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This project considers the lattice polygon analog of the square peg problem: given a simple closed lattice curve in the plane, we ask if there is always a square whose vertices are vertices of the lattice curve. We start by using the notion of taxicab distance to define a taxicab version of continuity. We use this definition to state and prove a theorem that is an analogue of the intermediate value theorem, but for integers. Then, through use of our integer intermediate value theorem we are able to find squares whose vertices are vertices of the lattice polygons. First, we show that every taxicab simple closed curve that is enclosed in a rectangle, with a corner still intact, has at least one horizontal square inscribed to its vertices. We then look at cases of symmetry and show that every taxicab simple closed curve that has a base, is enclosed, and is symmetric about a line has a square. Lastly, we show that every taxicab simple closed curve that is symmetric about two perpendicular lines has a square.
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Foy, Joseph Stephen, "Inscribed Squares in Taxicab Polygons" (2013). Senior Projects Spring 2013. 239.
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