The faces of a zonohedron can be grouped into zones. A zone is an encircling band of faces which share a common edge direction (and length). This is best understood by examining a model of the same "random" 42-face zonohedron as above, or a rhombic enneacontahedron, or a rhombic tricontahedron, in which one zone is colored. Each face is part of two zones. Because opposite edges of each parallelogram are parallel, the faces can be grouped into chains which all share a common direction in space. The chains must be closed cycles because there are only a finite number of faces.
Exercise: Look and count to see that each zone in the random 42-face zonohedron contains 12 faces, and that each zone in the rhombic enneacontahedron contains 18 faces. Also observe that any pair of zones intersect each other in exactly two places --- at two opposite equal faces.
Exercise: Count and see that these three zonohedra have 3, 4, and 6 zones, respectively. The zones consist of 4, 6, and 10 faces respectively.
These three zonohedra are closely related to the Platonic solids:
Observe that the zonohedra above have 6, 12, 30, 42, and 90 faces. Hindsight indicates that these numbers are of the form n(n-1) where n is 3, 4, 6, 7, or 10. This should suggest the case of n=5.
The number n is just the number of zones, and each pair of zones meets twice (once in front and once in back). There are n(n-1)/2 possible pairs we can choose from n zones, and so n(n-1) faces.
We can create a zonohedron from any set of three or more vectors or directions in space. Three vectors define a parallelepiped. Adding an additional vector to the set leads to enlarged zonohedra (with one more zone) by the reverse of the 6-zone-to-5-zone process referred to above.
To construct symmetric zonohedra by this process, we start with a symmetrically arranged set of vectors, for example the axes of symmetry of a symmetric polyhedron. These directions in space form what is called a star of vectors. Every zonohedron has an underlying star---its edge directions---and from any star we can construct a zonohedron having those edge directions:
There are several ways to generalize the above notion of zonohedron. One is to allow nonconvex polyhedra; a second is to allow polygons with more than four sides. A third is an original concept which I call a zonish polyhedron, in which an arbitrary polyhedron is used as a seed:
Good references on zonohedra are W.W. Rouse Ball, Mathematical Recreations and Essays, p. 140, and Coxeter, Regular Polytopes, p. 27.