  Understanding the structure of the matter distribution in the Universe due to
the action of the gravitational instability -- the cosmic web -- is complicated
by lack of direct analytic access to the nonlinear domain of structure
formation. Here, we suggest and apply a novel tessellation method designed for
cold dark matter (CDM) N-body cosmological simulations. The method is based on
the fact that the initial CDM state can be described by a 3-D manifold (in a
6-D phase space) that remains continuous under evolution. Our technique uses
the full phase space information and has no free parameters; it can be used to
compute multi-stream and density fields, the main focus of this paper. Using a
large-box LCDM simulation we carry out a variety of initial analyses with the
technique. These include studying the correlation between multi-streaming and
density, the identification of structures such as Zel'dovich pancakes and
voids, and statistical measurements of quantities such as the volume fraction
as a function of the number of streams -- where we find a remarkable scaling
relation. Cosmological implications are briefly discussed.
