Exploring the Cosmic Web in the Sloan Digital Sky Survey Data Release
  Seven using the Local Dimension
  It is possible to visualize the Cosmic Web as an interconnected network of
one-dimensional filaments, two-dimensional sheets and three-dimensional
volume-filling structures which we refer to as clusters. We have used the Local
Dimension D, which takes values D = 1, 2 and 3 for filaments, sheets and
clusters, respectively, to analyse the Cosmic Web in a three-dimensional
volume-limited galaxy sample from the Sloan Digital Sky Survey Data Release 7.
The analysis was carried out separately using three different ranges of
length-scales: 0.5-5, 1-10 and 5-50 $h^{-1} {\rm Mpc}$. We find that there is a
progressive increase in the D values as we move to larger length-scales. At the
smallest length-scale, the galaxies predominantly reside in filaments and
sheets. There is a shift from filaments to sheets and clusters at larger
scales. Filaments are completely absent at the largest length-scale (5-50
$h^{-1} {\rm Mpc}$). Considering the effect of the density environment on the
Cosmic Web, we find that the filaments preferentially inhabit regions with a
lower density environment as compared to sheets and clusters which prefer
relatively higher density environments. A similar length-scale dependence and
environment dependence was also found in a galaxy sample drawn from the
Millennium Simulation which was analysed in exactly the same way as the actual
data.
