A spiral in
is defined
as a set of the form
,
where
is
a spherical sequence. Such point sets have been extensively studied, in particular in the
planar case
,
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
visibility from discrete geometry are employed to characterise density properties
of such point sets. More precisely, necessary and sufficient conditions are
established for a spiral to be an
orchard (a “homogeneous” density property
defined by Pólya), a
uniform orchard (a concept introduced in this work), a
set with no visible point (implying that the point set is dense enough in a
suitable sense) and a
dense forest (a quantitative and uniform refinement of
the previous concept).
