are not on , and let the points be all points of except those in . 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 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 : and and and 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. , link http://cage.ugent.be/~bamberg/FGQ.pdf Incidence geometry Families of sets