78th Annual Meeting of the APS Division of Fluid Dynamics (Nov 23 — 25, 2025)

P032: Fractal Hasimoto Surfaces

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After an appropriate renormalisation, a triangular vortex core in an ideal fluid is expected to evolve under the vortex filament equation. The initial triangle, shown at the base, repeatedly reshapes into rotated polygons, a process observed experimentally in non-circular jets and known as axis switching. In the absence of viscosity, the evolving filament captures the limiting structure of such flows. Over time, its path sweeps out a fractal-like surface whose ridges echo the patterns of Romanesco.This behavior was first demonstrated rigorously by Robert Jerrard and Didier Smets (J. Eur. Math. Soc., 2015), who proved that polygonal filaments with corners can evolve globally under the vortex filament equation as weak solutions. Subsequent work by Valeria Banica and Luis Vega (Ann. PDE, 2020) revealed the multifractal character of the resulting trajectories, linking the dynamics to the celebrated Riemann non-differentiable function and to Talbot-type diffraction patterns.The presented visualization illustrates this evolution in the special case of a triangular vortex filament. Starting from a simple polygonal loop, the filament evolves through intermediate shapes that repeatedly approach rotated triangular configurations. As the motion unfolds, the visualization transitions to reveal the full trajectory surface, sometimes referred to as a Hasimoto surface, traced out by the evolving filaments. This surface exposes a striking hierarchy of nested spikes, making visible the intricate structures predicted by theory.We thank Luis Vega and Valeria Banica for their insightful discussions and guidance in preparing this work.