  We present the Smoothed Hessian Major Axis Filament Finder (SHMAFF), an
algorithm that uses the eigenvectors of the Hessian matrix of the smoothed
galaxy distribution to identify individual filamentary structures. Filaments
are traced along the Hessian eigenvector corresponding to the largest
eigenvalue, and are stopped when the axis orientation changes more rapidly than
a preset threshold. In both N-body simulations and the Sloan Digital Sky Survey
(SDSS) main galaxy redshift survey data, the resulting filament length
distributions are approximately exponential. In the SDSS galaxy distribution,
using smoothing lengths of 10 h^{-1} Mpc and 15 h^{-1} Mpc, we find filament
lengths per unit volume of 1.9x10^{-3} h^2 Mpc^{-2} and 7.6x10^{-4} h^2
Mpc^{-2}, respectively. The filament width distributions, which are much more
sensitive to non-linear growth, are also consistent between the real and mock
galaxy distributions using a standard cosmology. In SDSS, we find mean filament
widths of 5.5 h^{-1} Mpc and 8.4 h^{-1} Mpc on 10 h^{-1} Mpc and 15 h^{-1} Mpc
smoothing scales, with standard deviations of 1.1 h^{-1} Mpc and 1.4 h^{-1}
Mpc, respectively. Finally, the spatial distribution of filamentary structure
in simulations is very similar between z=3 and z=0 on smoothing scales as large
as 15 h^{-1} Mpc, suggesting that the outline of filamentary structure is
already in place at high redshift.
