# Difference between revisions of "Quadrangle"

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+ | $#C+1 = 34 : ~/encyclopedia/old_files/data/Q076/Q.0706000 Quadrangle | ||

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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. | ||

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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]] | + | 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 | + | 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 | + | 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 | + | 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 | + | A finite generalized quadrangle of order $ ( s, t) $ |

+ | is one that satisfies i) above and also | ||

− | ii) each point is incident with precisely | + | 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 | + | 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 | + | 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" /> | ||

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Figure: q076000b | Figure: q076000b | ||

− | This is also an example of a grid, which is an incidence structure | + | 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 | + | 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=48359