Visibility properties of spiral sets

Abstract

A spiral in $\mathbb{R}^{d+1}$ is defined as a set of the form $\left\{\sqrt[d+1]{n}\cdot\boldsymbol{u}_n\right\}_{n\ge 1},$ where $\left(\boldsymbol{u}_n\right)_{n\ge 1}$ is a spherical sequence. Such point sets have been extensively studied, in particular in the planar case $d=1$, as they then serve as natural models describing phyllotactic structures (i.e. structures representing configurations of leaves on a plant stem). Recent progress in this theory provides a fine analysis of the distribution of spirals (e.g., their covering and packing radii). Here, various concepts of visiblity from discrete geometry are employed to characterise density properties of such point sets. More precisely, necessary an sufficient conditions are established for a spiral to be (1) an orchard (a "homogeneous" density property defined by P\`olya), (2) a uniform orchard (a concept introduced in this work), (3) a set with no visible point (implying that the point set is dense enough in a suitable sense) and (4) a dense forest (a quantitative and uniform refinement of the previous concept).