Dense Near Polygons with two Types of Quads and Three Types of Hexes

Abstract

We show that the number of points at distance \( i \) from a given point \( x \) in a dense near polygon only depends on \( i \) and not on the point \( x \). We give a number of easy corollaries of this result. Subsequently, we look to the case of dense near polygons \( S \) with an order in which there are two possibilities for \( t_Q \), where \( Q \) is a quad of \( S \), and three possibilities for \( (t_H, v_H) \), where \( H \) is a hex of \( S \). Using the above-mentioned results, we will show that the number of quads of each type through a point is constant. We will also show that the number of hexes of each type through a point is constant if a certain matrix is nonsingular. If each hex is a regular near hexagon, a glued near hexagon or a product near hexagon, then that matrix turns out to be nonsingular in all but one of the eight possible cases. For the exceptional case, however, we provide an example of a near polygon that does not have a constant number of hexes of each type through each point.