Isodistance in Multi-Filament Packings: Parallel but not Straight

Assemblies of one-dimensional filaments appear in a wide range of physical systems, from biopolymer bundles, organogel fibers, columnar liquid crystals, vortex arrays, and nanotube yarns, to everyday macroscopic structures, like ropes and textiles. Interactions between the different chains or filaments in these diverse examples is dominated by the shortest distance between them, leading to common geometric constraints. When the bundle is twisted—which is often the case for self assembled chiral molecules—or has an otherwise nontrivial geometry, the packings of the filaments are frustrated. Frustration in filamentous assemblies is particularly insidious since there is in general neither uniform spacing between filaments at different points in the bundle or at different positions on the same fibers. In this talk I will describe geometric constraints on packing N >>1 curves such that the distance along their lengths remains constant, a property I call isodistance, and demonstrate that all such packings fall into two classes: the arbitrarily bent but untwisted "developable domains"; and "straight", helical bundles with constant twist. We also construct examples of twisted toroidal bundles to analyze the consequences for packings that are simultaneous twisted and bent.