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.