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 m_n means n directional vectors, each containing m coparallel congruent edges. Every type has order 2 C_i central inversion symmetry in general, but each has higher symmetry geometries as well.

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

Symmetries of 5 types

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

#	Polyhedron	Symmetry (order)	Vertices	Edges	Faces	Belts
1	Rhombohedron	C_i (2)	8	12	6	4₃
	Trigonal trapezohedron	D_3d (12)
	Parallelepiped	C_i (2)
	Rectangular cuboid	D_2h (8)
	Cube	O_h (24)
2	Hexagonal prism	C_i (2)	12	18	8	6₁, 4₃
2	Hexagonal prism	D_6h (24)	12	18	8	6₁, 4₃
3	Rhombic dodecahedron	D_4h (16)	14	24	12	6₄
		D_2h (8)
		O_h (24)
4	Elongated dodecahedron	D_4h (16)	18	28	12	6₄, 4₁
4	Elongated dodecahedron	D_2h (8)	18	28	12	6₄, 4₁
5	Truncated octahedron	O_h (24)	24	36	14	6₆

Examples

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

Cubic	Hexagonal prismatic	Rhombic dodecahedral	Elongated dodecahedral	Bitruncated cubic
General symmetry examples

Parallelotope

In higher dimensions a parallelohedron is called a parallelotope. There are 52 variations for 4-dimensional parallelotopes.[1][2]

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References

Crystal Symmetries: Shubnikov Centennial Papers, edited by B. K. Vainshtein, I. Hargittai
Once more about the 52 four-dimensional parallelotopes, Michel Deza, Viacheslav Grishukhin (2003)

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

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[1] Crystal Symmetries: Shubnikov Centennial Papers, edited by B. K. Vainshtein, I. Hargittai

[2] Once more about the 52 four-dimensional parallelotopes, Michel Deza, Viacheslav Grishukhin (2003)