Generalized quadrangle

In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles (yet containing many quadrangles). A generalized quadrangle is by definition a polar space of rank two. They are the generalized n-gons with n = 4 and near 2n-gons with n = 2. They are also precisely the partial geometries pg(s,t,α) with α = 1.

A generalized quadrangle is an incidence structure (P,B,I), with I ⊆ P × B an incidence relation, satisfying certain axioms. Elements of P are by definition the points of the generalized quadrangle, elements of B the lines. The axioms are the following:

(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 = {1,2,3,4,5,6,7,8,9} and L = {123, 456, 789, 147, 258, 369} 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), whose representation has been dubbed "the doily" by Stan Payne in 1973.

There are two interesting graphs that can be obtained from a generalized quadrangle.

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.

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 W(2) quadrangle and the unique GQ(2,4). These five quadrangles corresponds to the five root systems in the ADE classes A_n, D_n, E₆, E₇ and E₈ , i.e., the simply laced root systems. See and.

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