Geometry and Optimal Packing of Twisted Columns and Filaments

TL;DR

This review explores the geometric principles and constraints of packing twisted filamentous structures, linking non-Euclidean surface geometry to filament bundle configurations and defect formations.

Contribution

It introduces a mathematical framework connecting filament spacing and orientations to non-Euclidean geometries, advancing understanding of twisted and toroidal filament packing.

Findings

Connection between filament packing and non-Euclidean surface geometry.

Demonstration of defect states related to curvature and twisting.

Relation of filament bundle geometry to classical problems like the Thomson problem.

Abstract

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.