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.

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1. We state it this way to emphasize that we are thinking of \(\mathcal {P}\) and \(\mathbb {R}^2\) as different planes.

The authors would like to thank Dr. C. Kenneth Fan for his assistance in obtaining the results of the paper as well as helping to create and edit this paper.

Authors and Affiliations

1. Girls’ Angle, Cambridge, USA

   Milena Harned & Iris R. Liebman

2. Columbia University, New York, NY, USA

   Milena Harned

3. Yale University, New Haven, CT, USA

   Iris R. Liebman

Corresponding author

Correspondence to Milena Harned.

Conflicts of Interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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