Parallelohedron

In geometry a parallelohedron is a polyhedron that can be translated without rotations in 3-dimensional Euclidean space to fill space with a honeycomb in which all copies of the polyhedron meet face-to-face. There are five types of parallelohedron, first identified by Evgraf Fedorov in 1885 in his studies of crystallographic systems: the cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron.[1]

Every parallelohedron is a zonohedron, constructed as the Minkowski sum of between three and six line segments. Each of these line segments can have any positive real number as its length, and each edge of a parallelohedron is parallel to one of these generating segments, with the same length. If the length of a segments of a parallelohedron generated from four or more segments is reduced to zero, the result is that the polyhedron degenerates to a simpler form, a parallelohedron formed from one fewer segment.[2] As a zonohedron, these shapes automatically have 2 Ci central inversion symmetry,[1] but additional symmetries are possible with an appropriate choice of the generating segments.[3]

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.[1]

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.[3]

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.[5] Some of the mathematics in this book is faulty; for instance it includes an incorrect proof of a lemma stating that every monohedral tiling of the plane is eventually periodic, which remains unsolved as the einstein problem.[6] 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 to prove that parallelohedra are centrally symmetric.[1]

In two dimensions the analogous figure to a parallelohedron is a parallelogon, a polygon that can tile the plane edge-to-edge by translation. These are parallelograms and hexagons with opposite sides parallel and of equal length.[7]

A plesiohedron is a broader class of three-dimensional space-filling polyhedra, formed from the Voronoi diagrams of periodic sets of points.[7] As Boris Delaunay proved in 1929,[12] every parallelohedron can be made into a plesiohedron by an affine transformation,[1] but this remains open in higher dimensions,[2] 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.[7]