  We suggest a geometrical mechanism for the ordering of slender filaments
inside non-isotropic containers, using cortical microtubules in plant cells and
packing of viral genetic material inside capsids as concrete examples. We show
analytically how the shape of the cell affects the ordering of phantom,
non-self-avoiding, stiff rods. We find that for oblate cells the preferred
orientation is along the equator, while for prolate spheroids with an aspect
ratio close to one, the orientation is along the principal (long axis).
Surprisingly, at high enough aspect ratio, a configurational phase transition
occurs, and the rods no longer point along the principal axis, but at an angle
to it, due to high curvature at the poles. We discuss some of the possible
effects of self avoidance, using energy considerations. These results are
relevant to other packing problems as well, such as spooling of filament in the
industry or spider silk inside water droplets.
