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.
