  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.
