## Background
Some simple zonohedra are shown in Figure 1. All are easily generated by the algorithm presented below. In addition to their aesthetic quality, zonohedra are the form of certain mineral crystals. They are projections into 3-space of higher dimensional hypercubes. They pop up in the analysis of quasi-regular "Penrose" tilings of the plane. They have been proposed as architectural structures, and a number of familiar polyhedra happen to be zonohedra. One special type of zonohedron is the The zonohedra illustrated in Figure 1 are zonohedrifications of other polyhedra listed in the caption. The geometric relation between a polyhedron, , and its zonohedrification, , is the following: For every plane defined by the center, , of and (at least) two vertices, , , of , there are two equal opposite faces of parallel to that plane. Furthermore, these two opposite faces of have edges that are parallel and equal to the lines and . Furthermore, every edge of comes about in that manner, (i.e., every edge is congruent and parallel to a line for some vertex of ). As an example, given an octahedron, we can find three different planes each defined by two vertices and the octahedron's center. (Each plane happens to contain four vertices.) Evidently, the six faces of a (properly oriented and scaled) cube come in pairs parallel to these three orthogonal planes, and the edges of the cube are each equal and parallel to some line connecting a vertex of the octahedron with its center. We write , where indicates the zonohedrification of . (Coincidentally, the cube is also the dual of the octahedron; usually the dual and the zonohedrification are distinct.) To continue the example, we find that , i.e., , is the The definition above assumes that (the center of ) is well defined. If has a center of symmetry, it is a natural choice for , but the zonohedrification of an arbitrary irregular polyhedron will depend on what point we choose to call . The software below takes the origin as . Starting with a square pyramid (half an octahedron), the zonohedrification is still a cube if is chosen to be the center of the base, but an elongated dodecahedron (Figure 1h) if lies above or below the center of the base. The elongated dodecahedron is notable for being one of the five convex solids that are space-filling by translation (without rotation or reflection). These five solids, Figures 1a, b, e, h, and Figure 3b, are each the Voronoi cell of a three-dimensional lattice and so any sheared or stretched version is equally space filling. They were first enumerated by the Russian crystallographer E. S. Fedorov, who originally defined and studied zonohedra a century ago.
To appreciate zonohedra, we must see their From this circumhopping construction, it follows that a zonohedron has a number of zones equal to the number of different edge directions and, in fact, the set of edge directions determines the zonohedron. If we consider each edge direction vector to be rooted at the origin, we have what is called a
If a zonohedron is made entirely of parallelograms and has more than three zones, one can remove any zone from it and assemble the two disconnected pieces to obtain a smaller zonohedron, with one less zone. Figure 2 illustrates this process starting with (a) the 6-zone 30-face rhombic triacontahedron. (It is ; the twelve vertices of an icosahedron lie on six "long diagonal" lines which define the zonal edge directions.) By steps, it is first reduced to (b) a 5-zone 20-face rhombic icosahedron, which is a polar zonohedron, , then (c) the 4-zone 12-face rhombic dodecahedron of the second kind 1; and finally (d) a 3-zone 6-face parallelepiped. Observe in the previous paragraph that -zone zonohedra have faces. This is always the case when a zonohedron is bounded by parallelograms. The reason is easily seen if one observes that every pair of zones intersects twice, at two opposite faces. As there are choose possible pairs of zones, it follows that there are faces. In the general case, where some zonogon faces have more than four sides, there will be fewer faces, as some faces are the intersection of three or more zones. The combinatorics of counting faces, edges, and vertices is interesting, but not needed here. One can reverse the zone-removal process of Figure 2 and write an algorithm, which constructs a zonohedron zone-by-zone, but I have found that approach to be relatively slow. Before describing a faster method, let me first mention and dismiss an even slower method.
If an Converted by Mathematica
September 30, 1999
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