Some Constructions and Embeddings of the Tilde Geometry

Abstract

Some old and new onstructions of the tilde geometry (the flag transitive connected triple cover of the unique generalized quadrangle $ W(2) $ of order $ (2 ,2)$) are discussed. Using them,we prove some properties of that geometry.In particular,we compute its generating rank,we give an explicit description of its universal projective embedding and we determine its homogeneous embeddings.