Visibility Properties of Spiral Sets

TL;DR

This paper investigates the visibility and density properties of spiral point sets in Euclidean space, providing new criteria for various density and visibility concepts, with applications to models of natural phyllotactic structures.

Contribution

It introduces new conditions characterizing the visibility and density of spiral sets, including the concept of a uniform orchard and dense forest, extending previous geometric analyses.

Findings

Established necessary and sufficient conditions for spiral sets to be orchards and uniform orchards.

Provided criteria for spiral sets to have no visible points, indicating high density.

Introduced the concept of dense forests as a quantitative refinement of density properties.

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).