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 2n2n-gons of order (2,t)(2,t) with a nice chain of subpolygons, i.e. a chain F0⊂F1⊂⋯⊂FnF0⊂F1⊂⋯⊂Fn of convex subpolygons satisfying (i) FiFi, i∈{0,…,n}i∈{0,…,n}, is a near 2i2i-gon, and (ii) FiFi, i∈{0,…,n−1}i∈{0,…,n−1}, is big in Fi+1Fi+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)(3,t) with a nice chain of subpolygons.