A general theory for dense near polygons with a nice chain of subpolygons

In [B. De Bruyn, P. Vandecasteele, Near polygons with a nice chain of sub near polygons, J. Combin. Theory Ser. A 108 (2004) 297–311], we determined all dense near 2 n -gons of order ( 2 , t ) with a nice chain of subpolygons, i.e. a chain F 0 ⊂ F 1 ⊂ ⋯ ⊂ F n of convex subpolygons satisfying (i) F i , i ∈ { 0 , … , n } , is a near 2 i -gon, and (ii) F i , i ∈ { 0 , … , n − 1 } , is big in F i + 1 . In the present paper, we describe a method which can be used for classifying general dense near polygons with such a chain. We will use this method to determine all dense near polygons with a nice chain of subpolygons if every hex is either classical or glued. We will apply the latter result to determine all dense near octagons of order ( 3 , t ) with a nice chain of subpolygons.