Projective plane

From Groupprops
Jump to: navigation, search

Template:Incidence structure property


Symbol-free definition

A projective plane is an incidence structure satisfying the following properties (note that blocks are now called lines):

  1. To any two distinct points, there is a unique line incident with both of them
  2. To any two distinct lines, there is a unique point incident with both of them
  3. There exist four points of which no three are incident with the same line

A set of four points as described by the third condition is termed a quadrangle. Note that conditions (1) and (2) are dual to each other, and further, that these, along with (3), also imply the existence of quadrilaterals which are the dual notion to quadrangles.

A projective plane can also be viewed as a square 2-design with v_2 = 1.

Relation with other properties

Stronger properties

Weaker properties

Related properties