Geometry and Optimal Packing of Twisted Columns and Filaments

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.