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Understanding the structure of the matter distribution in the Universe due to the action of the gravitational instability---the creation of the ``cosmic web''---is complicated by lack of direct analytic access to the complex 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-dimensional manifold (in a 6-dimensional 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 multistream and density fields, the main focus of this paper. Using a large-box simulation we carry out a variety of illustrative initial analyses with the technique. These include studying the correlation between multistreaming 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.
Shandarin et al. (Wed,) studied this question.