  This review presents recent progress in understanding constraints and
consequences of close-packing geometry of filamentous or columnar materials
possessing non-trivial textures, focusing in particular on the common motifs of
twisted and toroidal structures. The mathematical framework is presented that
relates spacing between line-like, filamentous elements to their backbone
orientations, highlighting the explicit connection between the inter-filament
{\it metric} properties and the geometry of non-Euclidean surfaces. The
consequences of the hidden connection between packing in twisted filament
bundles and packing on positively curved surfaces, like the Thomson problem,
are demonstrated for the defect-riddled ground states of physical models of
twisted filament bundles. The connection between the "ideal" geometry of {\it
fibrations} of curved three-dimensional space, including the Hopf fibration,
and the non-Euclidean constraints of filament packing in twisted and toroidal
bundles is presented, with a focus on the broader dependence of metric geometry
on the simultaneous twisting and folded of multi-filament bundles.
