’ configuration in at least one instance. Thus “Desarguesian” describes a structural property, not merely a diagram or a particular drawing.

The algebraic model of a Desarguesian plane is PG(2,D), constructed from a three-dimensional vector space over a division ring D. A point is a one-dimensional subspace, written in homogeneous coordinates as [x:y:z], while a line is a two-dimensional subspace or, in a chosen convention, a homogeneous linear equation. Multiplying all coordinates by the same nonzero scalar leaves the represented point unchanged.¹

A division ring permits addition, multiplication, and division by every nonzero element, but multiplication need not be commutative. This possibility makes coordinate conventions significant: one must specify whether scalars act on the left or right. Desargues’ theorem is precisely what permits the geometric plane to recover such a coordinatization; arbitrary projective planes generally require more general ternary-ring coordinates.

When D is a finite division ring, Wedderburn’s little theorem makes D a field, so every finite Desarguesian plane is the classical plane PG(2,q) over the finite field with q elements.² It has q² + q + 1 points and the same number of lines; each line contains q + 1 points, and each point lies on q + 1 lines.

The familiar real projective plane PG(2,R) and complex projective plane PG(2,C) are infinite Desarguesian examples. Their homogeneous coordinates explain why parallel Euclidean lines acquire a common point at infinity. The finite planes PG(2,2), PG(2,3), and PG(2,4) provide small classical incidence structures, while planes of prime-power order that are not known or not constructed as coordinate planes motivate the broader study of finite geometry.

Desarguesianity is closely tied to the distinction between fields and division rings. Within a Desarguesian plane, Pappus’ theorem holds exactly when the coordinating division ring is commutative; therefore a plane may be Desarguesian without being Pappusian if noncommutative coordinates are allowed.

The plane also has a large group of projective transformations: invertible 3-by-3 matrices over D act on homogeneous coordinates, with scalar matrices acting trivially in the usual quotient description. This links incidence geometry to linear algebra and to groups such as the projective general linear group. Duality exchanges points and lines while preserving the incidence statements, so the dual of a Desarguesian plane is again Desarguesian. Non-Desarguesian planes remain important because they show exactly which geometric conclusions depend on linear coordinatization rather than on the projective axioms alone.³

Desargues’ theorem

The theorem that two triangles perspective from a point have corresponding side intersections lying on one line, with the dual converse formulation in a projective plane.

Division ring

A ring in which every nonzero element has a multiplicative inverse; multiplication is not required to be commutative.

Homogeneous coordinates

Coordinates defined up to multiplication by a common nonzero scalar, used to represent points and lines in projective geometry.

Pappus’ theorem

A projective incidence theorem whose validity in a Desarguesian plane corresponds to commutativity of its coordinating division ring.

Non-Desarguesian plane

A projective plane in which Desargues’ theorem fails for at least one configuration.