generalized quadrangle
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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a generalized quadrangle is an
incidence structure In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore al ...
whose main feature is the lack of any triangles (yet containing many quadrangles). A generalized quadrangle is by definition a
polar space In mathematics, in the field of geometry, a polar space of rank ''n'' (), or ''projective index'' , consists of a set ''P'', conventionally called the set of points, together with certain subsets of ''P'', called ''subspaces'', that satisfy these ax ...
of rank two. They are the with ''n'' = 4 and near 2n-gons with ''n'' = 2. They are also precisely the partial geometries pg(''s'',''t'',α) with α = 1.


Definition

A generalized quadrangle is an incidence structure (''P'',''B'',I), with I ⊆ ''P'' × ''B'' an
incidence relation In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is ''X'' and the second is ''Y'', the matrix has one row for each element ...
, satisfying certain
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s. Elements of ''P'' are by definition the ''points'' of the generalized quadrangle, elements of ''B'' the ''lines''. The axioms are the following: * There is an ''s'' (''s'' ≥ 1) such that on every line there are exactly ''s'' + 1 points. There is at most one point on two distinct lines. * There is a ''t'' (''t'' ≥ 1) such that through every point there are exactly ''t'' + 1 lines. There is at most one line through two distinct points. * For every point ''p'' not on a line ''L'', there is a unique line ''M'' and a unique point ''q'', such that ''p'' is on ''M'', and ''q'' on ''M'' and ''L''. (''s'',''t'') are the ''parameters'' of the generalized quadrangle. The parameters are allowed to be infinite. If either ''s'' or ''t'' is one, the generalized quadrangle is called ''trivial''. For example, the 3x3 grid with ''P'' = and ''B'' = is a trivial GQ with ''s'' = 2 and ''t'' = 1. A generalized quadrangle with parameters (''s'',''t'') is often denoted by GQ(''s'',''t''). The smallest non-trivial generalized quadrangle is
GQ(2,2) ''GQ'' (formerly ''Gentlemen's Quarterly'' and ''Apparel Arts'') is an American international monthly men's magazine based in New York City and founded in 1931. The publication focuses on fashion, style, and culture for men, though articles on ...
, whose representation has been dubbed "the doily" by Stan Payne in 1973.


Properties

* , P, =(s t+1)(s+1) * , B, =(s t+1)(t+1) * (s+t), st(s+1)(t+1) * s\neq 1 \Longrightarrow t\leq s^2 * t\neq 1 \Longrightarrow s\leq t^2


Graphs

There are two interesting graphs that can be obtained from a generalized quadrangle. * The ''collinearity graph'' having as vertices the points of a generalized quadrangle, with the collinear points connected. This graph is a
strongly regular graph In graph theory, a strongly regular graph (SRG) is defined as follows. Let be a regular graph with vertices and degree . is said to be strongly regular if there are also integers and such that: * Every two adjacent vertices have commo ...
with parameters ((s+1)(st+1), s(t+1), s-1, t+1) where (s,t) is the order of the GQ. * The ''incidence graph'' whose vertices are the points and lines of the generalized quadrangle and two vertices are adjacent if one is a point, the other a line and the point lies on the line. The incidence graph of a generalized quadrangle is characterized by being a
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
,
bipartite graph In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are ...
with
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
four and
girth Girth may refer to: ;Mathematics * Girth (functional analysis), the length of the shortest centrally symmetric simple closed curve on the unit sphere of a Banach space * Girth (geometry), the perimeter of a parallel projection of a shape * Girth ...
eight. Therefore, it is an example of a
Cage A cage is an enclosure often made of mesh, bars, or wires, used to confine, contain or protect something or someone. A cage can serve many purposes, including keeping an animal or person in captivity, capturing an animal or person, and displayin ...
. Incidence graphs of configurations are today generally called
Levi graph In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.. See in particulap. 181 From a collection of points and lines in an incidence geometry or a projective configuration, we fo ...
s, but the original Levi graph was the incidence graph of the GQ(2,2).


