2019/04/15 by Jemal Guven, Guven, Jemal
Engineering · Mathematics · Physics and Astronomy · #Advanced Materials and Mechanics #Biological Physics (physics.bio-ph) #Classical Physics (physics.class-ph) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Numerical methods for differential equations #Quantum chaos and dynamical systems #Soft Condensed Matter (cond-mat.soft)
paper · pdf · doi:10.48550/arxiv.1904.06876
openalex publication_date 2019/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Self-similar curves are a recurring motif in nature. The tension-free stationary states of conformally invariant energies describe the simplest curves of this form. Planar logarithmic spirals, for example, are associated with conformal arc-length; their unique properties reflect the symmetry and the manner of its breaking. Constructing their analogues in three-dimensions is not so simple. The qualitative behavior of these states is controlled by two parameters, the conserved scaling current S and the magnitude of the torque M. Their conservation determines the curvature and the torsion. If the spiral apex is located at the origin, the conserved special conformal current vanishes. Planar logarithmic spirals occur when M and S are tuned so that 4MS =1. More generally, the spiral exhibits internal structure, nutating between two fixed cones aligned along the torque axis. It expands monotonically as this pattern precesses about this axis. If the spiral is supercritical (4MS>1) the cones are identical and oppositely oriented. The torsion changes sign where the projection along the torque axis turns, the spiral twisting one way and then the other within each nutation. These elementary spirals provide templates for understanding a broad range of self-similar spatial spiral patterns occurring in nature. In particular, supercritical trajectories approximate rather well the nutating tip of the growing tendril in a climbing plant first described by Darwin.