of configurations are today generally called Levi graphs, 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 systems in the ADE classes An, Dn, E6, E7 and E8 , i.e., the simply laced root systems. See and. Classical generalized quadrangles When looking at the different cases for polar spaces of rank at least three, and extrapolating them to rank 2, one finds these (finite) generalized quadrangles : A hyperbolic quadric , a parabolic quadric and an elliptic quadric are the only possible quadrics in projective spaces over finite fields with projective index 1. We find these parameters respectively : (this is just a grid) A hermitian variety has projective index 1 if and only if n is 3 or 4. We find : A symplectic polarity in has a maximal isotropic subspace of dimension 1 if and only if . Here, we find a generalized quadrangle , with . The generalized quadrangle derived from is always isomorphic with the dual of , and they are both self-dual and thus isomorphic to each other if and only if is even. Non-classical examples Let O be a hyperoval in with q an even prime power, and embed that projective (desarguesian) plane into . Now consider the incidence structure where the points are all points not in , the lines are those not on , intersecting 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 (odd or even) and consider a symplectic polarity in . Choose an arbitrary point p and define . Let the lines of our incidence structure be all absolute lines not on together with all lines through p which