Zonohedron

Consider any star of line segments through one point in space such that no three lines are coplanar. Then there exists a polyhedron, known as a zonohedron, whose faces consist of rhombi and whose edges are parallel to the given lines in sets of . Furthermore, for every pair of the lines, there is a pair of opposite faces whose sides lie in those directions (Ball and Coxeter 1987, p. 141). A zonohedron is a therefore a polyhedron in which every face is centrally symmetric (Towle 1996, Eppstein).

There is some confusion in the definition of zonotopes (Eppstein 1996). Wells (1991, pp. 274-275) requires the generating vectors to be in general position (all -tuples of vectors must span the whole space), so that all the faces of the zonotope are parallelotopes. Others (Bern et al. 1995; Ziegler 1995, pp. 198-208; Eppstein 1996) do not make this restriction. Coxeter (1973) starts with one definition but soon switches to the other.

While all zonohedra have Dehn invariant 0, only zonohedra that are parallelohedraParalleohedron are space-filling.

The figures above illustrate the Archimedean solids together with the zonohedra determined by their nonparallel vertices.

Similarly, the figures above illustrate the Platonic solids together with the zonohedra determined by the subsets of their vertices that are not antiparallel.

The combinatorics of the faces of a zonohedron are equivalent to those of line arrangements in the plane (Eppstein 1996).

If the line segments are all of equal length, the zonohedron is known as an equilateral zonohedron (Coxeter 1973, p. 29).

There exist parallelograms in a nonsingular zonohedron, where is the number of different directions in which polyhedron edges occur (Ball and Coxeter 1987, pp. 141-144).

Every convex polyhedron bounded solely by parallelograms is a zonohedron (Coxeter 1973, p. 27), as is every convex polyhedron whose faces are parallel-sided -gons (Coxeter 1973, p. 29)

In addition to the equilateral and polar zonohedra, the parallelepipeds, primary parallelohedra (Coxeter 1973, pp. 29-30), and rhombohedra are zonohedra.