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Concept
Parallelohedron
Summary
In geometry, a parallelohedron is a polyhedron that can be translated without rotations in 3-dimensional Euclidean space to fill space with 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.
Every parallelohedron is a zonohedron, a centrally symmetric polyhedron with centrally symmetric faces. Like any zonohedron, it can be constructed as the Minkowski sum of line segments, one segment for each parallel class of edges of the polyhedron. For parallelohedra, there are between three and six of these parallel classes. The lengths of the segments can be adjusted arbitrarily; doing so extends or shrinks the corresponding edges of the parallelohedron, without changing its combinatorial type or its property of tiling space. As a limiting case, for a parallelohedron with more than three parallel classes of edges, the length of any one of these classes can be adjusted to zero, producing another parallelohedron of a simpler form, with one fewer class of parallel edges. As with all zonohedra, these shapes automatically have 2 Ci central inversion symmetry, but additional symmetries are possible with an appropriate choice of the generating segments.
The five types of parallelohedron are:
A parallelepiped, generated from three line segments that are not all parallel to a common plane. Its most symmetric form is the cube, generated by three perpendicular unit-length line segments. It tiles space to form the cubic honeycomb.
A hexagonal prism, generated from four line segments, three of them parallel to a common plane and the fourth not. Its most symmetric form is the right prism over a regular hexagon. It tiles space to form the hexagonal prismatic honeycomb.
The rhombic dodecahedron, generated from four line segments, no two of which are parallel to a common plane.
Official source
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