Parallelohedron

Every parallelohedron is a zonohedron, a centrally symmetric polyhedron with centrally symmetric faces. Like any zonohedron, it can be constructed as the Minkowski sum of line segments, one segment for each parallel class of edges of the polyhedron. For parallelohedra, there are between three and six of these parallel classes. 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. As a limiting case, for a parallelohedron with more than three parallel classes of edges, the length of any one of these classes can be adjusted to zero, producing another parallelohedron of a simpler form, with one fewer class of parallel edges.^[1] Every parallelotope can be obtained in this way as a degenerate truncated octahedron.^[2] As with all zonohedra, every parallelohedron has central inversion symmetry,^[3] but additional symmetries are possible with an appropriate choice of the generating segments.^[4]

The five types of parallelohedron are:^[3]

• A parallelepiped, generated from three line segments that are not all parallel to a common plane. Its most symmetric form is the cube, generated by three perpendicular unit-length line segments.^[3] It tiles space to form the cubic honeycomb.
• A hexagonal prism, generated from four line segments, three of them parallel to a common plane and the fourth not. Its most symmetric form is the right prism over a regular hexagon.^[3] It tiles space to form the hexagonal prismatic honeycomb.
• The rhombic dodecahedron, generated from four line segments, no two of which are parallel to a common plane. Its most symmetric form is generated by the four long diagonals of a cube.^[3] It tiles space to form the rhombic dodecahedral honeycomb.
• The elongated dodecahedron, generated from five line segments, with two triples of coplanar segments. It can be generated by using an edge of the cube and its four long diagonals as generators.^[3] It tiles space to form the elongated dodecahedral honeycomb.
• The truncated octahedron, generated from six line segments with four triples of coplanar segments. It can be embedded in four-dimensional space as the 4-permutahedron, whose vertices are all permutations of the counting numbers (1,2,3,4). In three-dimensional space, its most symmetric form is generated from six line segments parallel to the face diagonals of a cube.^[3] It tiles space to form the bitruncated cubic honeycomb.

Any zonohedron whose faces have the same combinatorial structure as one of these five shapes is a parallelohedron, regardless of its particular angles or edge lengths. For example, any affine transformation of a parallelohedron will produce another parallelohedron of the same type.^[3] Senechal & Taylor (2023) define a "belt" of a zonohedron to be the cycle of faces that contain all parallel copies of one edge. The number of faces in a belt must be even. As Federov and many others showed, the parallelohedra are exactly the zonohedra all of whose belts consist of four or six faces.^[2]

When further subdivided according to their symmetry groups, there are 22 forms of the parallelohedra. For each form, the centers of its copies in its honeycomb form the points of one of the 14 Bravais lattices. Because there are fewer Bravais lattices than symmetric forms of parallelohedra, certain pairs of parallelohedra map to the same Bravais lattice.^[4]

By placing one endpoint of each generating line segment of a parallelohedron at the origin of three-dimensional space, the generators may be represented as three-dimensional vectors, the positions of their opposite endpoints. 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. A parallelohedron with ${\displaystyle g}$  vectors can in this way be parameterized by ${\displaystyle 3g}$  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.^[6] Federov was working under an incorrect theory of the structure of crystals, according to which every crystal has a repeating structure in the shape of a parallelohedron, which in turn is formed from one or more molecules that all take the same shape (a stereohedron). This theory was falsified by the 1913 discovery of the structure of halite (table salt) which is not partitioned into separate molecules, and more strongly by the much later discovery of quasicrystals.^[2]

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. The classification of parallelohedra was later placed on a firmer footing by Hermann Minkowski, who used his uniqueness theorem for polyhedra with given face normals and areas to prove that parallelohedra are centrally symmetric.^[3]

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.^[9]

There are multiple non-convex polyhedra that tile space by translation, beyond the five Federov parallelohedra. These are not zonohedra and need not be centrally symmetric. For instance, some of these can be obtained from a rhombic triacontahedron by replacing certain triples of faces by indentations. According to a conjecture of Branko Grünbaum, for every polyhedron that is topologically a sphere and can tile space by translation, it is possible to group its faces into patches (unions of connected subsets of faces) so that the combinatorial structure of these patches is the same as the combinatorial structure of the faces of one of the five Federov parallelohedra. This conjecture remains unproven.^[2]

In higher dimensions a convex polytope that tiles space by translation is called a parallelotope. There are 52 different four-dimensional parallelotopes, first enumerated by Boris Delaunay (with one missing parallelotope, later discovered by Mikhail Shtogrin),^[10] and more than 100,000 types in five dimensions.^[11][12] Unlike the case for three dimensions, not all of them are zonotopes. 17 of the four-dimensional parallelotopes are zonotopes, one is the regular 24-cell, and the remaining 34 of these shapes are Minkowski sums of zonotopes with the 24-cell.^[13] A ${\displaystyle d}$ -dimensional parallelotope can have at most ${\displaystyle 2^{d+1}-2}$  facets, with the permutohedron achieving this maximum.^[1]

A plesiohedron is a broader class of three-dimensional space-filling polyhedra, formed from the Voronoi diagrams of periodic sets of points.^[9] As Boris Delaunay proved in 1929,^[14] every parallelohedron can be made into a plesiohedron by an affine transformation,^[3] but this remains open in higher dimensions,^[1] and in three dimensions there also exist other plesiohedra that are not parallelohedra. The tilings of space by plesiohedra have symmetries taking any cell to any other cell, but unlike for the parallelohedra, these symmetries may involve rotations, not just translations.^[9]

• Weisstein, Eric W. "Primary parallelohedron". MathWorld.