  We investigate with experiments and novel mapping the structure of a
hexagonally ordered filament bundle that is held near its ends and
progressively twisted around its central axis. The filaments are free to slide
relative to each other and are further held under tension-free boundary
conditions. Measuring the bundle packing with micro x-ray imaging, we find that
the filaments develop the helical rotation $\Omega$ imposed at the boundaries.
We then show that the observed structure is consistent with a mapping of the
filament positions to disks packed on a dual non-Euclidean surface with a
Gaussian curvature which increases with twist. We further demonstrate that the
mean inter-filament distance is minimal on the surface which can be
approximated by a hemisphere with an effective curvature $K_{eff} = 3\Omega^2$.
Examining the packing on the dual surface, we analyze the geometric frustration
of packing in twisted bundles and find the core to remain relatively
hexagonally ordered with inter-filament strains growing from the bundle center,
driving the formation of defects at the exterior of highly twisted bundles.
