Polygram extrapolation refers to the process of approximating the shape of a polygram—such as a star polygon—using mathematical tools like Fourier series. A polygram, defined by connecting non-adjacent vertices of a regular polygon in a cyclic pattern, exhibits sharp angles and symmetry, making it a compelling subject for harmonic analysis. By breaking down its outline into a sum of rotating vectors (epicycles), one can recreate the entire shape using only trigonometric components. This extrapol