Parallelohedron

Summary

A parallelohedron is a convex polytope whose translated copies fill space face to face, with no rotations allowed. In the plane the parallelohedra are the parallelograms and the centrally symmetric hexagons. In space there are five combinatorial types, found by Fedorov: the parallelepiped, the hexagonal prism, the rhombic dodecahedron, the elongated dodecahedron and the truncated octahedron. The Voronoi cell of any lattice is a parallelohedron, and Voronoi's conjecture asks whether every parallelohedron arises this way up to an affine map.