  Dusty magnetized structures observable in the far-infrared (FIR) at high
Galactic latitudes are ubiquitous and found to be closely related to HI
filaments with coherent velocity structures. Considering dimensionless
morphological characteristics based on Minkowski functionals, we determine the
distribution of filamentarities $F$ and aspect ratios $A$ for these structures.
Our data are based on Planck FIR and HI4PI HI observations. Filaments have
previously been extracted by applying the Hessian operator. We trace individual
filamentary structures along the plane of the sky and determine $A$ and $F$.
Filaments in the diffuse interstellar medium (ISM) are seldom isolated
structures, but are rather part of a network of filaments with a well-defined,
continuous distribution in $A$ and $F$. This distribution is self-replicating,
and the merger or disruption of individual filamentary structures leads only to
a repositioning of the filament in $A$ and $F$ without changing the course of
the distribution. FIR and HI filaments identified at high Galactic latitudes
are a close match to model expectations for narrow filaments with approximately
constant widths. This distribution is continuous without clear upper limits on
the observed aspect ratios. Filaments are associated with enhanced column
densities of CO-dark $H_2$. Radial velocities along the filaments are coherent
and mostly linear with typical dispersions of $\Delta v_{\mathrm{LSR}} = 5.24 $
km/s. The magnetic field strength in the diffuse turbulent ISM scales with
hydrogen volume density as $B \propto n_{\mathrm{H}}^{0.58} $. At high Galactic
latitudes, we determine an average turbulent magnetic field strength of
$\langle \delta B \rangle = 5.3 ~\mu$G and an average mean strength of the
magnetic field in the plane of the sky of $\langle B_{\mathrm{POS}} \rangle =
4.4 ~\mu$G.
