A generalized quadrangle is an incidence structure

Incidence structure

In mathematics, an incidence structure is a tripleC=.\,where P is a set of "points", L is a set of "lines" and I \subseteq P \times L is the incidence relation. The elements of I are called flags. If \in I,...

. A generalized quadrangle is by definition a polar space

Polar space

In mathematics, in the field of combinatorics, a polar space of rank n , or projective index n−1, consists of a set P, conventionally the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms :* Every subspace, together with its own subspaces, is isomorphic...

 of rank two. They are the generalized n-gon

Generalized n-gon

In combinatorial theory, a generalized polygon is an incidence structure introduced by Jacques Tits. Generalized polygons encompass as special cases projective planes and generalized quadrangles , which form the most complex kinds of axiomatic projective and polar spaces...

s with . They are also precisely the partial geometries

Partial geometry

An incidence structure C= consists of points P, lines L, and flags I \subseteq P \times L where a point p is said to be incident with a line l if \in I...

  with .

Definition

A generalized quadrangle is an incidence structure , with an incidence relation, satisfying certain axiom

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

s. Elements of are by definition the points of the generalized quadrangle, elements of the lines. The axioms are the following:

• There is a () such that on every line there are exactly points. There is at most one point on two distinct lines.
• There is a () such that through every point there are exactly lines. There is at most one line through two distinct points.
• For every point not on a line , there is a unique line and a unique point , such that is on , and on and .

are the parameters of the generalized quadrangle.

Duality

If is a generalized quadrangle with parameters '', then , with the inverse incidence relation, is also a generalized quadrangle. This is the dual generalized quadrangle. Its parameters are ''. Even if , the dual structure need not be isomorphic with the original structure.

Properties

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• 
• When constructing a graph with as vertices the points of a generalized quadrangle, and with the collinear points connected, one finds a strongly regular graph
  Strongly regular graph
  In graph theory, a discipline within mathematics, a strongly regular graph is defined as follows. Let G = be a regular graph with v vertices and degree k...
  
  .
• 
• 
• 

Classical generalized quadrangles

When looking at the different cases for polar space

Polar space

In mathematics, in the field of combinatorics, a polar space of rank n , or projective index n−1, consists of a set P, conventionally the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms :* Every subspace, together with its own subspaces, is isomorphic...

s of rank at least three, and extrapolating them to rank 2, one finds these (finite) generalized quadrangles :

• A hyperbolic quadric
  Quadric
  In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
  
   , a parabolic quadric and an elliptic quadric are the only possible quadrics in projective spaces over finite fields with projective index 1. We find these parameters respectively :

(this is just a grid)

• A hermitian variety has projective index 1 if and only if n is 3 or 4. We find :

• A symplectic polarity in has a maximal isotropic subspace of dimension 1 if and only if . Here, we find a generalized quadrangle , with .

The generalized quadrangle derived from is always isomorphic with the dual of , and they are both self-dual and thus isomorphic to each other if and only if is even.

Non-classical examples

• Let O be a hyperoval in with q an even prime power
  Prime power
  In mathematics, a prime power is a positive integer power of a prime number.For example: 5=51, 9=32 and 16=24 are prime powers, while6=2×3, 15=3×5 and 36=62=22×32 are not...
  
  , and embed that projective (desarguesian) plane into . Now consider the incidence structure where the points are all points not in , the lines are those not on , intersecting in a point of O, and the incidence is the natural one. This is a (q-1,q+1)-generalized quadrangle.
• Let q be an integer (odd or even) and consider a symplectic polarity in . Choose a random point p and define . Let the lines of our incidence structure be all absolute lines not on together with all lines through p which are not on , and let the points be all points of except those in . The incidence is again the natural one. We obtain once again a (q-1,q+1)-generalized quadrangle

Restrictions on parameters

By using grids and dual grids, any integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

 , allows generalized quadrangles with parameters and . Apart from that, only the following parameters have been found possible until now, with an arbitrary prime power

Prime power

In mathematics, a prime power is a positive integer power of a prime number.For example: 5=51, 9=32 and 16=24 are prime powers, while6=2×3, 15=3×5 and 36=62=22×32 are not...

:

and

and

and