Parallelohedron


In geometry a parallelohedron is a polyhedron that can tessellate 3-dimensional spaces with face-to-face contacts via translations. This requires all opposite faces be congruent. Parallelohedra can only have parallelogonal faces, either parallelograms or hexagons with parallel opposite edges.
There are 5 types, first identified by Evgraf Fedorov in his studies of crystallographic systems.

Topological types

The vertices of parallelohedra can be computed by linear combinations of 3 to 6 vectors. Each vector can have any length greater than zero, with zero length becoming degenerate, or becoming a smaller parallelohedra.
The greatest parallelohedron is a truncated octahedron which is also called a 4-permutahedron and can be represented with in a 4D in a hyperplane coordinates as all permutations of the counting numbers.
A belt mn means n directional vectors, each containing m coparallel congruent edges. Every type has order 2 Ci central inversion symmetry in general, but each has higher symmetry geometries as well.
NameCube
Hexagonal prism
Elongated cube		Rhombic dodecahedronElongated dodecahedronTruncated octahedron
Images
Edge
types		3 edge-lengths3+1 edge-lengths4 edge-lengths4+1 edge-lengths6 edge-lengths
Belts4343, 616464, 4166

Symmetries of 5 types

There are 5 types of parallelohedra, although each has forms of varied symmetry.
#PolyhedronSymmetry
ImageVerticesEdgesFacesBelts
1RhombohedronCi 812643
1Trigonal trapezohedronD3d 812643
1ParallelepipedCi 812643
1Rectangular cuboidD2h 812643
1CubeOh 812643
2Hexagonal prismCi 1218861, 43
2Hexagonal prismD6h 1218861, 43
3Rhombic dodecahedronD4h 14241264
3Rhombic dodecahedronD2h 14241264
3Rhombic dodecahedronOh 14241264
4Elongated dodecahedronD4h 18281264, 41
4Elongated dodecahedronD2h 18281264, 41
5Truncated octahedronOh 24361466

Examples

Parallelotope

In higher dimensions a parallelohedron is called a parallelotope. There are 52 variations for 4-dimensional parallelotopes.