Generalized square

Generalized quadrilateral is a name for certain incidence structures that are examined in particular in finite geometry .

A small non-trivial generalized square: the “doily”, the only one except for isomorphism . ${\displaystyle GQ(2,2)}$

definition

An incidence structure is called a generalized quadrilateral if the following axioms hold: ${\displaystyle {\mathcal {I}}=({\mathfrak {p}},{\mathfrak {B}},I)}$

1. There is a natural number , so that each block contains exactly s points - here blocks are usually referred to as straight lines . ${\displaystyle s\geq 1}$ ${\displaystyle b\in {\mathfrak {B}}}$
2. There is a natural number such that exactly t straight lines go through every point . ${\displaystyle t\geq 1}$ ${\displaystyle p\in {\mathfrak {p}}}$
3. There is at most one straight line through two different points .
4. For every point p that does not lie on a straight line g , there is exactly one straight line h that intersects g .

More generally, it is also allowed that one of the numbers in the first two axioms is a fixed infinite number . ${\displaystyle s,t}$

order

The number of points on any straight line is summarized together with the number of straight lines through any point and the pair of numbers is called the order of the generalized quadrilateral. One then also writes that the square is a . ${\displaystyle s=v_{1}}$ ${\displaystyle t=b_{1}}$ ${\displaystyle (s-1,t-1)}$ ${\displaystyle GQ(s-1,t-1)}$

properties

• If there is more than one point and more than one line, the structure is simple. that is, two straight lines are equal if and only if they contain the same points.
• The dual incidence structure of a , which arises by swapping the set of points with the set of lines and reversing the incidence relation, is a . It holds more generally (as the statement also un finite generalized quadrangles applies): The class of all generalized quadrangles is dual to itself. ${\displaystyle GQ(s-1,t-1)}$ ${\displaystyle GQ(t-1,s-1)}$
• In this case , too , the generalized square need not be isomorphic to its dual square. ${\displaystyle s=t}$
• Every finite generalized quadrangle fulfills the regularity conditions and and is therefore a tactical configuration . ${\displaystyle (P_{1})}$ ${\displaystyle (B_{1})}$
• If the number of points and the number of straight lines exist, then there are pairs of points without connecting straight lines , so the generalized quadrangle is not an incidence geometry and also not a 2- block diagram . ${\displaystyle v_{0}\geq 2}$ ${\displaystyle b_{0}\geq 2}$

Numbers of points and lines

If is, then: ${\displaystyle s,t\geq 1}$

• One contains exactly points. ${\displaystyle GQ(s-1,t-1)}$ ${\displaystyle (s+1)\cdot (st+1)}$
• One contains exactly straight lines. ${\displaystyle GQ(s-1,t-1)}$ ${\displaystyle (t+1)\cdot (st+1)}$

Examples

• Trivial examples are:
  1. Structures with a straight line, all of which contains points , ${\displaystyle s}$ ${\displaystyle (GQ(s-1,0))}$
  2. dual to the previous: structures with a point through which all straight lines pass , ${\displaystyle t}$ ${\displaystyle (GQ(0,t-1))}$
• The ordinary square (corner points as points and sides as blocks) is the only GQ apart from isomorphism with exactly 4 points and isomorphic to its dual structure. ${\displaystyle GQ(1,1)}$
• More common is a square grid . ${\displaystyle GQ(n,1)}$
• The "doily" is a . It was so named by Payne, and the doily diagram shown in the introduction was chosen as the cover picture of the proceedings. ${\displaystyle GQ(2,2)}$

On a hyperboloid

→ Main article : Quadratic set

The figure shows an (affine) single-shell hyperboloid with some straight lines contained in this quadric, which can be divided into two disjoint families.

On a hyperboloid in a three-dimensional affine or projective space, a generalized square can be explained as follows: The points are the points on the hyperboloid surface, the straight lines are the straight lines contained entirely in the hyperboloid. These straight lines form two families, the straight lines of such a family are skewed to each other in pairs . Exactly two straight lines go through each point . ${\displaystyle (t=2)}$

In a finite projective space over the finite body , every straight line contains points. So this generalized square is a . It is isomorphic to a square lattice. ${\displaystyle \mathbb {P} ^{3}(\mathbb {F_{q}} )}$ ${\displaystyle \mathbb {F_{q}} }$ ${\displaystyle s=q+1}$ ${\displaystyle GQ(q,1)}$

literature

• Albrecht Beutelspacher , Ute Rosenbaum: Projective geometry . From the basics to the applications (=  Vieweg Studium: advanced course in mathematics ). 2nd, revised and expanded edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X ( table of contents [accessed April 1, 2012] generalized quadrilaterals on quadrics).
• SE Payne, Joseph A. Thas : Finite generalized quadrangles . In: Research Notes in Mathematics . Pitman (Advanced Publishing Program), Boston 1984, ISBN 0-273-08655-3 .
• SE Payne: Finite generalized quadrangles . a survey. In: Proceedings of the International Conference on Projective Planes . Pullman, Washington 1973, p. 219-261 .
• Burkhard Polster: A geometrical picture book . 1st edition. Springer, New York / Berlin / Heidelberg 1998, ISBN 0-387-98437-2 .
• Koen Thas: Symmetry in finite generalized quadrangles . In: Frontiers in Mathematics . Birkhäuser Verlag, Basel 2004, ISBN 3-7643-6158-1 .

Web links

• Eric W. Weisstein : Generalized Quadrangle . In: MathWorld (English).

References and comments

1. ^ Payne and Thas (1984)
2. ↑ ^{a b c} Polster (1991), 4th Generalized Quadrangles
3. ↑ ^{a b} Payne (1973) - The English word doily roughly describes a doily