Properties of parallelotopes equivalent to Voronoi's conjecture

A parallelotope is a polytope whose translation copies fill space without gaps and intersections by interior points. Voronoi conjectured that each parallelotope is an affine image of the Dirichlet domain of a lattice, which is a Voronoi polytope. We give several properties of a parallelotope and prove that each of them is equivalent to it is an affine image of a Voronoi polytope.Comment: 18 pages (submitted

Clusters of Cycles

A {\it cluster of cycles} (or {\it $(r,q)$-polycycle}) is a simple planar 2--co nnected finite or countable graph $G$ of girth $r$ and maximal vertex-degree $q$, which admits {\it $(r,q)$-polycyclic realization} on the plane, denote it by $P(G)$, i.e. such that: (i) all interior vertices are of degree $q$, (ii) all interior faces (denote their number by $p_r$) are combinatorial $r$-gons and (implied by (i), (ii)) (iii) all vertices, edges and interior faces form a cell-complex. An example of $(r,q)$-polycycle is the skeleton of $(r^q)$, i.e. of the $q$-valent partition of the sphere $S^2$, Euclidean plane $R^2$ or hyperbolic plane $H^2$ by regular $r$-gons. Call {\it spheric} pairs $(r,q)=(3,3),(3,4),(4,3),(3,5),(5,3)$; for those five pairs $P(r^q)$ is $(r^q)$ without the exterior face; otherwise $P(r^q)=(r^q)$. We give here a compact survey of results on $(r,q)$-polycycles.Comment: 21. to in appear in Journal of Geometry and Physic