**Plane Duality**

A projective plane is defined axiomatically as an incidence structure, in terms of a set *P* of points, a set *L* of lines, and an incidence relation *I* that determines which points lie on which lines. As P and L are only sets one can interchange their roles and define a **plane dual structure**.

By interchanging the role of "points" and "lines" in

- C=(P,L,I)

we obtain the dual structure

- C* =(L,P,I*),

where *I** is the inverse relation of *I*.

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 C, then the plane dual of that statement must be true in the dual plane C*. This follows since dualizing each statement in the proof "in C" gives a statement of the proof "in C*."

In the projective plane C, it can be shown that there exist four lines, no three of which are concurrent. Dualizing this theorem and the first two axioms in the definition of a projective plane shows that the plane dual structure C* is also a projective plane, called the **dual plane** of C.

If C and C* are isomorphic, then C is called * self-dual*. The projective planes PG(2,

*K*) for any division ring

*K*are 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.

The * Principle of Plane Duality* says that dualizing any theorem in a self-dual projective plane C produces another theorem valid in C.

Read more about this topic: Projective Plane

### Other articles related to "plane, duality":

... A (

**plane**)

**duality**is a map from a projective

**plane**C = (P,L,I) to its dual

**plane**C* = (L,P,I*) (see above) which preserves incidence ... That is, a (

**plane**)

**duality**σ will map points to lines and lines to points (Pσ = L and Lσ = P) in such a way that if a point Q is on a line m ( denoted by Q I m) then Qσ I* m ... A (

**plane**)

**duality**which is an isomorphism is called a correlation ...

### Famous quotes containing the word plane:

“We’ve got to figure these things a little bit different than most people. Y’know, there’s something about going out in a *plane* that beats any other way.... A guy that washes out at the controls of his own ship, well, he goes down doing the thing that he loved the best. It seems to me that that’s a very special way to die.”

—Dalton Trumbo (1905–1976)