  It is now well accepted that the galaxies are distributed in filaments,
sheets and clusters all of which form an interconnected network known as the
Cosmic Web. It is a big challenge to quantify the shapes of the interconnected
structural elements that form this network. Tools like the Minkowski
functionals which use global properties, though well suited for an isolated
object like a single sheet or filament, are not suited for an interconnected
network of such objects. We consider the Local Dimension $D$, defined through
$N(R)=A R^D$, where $N(R)$ is the galaxy number count within a sphere of
comoving radius $R$ centered on a particular galaxy, as a tool to locally
quantify the shape in the neigbourhood of different galaxies along the Cosmic
Web. We expect $D \sim 1,2$ and 3 for a galaxy located in a filament, sheet and
cluster respectively. Using LCDM N-body simulations we find that it is possible
to determine $D$ through a power law fit to $N(R)$ across the length-scales 2
to $10 {\rm Mpc}$ for $\sim 33 %$ of the galaxies. We have visually identified
the filaments and sheets corresponding to many of the galaxies with $D \sim 1$
and 2 respectively. In several other situations the structure responsible for
the $D$ value could not be visually identified, either due to its being tenuous
or due to other dominating structures in the vicinity. We also show that the
global distribution of the $D$ values can be used to visualize and interpret
how the different structural elements are woven into the Cosmic Web.
