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The circle, the set of points equal distance from a given point, is a round, perfectly symmetrical object in two dimensional Euclidian Space. What if we redefined distance, the metric, in such a way that circles were not circles, but rather cornered circles? What would happen to the circumference? Of course, because circumference is dependent on measuring distance the change in metric would also affect the circumference measurement. Given a symmetric convex set, it is possible to define a metric space that has the boundary of the set as a unit circle. The main result of this paper is to find the circumference of every even edged regular polygon unit circle in the corresponding metric space.
Reed, Samuel, "Cornered Circles" (2014). Mathematics Honors Theses. 7.