  We explore the asymptotic behavior of the centroids of random polygons
constructed from regular polygons with vertices on the unit circle by extending
the rays so that their lengths form a random permutation of the first (n)
integers. Surprisingly, this question has connections to diverse mathematical
contexts, including random matrix theory and discrete Fourier transforms.
Through rigorous analysis, we establish that the sequence of the suitably
rescaled centroids converges to a circularly-symmetric complex normal
distribution with variance (\frac{1}{12}). This result is a manifestation of
central limit behavior in a setting involving sums of heavily dependent random
variables.
