In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a duality. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry.
Principle of duality
A projective plane may be defined axiomatically as an incidence structure, in terms of a set of points, a set of lines, and an incidence relation that determines which points lie on which lines. These sets can be used to define a plane dual structure.
Interchange the role of "points" and "lines" in
to obtain the dual structure
, where is the converse relation of . is also a projective plane, called the dual plane of .
If and are isomorphic, then is called self-dual. The projective planes for any field (or, more generally, for every division ring (skewfield) isomorphic to its dual) are self-dual. In particular, Desarguesian planes of finite order are always self-dual. However, there are non-Desarguesian planes which are not self-dual, such as the Hall planes and some that are, such as the Hughes planes.
In a projective plane a statement involving points, lines and incidence between them that is obtained from another such statement by interchanging the words "point" and "line" and making whatever grammatical adjustments that are necessary, is called the plane dual statement of the first. The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". Forming the plane dual of a statement is known as dualizing the statement.
If a statement is true in a projective plane , then the plane dual of that statement must be true in the dual plane . This follows since dualizing each statement in the proof "in " gives a corresponding statement of the proof "in ".
The principle of plane duality says that dualizing any theorem in a self-dual projective plane produces another theorem valid in .
The above concepts can be generalized to talk about space duality, where the terms "points" and "planes" are interchanged (and lines remain lines). This leads to the principle of space duality.
These principles provide a good reason for preferring to use a "symmetric" term for the incidence relation. Thus instead of saying "a point lies on a line" one should say "a point is incident with a line" since dualizing the latter only involves interchanging point and line ("a line is incident with a point").
The validity of the principle of plane duality follows from the axiomatic definition of a projective plane. The three axioms of this definition can be written so that they are self-dual statements implying that the dual of a projective plane is also a projective plane. The dual of a true statement in a projective plane is therefore a true statement in the dual projective plane and the implication is that for self-dual planes, the dual of a true statement in that plane is also a true statement in that plane.
Dual theorems
As the real projective plane, , is self-dual there are a number of pairs of well known results that are duals of each other. Some of these are:
Desargues' theorem ⇔ Converse of Desargues' theorem
Pascal's theorem ⇔ Brianchon's theorem
Menelaus' theorem ⇔ Ceva's theorem
Dual configurations
Not only statements, but also systems of points and lines can be dualized.
A set of points and lines is called an configuration if of the lines pass through each point and of the points lie on each line. The dual of an configuration, is an configuration. Thus, the dual of a quadrangle, a (43, 62) configuration of four points and six lines, is a quadrilateral, a (62, 43) configuration of six points and four lines.
The set of all points on a line, called a projective range has as its dual a pencil of lines, the set of all lines on a point.