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Self-similar curves are a recurring motif in nature. The tension-freestationary states of conformally invariant energies describe the simplestcurves of this form. Planar logarithmic spirals, for example, are associatedwith conformal arc-length; their unique properties reflect the symmetry and themanner of its breaking. Constructing their analogues in three-dimensions is notso simple. The qualitative behavior of these states is controlled by twoparameters, the conserved scaling current $S$ and the magnitude of the torque$M.$ Their conservation determines the curvature and the torsion. If the spiralapex is located at the origin, the conserved \textit{special} conformal currentvanishes. Planar logarithmic spirals occur when $M$ and $S$ are tuned so that$4MS =1$. More generally, the spiral exhibits internal structure, nutatingbetween two fixed cones aligned along the torque axis. It expands monotonicallyas this pattern precesses about this axis. If the spiral is supercritical($4MS>1$) the cones are identical and oppositely oriented. The torsion changessign where the projection along the torque axis turns, the spiral twisting oneway and then the other within each nutation. These elementary spirals providetemplates for understanding a broad range of self-similar spatial spiralpatterns occurring in nature. In particular, supercritical trajectoriesapproximate rather well the nutating tip of the growing tendril in a climbingplant first described by Darwin.

Conformal symmetry breaking and self-similar spirals