I shall present some recent work done in collaboration with F. De la Hoz about the the evolution of regular polygons within the so-called Vortex Filament Equation. Each corner of the polygon generates some Kelvin waves that interact in a non-linear way that is closely related to the (linear) Talbot effect in optics. The question of the possible connection between the Talbot effect and turbulence will be also addressed, and in particular the appearance of multi-fractals and their relation with the so called Frisch-Parisi conjecture.