  Helically close-packed states of filaments are common in natural and
engineered material systems, ranging from nanoscopic biomolecules to
macroscopic structural components. While the simplest models of helical
close-packing, described by the ideal rope model, neglect anisotropy
perpendicular to the backbone, physical filaments are often quite far from
circular in their cross-section. Here, we consider an anisotropic
generalization of the ideal rope model and show that cross-section anisotropy
has a strongly non-linear impact on the helical close-packing configurations of
helical filaments. We show that the topology and composition of the
close-packing landscape depends on the cross-sectional aspect ratio and is
characterized by several distinct states of self-contact. We characterize the
local density of these distinct states based on the notion of confinement
within a 'virtual' cylindrical capillary, and show that states of optimal
density vary strongly with the degree of anisotropy. While isotropic filaments
are densest in a straight configuration, any measure of anisotropy leads to
helicity of the maximal density state. We show the maximally dense states
exhibit a sequence of transitions in helical geometry and cross-sectional tilt
with increasing anisotropy, from spiral tape to spiral screw packings.
Furthermore, we show that maximal capillary density saturates in a lower bound
for volume fraction of $\pi/4$ in the large-anisotropy, spiral-screw limit.
While cross-sectional anisotropy is well-known to impact the mechanical
properties of filaments, our study shows its strong effects to shape the
configuration space and packing efficiency of this elementary material motif.
