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This project begins by considering the unsolved square peg problem, posed a century ago by Otto Toeplitz, which asks if every simple closed planar curve contains four points that define a perfect square. The focus here is on simple closed curves on the integer lattice on the Euclidean Plane. Instead of trying to prove the existence of at least one square on any simple closed lattice curve, this project explores the antipodal version of it: what can be said about the number of squares in a given lattice curve? Are there any specific equations or bounds? I wrote a program for a Python Graphics User Interface to generate some examples. Based on the patterns noticed, I was able to come up with some equations, which were then used to prove conjectures posed from the program data. Then the question turned into a maximization problem, where an hypothesis was formulated about the maximal arrangement of a given number of vertices that would yield the highest possible ratio of squares to vertices.
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Mitkova, Mariya, "Squares on Lattices: an Investigation Inspired by the Square Peg Problem" (2012). Senior Projects Spring 2012. 298.