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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.

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.

# Polyhedron Symmetry
(order)
Image Vertices Edges Faces Belts
1 Rhombohedron Ci (2) 8 12 6 43
Trigonal trapezohedron D3d (12) Parallelepiped Ci (2) Rectangular cuboid D2h (8) Cube Oh (24) 2 Hexagonal prism Ci (2) 8 18 12 61, 43
D6h (24) 3 Rhombic dodecahedron D2h (8) 14 24 12 64
Oh (24) 4 Elongated dodecahedron D4h (16) 18 28 12 64, 41
D2h (8) 5 Truncated octahedron Oh (24) 24 36 14 66

High symmetric examples

Pm3m (221) Im3m (229) Fm3m (225)     Cubic                  Hexagonal prismatic         Rhombic dodecahedral     Elongated dodecahedral Bitruncated cubic       General symmetry examples

parallelogon - analogous space-filling polygons in 2D, with parallelograms and hexagons
parallelotope

References

The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.117
Coxeter, H. S. M. Regular polytopes (book), 3rd ed. New York: Dover, pp. 29-30, p.257, 1973.
Tutton, A. E. H. Crystallography and Practical Crystal Measurement, 2nd ed. London: Lubrecht & Cramer, 1964.
Weisstein, Eric W., "Primary parallelohedron", MathWorld.
Weisstein, Eric W., "Space-filling polyhedron", MathWorld.
E. S. Fedorov, Nachala Ucheniya o Figurah. [In Russian] (Elements of the theory of figures) Notices Imper. Petersburg Mineralog. Soc., 2nd ser.,24(1885), 1 – 279. Republished by the Acad. Sci. USSR, Moscow 1953.
Fedorov's five parallelohedra in R³
Fedorov's Five Parallelohedra

Mathematics Encyclopedia