Geometry of lifts of tilings of euclidean spaces

TL;DR

This paper introduces a new combinatorial-geometrical method for constructing a generatriss of Euclidean tilings, providing a simplified proof of Voronoi's Conjecture for parallelotopes through canonical scalings.

Contribution

It presents a novel approach for tiling construction and offers a new geometrical proof of a fundamental theorem in the theory of parallelotopes.

Findings

Voronoi's Conjecture holds if and only if the tiling admits a canonical scaling.

A new combinatorial-geometrical approach for constructing generatriss.

Explicit justification for canonical scalings of Euclidean space tilings.

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.