Namespaces
Variants
Actions

Difference between revisions of "Quadrangle"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
q0760001.png
 +
$#A+1 = 34 n = 0
 +
$#C+1 = 34 : ~/encyclopedia/old_files/data/Q076/Q.0706000 Quadrangle
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
In elementary geometry a quadrangle is a figure consisting of four segments intersecting in four (corner) points.
 
In elementary geometry a quadrangle is a figure consisting of four segments intersecting in four (corner) points.
  
Line 7: Line 19:
 
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.
 
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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q0760001.png" />, i.e. a (symmetric) incidence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q0760002.png" /> between points (the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q0760003.png" />) and lines (or blocks, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q0760004.png" />) such that
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q0760005.png" /> and line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q0760006.png" /> not passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q0760007.png" /> there is precisely one pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q0760008.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q0760009.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600013.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600014.png" /> and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600016.png" /> connected if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600017.png" /> is on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600018.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600020.png" /> one obtains the dual generalized quadrangle.
+
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 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600022.png" /> is one that satisfies i) above and also
+
A finite generalized quadrangle of order $  ( s, t) $
 +
is one that satisfies i) above and also
  
ii) each point is incident with precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600023.png" /> lines and there is at most one line through two distinct points;
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600024.png" /> points and two lines intersect in at most one point.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600025.png" /> is depicted below
+
A simple example of a finite generalized quadrangle of order $  ( 1, 2) $
 +
is depicted below
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/q076000b.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/q076000b.gif" />
Line 29: Line 63:
 
Figure: q076000b
 
Figure: q076000b
  
This is also an example of a grid, which is an incidence structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600026.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600028.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600029.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600030.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600033.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600034.png" />.
+
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]]).
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076000/q07600035.png" />-gons, [[#References|[a4]]].
+
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====
 
====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>
 
<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>

Latest revision as of 08:08, 6 June 2020


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

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 $ ( 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), 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

Figure: q076000b

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).

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

References

[a1] J. Tits, "Sur la trialité et certain groupes qui s'en déduisent" Publ. Math. IHES , 2 (1959) pp. 14–60
[a2] R. Dembowski, "Finite geometries" , Springer (1968) pp. 254
[a3] S.E. Payne, J.A. Thas, "Finite generalized quadrangles" , Pitman (1984)
[a4] E.E. Shult, "Characterizations of the Lie incidence geometries" K. Lloyd (ed.) , Surveys in Combinatorics , Cambridge Univ. Press (1983) pp. 157–186
How to Cite This Entry:
Quadrangle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadrangle&oldid=13574