Duality

If (''P'',''B'',I) is a generalized quadrangle with parameters (''s'',''t''), then (''B'',''P'',I−1), with I−1 the inverse incidence relation, is also a generalized quadrangle. This is the ''dual generalized quadrangle''. Its parameters are (''t'',''s''). Even if ''s'' = ''t'', the dual structure need not be isomorphic with the original structure.


Generalized quadrangles with lines of size 3

There are precisely five (possible degenerate) generalized quadrangles where each line has three points incident with it, the quadrangle with empty line set, the quadrangle with all lines through a fixed point corresponding to the windmill graph Wd(3,n), grid of size 3x3, the GQ(2,2) quadrangle and the unique GQ(2,4). These five quadrangles corresponds to the five
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representati ...
s in the ADE classes ''A''''n'', ''D''''n'', ''E''''6'', ''E''''7'' and ''E''''8'' , i.e., the simply laced root systems. See and.http://www.win.tue.nl/~aeb/2WF02/genq.pdf


Classical generalized quadrangles

When looking at the different cases for
polar space In mathematics, in the field of geometry, a polar space of rank ''n'' (), or ''projective index'' , consists of a set ''P'', conventionally called the set of points, together with certain subsets of ''P'', called ''subspaces'', that satisfy these ax ...
s of rank at least three, and extrapolating them to rank 2, one finds these (finite) generalized quadrangles : * A hyperbolic
quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
Q^+(3,q), a parabolic quadric Q(4,q) and an elliptic quadric Q^-(5,q) are the only possible quadrics in projective spaces over finite fields with projective index 1. We find these parameters respectively : : Q(3,q) :\ s=q,t=1 (this is just a grid) : Q(4,q) :\ s=q,t=q : Q(5,q) :\ s=q,t=q^2 * A hermitian variety H(n,q^2) has projective index 1 if and only if n is 3 or 4. We find : : H(3,q^2) :\ s=q^2,t=q : H(4,q^2) :\ s=q^2,t=q^3 * A symplectic polarity in PG(2d+1,q) has a maximal isotropic subspace of dimension 1 if and only if d=1. Here, we find a generalized quadrangle W(3,q), with s=q,t=q. The generalized quadrangle derived from Q(4,q) is always isomorphic with the dual of W(3,q), and they are both self-dual and thus isomorphic to each other if and only if q is even.


Non-classical examples

* Let ''O'' be a
hyperoval In projective geometry an oval is a point set in a plane that is defined by incidence properties. The standard examples are the nondegenerate conics. However, a conic is only defined in a pappian plane, whereas an oval may exist in any type of ...
in PG(2,q) with ''q'' an even
prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...
, and embed that projective (desarguesian) plane \pi into PG(3,q). Now consider the incidence structure T_2^(O) where the points are all points not in \pi, the lines are those not on \pi, intersecting \pi in a point of ''O'', and the incidence is the natural one. This is a ''(q-1,q+1)''-generalized quadrangle. * Let ''q'' be a
prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...
(odd or even) and consider a symplectic polarity \theta in PG(3,q). Choose an arbitrary point ''p'' and define \pi=p^. Let the lines of our incidence structure be all absolute lines not on \pi together with all lines through ''p'' which are not on \pi, and let the points be all points of PG(3,q) except those in \pi. 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 An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''z'', ''z'' ≥ 1 allows generalized quadrangles with parameters (1,''z'') and (''z'',1). Apart from that, only the following parameters have been found possible until now, with ''q'' an arbitrary
prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...
: : (q,q) : (q,q^2) and (q^2,q) : (q^2,q^3) and (q^3,q^2) : (q-1,q+1) and (q+1,q-1)


References

* S. E. Payne and J. A. Thas. Finite generalized quadrangles. Research Notes in Mathematics, 110. Pitman (Advanced Publishing Program), Boston, MA, 1984. vi+312 pp. {{ISBN, 0-273-08655-3, link http://cage.ugent.be/~bamberg/FGQ.pdf Incidence geometry Families of sets