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In elementary geometry a quadrangle is a figure consisting of four segments intersecting in four (corner) points.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/q076000a.gif" />

Figure: q076000a

Note that each point is incident with 2 lines, each line is incident with 2 points, that there is at most one line passing through two distinct points, that two lines intersect in at most one point, and that for a point and a line not incident with that point there is a unique line through that point intersecting the given line.

These properties exemplify the simplest case of a generalized quadrangle. This is an [[Incidence system|incidence system]]  $  ( P, B, I) $, 
i.e. a (symmetric) incidence relation  $  I \subset  P \times B $
between points (the set  $  P $) 
and lines (or blocks, the set  $  B $) 
such that

i) for each point  $  M $
and line  $  p $
not passing through  $  M $
there is precisely one pair  $  ( N, n) $
with  $  M $
on  $  n $, 
$  N $
on  $  n $
and  $  p $.

A generalized quadrangle can be seen as a very special kind of bipartite graph (cf. [[Graph, bipartite|Graph, bipartite]]), obtained by taking as its vertex set the disjoint union  $  P\amalg B $
and with  $  M \in P $, 
$  m \in B $
connected if and only if  $  M $
is on  $  m $.

Interchanging  $  P $
and  $  B $
one obtains the dual generalized quadrangle.

A generalized quadrangle is non-degenerate if there is no point that is collinear with all others, where two points are collinear if they are on a common line.

A finite generalized quadrangle of order  $  ( s, t) $
is one that satisfies i) above and also

ii) each point is incident with precisely  $  t + 1 $
lines and there is at most one line through two distinct points;

iii) each line has  $  s+ 1 $
points and two lines intersect in at most one point.

A simple example of a finite generalized quadrangle of order $(1, 2)$ is depicted below

[[File:Generalized quadrangle.svg|center|300px|Finite Generalized quadrangle]]

This is also an example of a grid, which is an incidence structure  $  ( P, B, I ) $
with  $  P = \{ {x _ {ij} } : {i= 1 \dots s _ {1} ,  j = 1 \dots s _ {2} } \} $, 
$  B = \{ l _ {1} \dots l _ {s _ {1}  } ;  m _ {1} \dots m _ {s _ {2}  } \} $
with  $  x _ {ij} $
on  $  l _ {k} $
if and only if  $  i = k $
and  $  x _ {ij} $
on  $  m _ {k} $
if and only if  $  j = k $.

There are three known families of generalized quadrangles associated with the classical groups; these are known as classical generalized quadrangles. There are also others, for instance coming from ovoids (cf. [[Ovoid(2)|Ovoid]]).

Generalized quadrangles were introduced by J. Tits [[#References|[a1]]] and he also described the classical ones and the first non-classical ones. More generally one considers generalized  $  m $-
gons, [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Tits,   "Sur la trialité et certain groupes qui s'en déduisent"  ''Publ. Math. IHES'' , '''2'''  (1959)  pp. 14–60</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Dembowski,   "Finite geometries" , Springer  (1968)  pp. 254</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.E. Payne,   J.A. Thas,   "Finite generalized quadrangles" , Pitman  (1984)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.E. Shult,   "Characterizations of the Lie incidence geometries"  K. Lloyd (ed.) , ''Surveys in Combinatorics'' , Cambridge Univ. Press  (1983)  pp. 157–186</TD></TR></table>