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The Square Peg Problem, also known as the inscribed square problem poses a question: Does every simple closed curve contain all four points of a square? This project introduces a new approach in proving the square peg problem in 2-dimensional lattice.
To accomplish the result, this research first defines the simple closed curve on 2-dimensional lattice. Then we identify the existence of inscribed half-squares, which are the set of three points of a square, in a lattice simple closed curve. Then we finally add a last point to form a half-square into a square to examine whether all four points of a square exist in a lattice simple closed curve. A sage program was used to find all missing corners of all inscribed half-squares. This has enabled us to look at the pattern of sets of all missing corners in specific shapes like rectangles.
By the end, we were able to conjecture that there exist missing corners in the interior and the exterior of the lattice simple closed curve unless the shape is a square. It is obvious that the square has an inscribed square. Hence if we could prove that the set of all missing corners is connected, we could give a new proof of the square peg problem in 2-dimensional lattice.
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Matsubara, Nathan M., "Square Peg Problem in 2-Dimensional Lattice" (2020). Senior Projects Fall 2020. 36.
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