Conformal symmetry breaking and self-similar spirals

Abstract: 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 \textit{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.