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 (1,2,3,4).

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.

Parallelohedra with edges colored by direction
Name Cube
(parallelepiped)
Rhombic dodecahedron Hexagonal prism
Elongated cube
Elongated dodecahedron Truncated octahedron
Images
Edge
types
3 edge-lengths 4 edge-lengths 3+1 edge-lengths 4+1 edge-lengths 6 edge-lengths
Belts 43 64 43, 61 64, 41 66

Symmetries of 5 types

There are 5 types of parallelohedra, although each has forms of varied symmetry.

#PolyhedronSymmetry
(order)
ImageVerticesEdgesFacesBelts
1 RhombohedronCi (2) 8 12 6 43
Trigonal trapezohedronD3d (12)
ParallelepipedCi (2)
Rectangular cuboidD2h (8)
CubeOh (24)
2 Hexagonal prismCi (2) 8181261, 43
D6h (24)
3 Rhombic dodecahedronD2h (8) 14241264
Oh (24)
4 Elongated dodecahedronD4h (16)18281264, 41
D2h (8)
5 Truncated octahedronOh (24)24361466

Examples

High symmetric examples
Pm3m (221) Im3m (229) Fm3m (225)
Cubic

Hexagonal prismatic
Rhombic dodecahedral
Elongated dodecahedral Bitruncated cubic
General symmetry examples

See also

References