Parallelohedron

In geometry, a parallelohedron or Fedorov polyhedron is a convex polyhedron that can be translated without rotations to fill Euclidean space, producing a honeycomb in which all copies of the polyhedron meet face-to-face. Evgraf Fedorov identified the five types of parallelohedron in 1885 in his studies of crystallographic systems. They are the cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron.

Each parallelohedron is centrally symmetric with symmetric faces, making it a special case of a zonohedron. Each parallelohedron is also a stereohedron, a polyhedron that tiles space so that all tiles are symmetric. The centers of the tiles in a tiling of space by parallelohedra form a Bravais lattice, and every Bravais lattice can be formed in this way. Adjusting the lengths of parallel edges in a parallelohedron, or performing an affine transformation of the parallelohedron, results in another parallelohedron of the same combinatorial type. It is possible to choose this adjustment so that the tiling by parallelohedra is the Voronoi diagram of its Bravais lattice, and so that the resulting parallelohedra become special cases of the plesiohedra.

The three-dimensional parallelohedra are analogous to two-dimensional parallelogons and higher-dimensional parallelotopes.