Geometry of lifts of tilings of euclidean spaces

Abstract

This paper provides explicit justification for a method of canonical scalings of tilings of euclidean spaces. We present a new combinatorially-geometrical approach for constructing a generatriss of a tiling. The approach is based on an operation of lifting of a tile up to a lifted neighbour. We use this approach and give a new short geometrical proof of a fundamental theorem of theory of parallelotopes: Voronoi's Conjecture holds for a given parallelotope $P$ if and only if the corresponding tiling $\mathcal T_P$ admits a canonical scaling.