Classical Polygons

Abstract

The notion of a generalized polygon arose from the classification of trialities of the geometry of the quadric Q(7,\( {\mathbb K} \)) in PG(7,\( {\mathbb K} \)), \( {\mathbb K} \) any field (see Subsection 2.4.2 below) in Tits [1959]. But already in Tits [1954], [1955] generalized polygons arise as a geometrical interpretation of complex Lie groups. This generalized the well-known connection between the classical complex Lie groups and the complex projective spaces and quadrics. It was noticed by Tits [1959] that an axiomatization of these geometries in the relative rank 2 case gave rise to geometries (generalized polygons) corresponding to the exceptional groups of Lie type G ₂ (Dickson’s groups).