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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.
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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.
Symmetries of 5 types
There are 5 types of parallelohedra, although each has forms of varied symmetry.
Parallelotope
In higher dimensions a parallelohedron is called a parallelotope. There are 52 variations for 4-dimensional parallelotopes.