Mathematicians finally proved that smoke-ring-style vortexes can 'leapfrog' around each other in a perfect loop forever.
March 24, 2026
Original Paper
Time-periodic leapfrogging vortex rings in the 3D Euler equations
arXiv · 2603.21644
The Takeaway
In 1858, physicist Hermann von Helmholtz predicted that two smoke-like rings could pass through each other repeatedly in a never-ending cycle. While this has been observed in experiments for decades, it took over 160 years to mathematically prove that the laws of physics allow these fluid structures to exist in this stable, periodic state without breaking apart.
From the abstract
We prove the existence of time-periodic leapfrogging vortex rings for the three-dimensional incompressible Euler equations, thereby providing a rigorous realization of a phenomenon first conjectured by Helmholtz (1858). In the leapfrogging motion, two coaxial vortex rings periodically exchange positions, a striking behavior repeatedly observed in experiments and numerical simulations, yet lacking complete mathematical justification. Our construction relies on a desingularization of two interacti