 Cube |
 Hexagonal prism |
 Rhombic dodecahedron |
 Elongated dodecahedron |
 Truncated octahedron |
In geometry, a parallelohedron is a convex polyhedron that can be translated without rotations to fill Euclidean space. This produces a honeycomb in which all copies of the polyhedron meet face-to-face. There are five types of parallelohedron, first identified by Evgraf Fedorov in 1885 in his studies of crystallographic systems: the cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron.
Each parallelohedron is a zonohedron and a plesiohedron. It has point reflection symmetry, as do its faces. Its edges can be grouped into subsets of equal-length parallel edges; the parallelohedron itself is the Minkowski sum of a set of line segments selected by choosing one representative from each parallel subset. Adjusting the lengths of the edges in any of these subsets, or more generally performing an affine transformation of a parallelohedron, results in another parallelohedron of the same combinatorial type. 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.
The three-dimensional parallelohedra are analogous to two-dimensional parallelogons and higher-dimensional parallelotopes.