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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
Rhombic dodecahedron Hexagonal prism
Elongated cube
Elongated dodecahedron Truncated octahedron
Images Parallelohedron edges cube.png Parallelohedron edges rhombic dodecahedron.png Parallelohedron edges hexagonal prism.png Parallelohedron edges elongated rhombic dodecahedron.png Parallelohedron edge truncated octahedron.png
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
Image Vertices Edges Faces Belts
1 Rhombohedron Ci (2) Rhombohedron.svg 8 12 6 43
Trigonal trapezohedron D3d (12) Acute golden rhombohedron.png
Parallelepiped Ci (2) Parallelepiped 2013-11-29.svg
Rectangular cuboid D2h (8) Cuboid simple.svg
Cube Oh (24) Parallelohedron edges cube.png
2 Hexagonal prism Ci (2) Skew hexagonal prism parallelohedron.png 8 18 12 61, 43
D6h (24) Parallelohedron edges hexagonal prism.png
3 Rhombic dodecahedron D2h (8) Squared rhombic dodecahedron.png 14 24 12 64
Oh (24) Parallelohedron edges rhombic dodecahedron.png
4 Elongated dodecahedron D4h (16) Elongated dodecahedron parallelohedron.png 18 28 12 64, 41
D2h (8) Contracted truncated octahedron.png
5 Truncated octahedron Oh (24) Parallelohedron edge truncated octahedron.png 24 36 14 66

High symmetric examples

Pm3m (221) Im3m (229) Fm3m (225)
Partial cubic honeycomb.png Hexagonal prismatic honeycomb.png Rhombic dodecahedra.png Rhombo-hexagonal dodecahedron tessellation.png HC-A4.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
Hexagonal prismatic
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
Rhombic dodecahedral
CDel node f1.pngCDel 3.pngCDel node.pngCDel split1-43.pngCDel nodes.png
Elongated dodecahedral Bitruncated cubic
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png

General symmetry examples

Rhombohedral prism honeycomb.png Skew hexagonal prism honeycomb.png Elongated rhombic dodecahedron honeycomb.png

See also

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


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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