  We analyze a volume limited galaxy sample from the SDSS to study the
environments of galaxies on different length scales in the local Universe. We
measure the local dimension of the SDSS galaxies on different length scales and
find that the sheets or sheetlike structures are the most prevalent pattern in
the cosmic web throughout the entire length scales. The abundance of sheets
peaks at $30 \, h^{-1}\, {\rm Mpc}$ and they can extend upto a length scales of
$90 \, h^{-1}\, {\rm Mpc}$ . Analyzing mock catalogues, we find that the sheets
are non-existent beyond $30 \, h^{-1}\, {\rm Mpc}$ in the Poisson
distributions. We find that the straight filaments in the SDSS galaxy
distribution can extend only upto a length scale of $30 \, h^{-1}\, {\rm Mpc}$.
Our results indicate that the environment of a galaxy exhibits a gradual
transition towards higher local dimension with increasing length scales finally
approaching a nearly homogeneous network on large scales. We compare our
findings with a semi analytic galaxy catalogue from the Millennium Run
simulation which are in fairly good agreement with the observations. We also
test the effects of the number density of the sample and the cut-off in the
goodness of fit which shows that the results are nearly independent of these
factors. Finally we apply the method to a set of simulations of the segment Cox
process and find that it can characterize such distributions.
