Constant spacing in filament bundles

Abstract: Assemblies of filaments appear in a wide range of systems: from biopolymer bundles, columnar liquid crystals, and superconductor vortex arrays; to familiar macroscopic materials, like ropes, cables and textiles. Interactions between the constituent filaments in such systems are most sensitive to the {\it distance of closest approach} between the central curves which approximate their configuration, subjecting these distinct assemblies to common geometric constraints. In this paper, we consider two distinct notions of constant spacing in multi-filament packings in $\mathbb{R}^3$: {\it equidistance}, where the distance of closest approach is constant along the length of filament pairs; and {\it isometry}, where the distances of closest approach between all neighboring filaments are constant and equal. We show that, although any smooth curve in $\mathbb{R}^3$ permits one dimensional families of collinear equidistant curves belonging to a ruled surface, there are only two families of tangent fields with mutually equidistant integral curves in $\mathbb{R}^3$. The relative shapes and configurations of curves in these families are highly constrained: they must be either (isometric) developable domains, which can bend, but not twist; or (non-isometric) constant-pitch helical bundles, which can twist, but not bend. Thus, simultaneously bent and twisted filament textures, such as twisted toroids of condensed DNA plasmids or wire ropes, are doubly frustrated: twist frustrates constant neighbor spacing in the cross-section, while non-equidistance requires longitudinal variations of inter-filament spacing. To illustrate the consequences of the failure of equidistance, we compare spacing in three "almost equidistant" ansatzes for twisted toroidal bundles and use our formulation of equidistance to construct upper bounds on the growth of longitudinal variations of spacing with bundle thickness.