An Unexpected Class of 5+gon-free Line Patterns

Let $S$ be a finite subset of ${\mathbb R}^2 \setminus (0,0)$. Generally, one would expect the pattern of lines $Ax + By = 1$, where $(A, B) \in S$ to contain polygons of all shapes and sizes. We show, however, that when $S$ is a rectangular subset of the integer lattice or a closely related set, no polygons with more than 4 sides occur. In the process, we develop a general theorem that explains how to find the next side as one travels around the boundary of a cell.