TY - JOUR

T1 - Constant spacing in filament bundles

AU - Atkinson, Daria W.

AU - Santangelo, Christian D.

AU - Grason, Gregory M.

N1 - Funding Information:
Weare grateful to R Kusner and participants of the 'Packing of Continua' workshop at the Aspen Center for Physics (NSF PHY 1607611) for numerous helpful discussions.Wealso thankDHall, J Selinger, andHWu for comments on this manuscript. This work was supported by the National Science Foundation under grant Nos. DMR-1608862 and DMR-1507377.
Publisher Copyright:
© 2019 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft.

PY - 2019/6/6

Y1 - 2019/6/6

N2 - Assemblies of one-dimensional filaments appear in a wide range of physical 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 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 : equidistance, where the distance of closest approach is constant along the length of filament pairs; and isometry, where the distances of closest approach between all neighboring filaments are constant and equal. We show that, although any smooth curve in 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 . 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, filament textures that are simultaneously bent and twisted, 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 additional longitudinal variations of spacing along the filaments. 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.

AB - Assemblies of one-dimensional filaments appear in a wide range of physical 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 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 : equidistance, where the distance of closest approach is constant along the length of filament pairs; and isometry, where the distances of closest approach between all neighboring filaments are constant and equal. We show that, although any smooth curve in 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 . 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, filament textures that are simultaneously bent and twisted, 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 additional longitudinal variations of spacing along the filaments. 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.

KW - filaments and fibers

KW - material geometry

KW - packing

UR - http://www.scopus.com/inward/record.url?scp=85073659023&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85073659023&partnerID=8YFLogxK

U2 - 10.1088/1367-2630/ab1c2d

DO - 10.1088/1367-2630/ab1c2d

M3 - Article

AN - SCOPUS:85073659023

VL - 21

JO - New Journal of Physics

JF - New Journal of Physics

SN - 1367-2630

IS - 6

M1 - 062001

ER -