Lost in the fog: Conformal arc-length, its tension-free states and spatial analogs of the logarithmic spiral

Jemal Guven (ICN-UNAM)

RESUMEN: The conformal arc-length is the simplest conformal invariant of a space curve. One would expect it to play a role in modeling a linear system without a natural length scale. Its equilibrium states are characterized not only by the two conserved currents, tension and torque, associated with its Euclidean invariance, but also those associated with its conformal invariance, scaling and special conformal currents. Unusually, there exist non-trivial states in which the tension vanishes. On a plane these are logarithmic spirals. We claim that the natural counterpart of a logarithmic spiral in three-dimensions should also be tension-free. We show how the conserved currents can be used to construct such states explicitly. We find that their qualitative behavior is controlled by two parameter, the scaling current S and the magnitude of the torque M. The naive analog of Bernoulli's spiral, a logarithmically expanding helix, spiraling ever upwards about the torque axis, involves fine tuning; as M increases, these helices begin to precess about this axis. Beyond a critical value of M, independent of S, these precessions overwhelm the helical motion, forming a logarithmic rosette crossing, as it grows, every plane in its three-dimensional environment. We argue that these trajectories approximate rather well the nutating tip of a growing tendril in a climbing plant and conjecture that supercritical spirals offer a clue to the solution of a not-unrelated three-dimensional analog of a not-quite-settled two-dimensional problem: lost in the fog at sea, what trajectory should the hapless swimmer follow to optimize their chance of finding the shore.