The trigonometric functions cosine and sine report the coordinates of a point on the unit circle at a given angle. The circle and its associated functions have perfect symmetry, which we will study by seeing what happens when we break it. The Euclidean unit circle has equation x^2+y^2=1, but we will change the 2's to p's in the distance function and study the curve with equation |x|^p+|y|^p=1. As p gets larger, the circle flattens out and looks more and more like a square; we thus call these curves "squircles." We will look at the analogues of the trigonometric functions in this family of geometries, called p-norms, and explore some interesting connections to special functions, calculus, and number theory.
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