In geometry, a parallelohedron is a convex polyhedron that can be translated without rotations to fill Euclidean space.
The lengths of the segments can be adjusted arbitrarily; doing so extends or shrinks the corresponding edges of the parallelohedron, without changing its combinatorial type or its property of tiling space.
[2] Senechal & Taylor (2023) define a "belt" of a zonohedron to be the cycle of faces that contain all parallel copies of one edge.
For this placement of the segments, one vertex of the parallelohedron will itself be at the origin, and the rest will be at positions given by sums of certain subsets of these vectors.
coordinates, three for each vector, but only some of these combinations are valid (because of the requirement that certain triples of segments lie in parallel planes, or equivalently that certain triples of vectors are coplanar) and different combinations may lead to parallelohedra that differ only by a rotation, scaling transformation, or more generally by an affine transformation.
When affine transformations are factored out, the number of free parameters that describe the shape of a parallelohedron is zero for a parallelepiped (all parallelepipeds are equivalent to each other under affine transformations), two for a hexagonal prism, three for a rhombic dodecahedron, four for an elongated dodecahedron, and five for a truncated octahedron.
[5] The classification of parallelohedra into five types was first made by Russian crystallographer Evgraf Fedorov, as chapter 13 of a Russian-language book first published in 1885, whose title has been translated into English as An Introduction to the Theory of Figures.
[4] Some of the mathematics in Federov's book is faulty; for instance it includes an incorrect proof of a lemma stating that every monohedral tiling of the plane is eventually periodic,[7] proven to be false in 2023 as part of the solution to the einstein problem.
[8] In the case of parallelohedra, Fedorov assumed without proof that every parallelohedron is centrally symmetric, and used this assumption to prove his classification.
[2] In two dimensions the analogous figure to a parallelohedron is a parallelogon, a polygon that can tile the plane edge-to-edge by translation.
There are two kinds of parallelogons: the parallelograms and the hexagons in which each pair of opposite sides is parallel and of equal length.
[1] A plesiohedron is a broader class of three-dimensional space-filling polyhedra, formed from the Voronoi diagrams of periodic sets of points.