Zonotope

A zonohedron is a convex polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180°. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as the three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorov, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.

The original motivation for studying zonohedra is that the Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way can tessellate 3-dimensional space and is called a primary parallelohedron. Each primary parallelohedron is combinatorially equivalent to one of five types: the rhombohedron (including the cube), hexagonal prism, truncated octahedron, rhombic dodecahedron, and the rhombo-hexagonal dodecahedron.

Let {v₀, v₁, ...} be a collection of three-dimensional vectors. With each vector v_i we may associate a line segment {x_iv_i|0≤x_i≤1}. The Minkowski sum {Σx_iv_i|0≤x_i≤1} forms a zonohedron, and all zonohedra that contain the origin have this form. The vectors from which the zonohedron is formed are called its generators. This characterization allows the definition of zonohedra to be generalized to higher dimensions, giving zonotopes.

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