Geometry of lifts of tilings of Euclidean spaces

The method of canonical scalings is substantiated for lifts of tilings of Euclidean space. A new combinatorial-geometric approach to the construction of a generatrix of a tiling is proposed. The construction is based on a simple and geometrically transparent operation of lifting a face to an earlier lifted neighbor. The approach is applied to the fundamental theorem of polytope theory related to the classical problem of validity of Voronoi's conjecture for parallelotopes. So far Voronoi's conjecture has been proved only for some special families of parallelotopes, and the theorem states that Voronoi's conjecture holds for a given parallelotope $P$ if and only if there exists a canonical scaling for the corresponding tiling $\mathcal T_P$. A new, significantly shortened (compared with the available ones), geometric proof of this fundamental theorem is given.