We consider the geometric flow Xt = κb, where κ is the curvature and b is the binormal component of the Frenet-Serret formulae. It can be expressed as Xt = Xs ∧ Xss, (1) where ∧ is the usual cross-product, and s denotes the arc-length parameterization. This equation, known as the vortex filament equation (VFE), appears after applying a localized induction approximation (LIA) to a vortex filament. Since the tangent vector T = Xs remains with constant length, we can assume that it takes values on the unit sphere. Differentiating (1), we get the so-called Schr¨odinger map equation on the sphere: Tt = T ∧ Tss. (2) We study the evolution of (1) and (2), taking a planar regular polygon of M sides as X(s, 0). Assuming uniqueness and bearing in mind the invariances and symmetries of (1) and (2), we are able to fully characterize, by algebraic means, X(s, t) and T(s, t), at rational multiples of t = 2π/M2 . We show that the values of X and T at those points are intimately related to the generalized quadratic Gauß sums: G(a, b, c) = Xc−1 l=0 e 2πi(al2+bl)/c . (3) We also discuss some related topics. All the results are fully supported by numerical simulations.