Dense near polygons with two types of quads and three types of hexes

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 tQ, where Q is a quad of S, and three possibilities for (tH , vH), 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.