Jiří Minarčík - Fractal Hasimoto Surfaces

The Vortex Filament Equation approximates the evolution of vortex filaments in an ideal fluid and is obtained from the Biot–Savart law under the localized induction approximation. As the filament evolves, it traces out a two-dimensional surface sometimes called a Hasimoto surface. For smooth initial curves these surfaces remain smooth, while polygons with sharp corners evolve through explicit self-similar solutions that recreate the polygonal structure at rational times. For regular polygons, point trajectories exhibit multifractal spectra. In this talk we study the associated Hasimoto surfaces, which develop cascades of singular spikes and romanesco-like fractal patterns. We will show new visualizations and discuss open questions about the limiting fractal geometry.