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Background

Zonohedra are beautiful, interesting polyhedra bounded by zonogons, where a zonogon is a polygon in which the edges come in equal opposite parallel pairs. The faces and the solid are necessarily centrally symmetric, and may have additional symmetries. In the special case where all the faces happen to be four-sided, a zonohedron is bounded by parallelograms.

[Graphics:../Images/index_gr_1.gif]

Figure 1. Some familiar polyhedra which are zonohedra: (a) cube, (b) truncated octahedron, (c) truncated cuboctahedron, (d) truncated icosidodecahedron, (e) rhombic dodecahedron, (f) octagonal prism, (g) truncated rhombic dodecahedron, (h) elongated dodecahedron, (i) rhombic enneacontahedron. These have respectively 3, 6, 9, 15, 4, 5, 7, 5, and 10 zones, and are the zonohedrifications of an octahedron, cuboctahedron, tetrakis cube (i.e., replace each face of a cube with a low square pyramid), icosidodecahedron, cube, octagonal pyramid, square pyramid, and dodecahedron.

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 polar zonohedron, which was illustrated in the Spring, 1996 issue of this journal [1]. Three excellent references [2, 3, 4] discuss the general case at the recreational, undergraduate, and graduate levels respectively. From a mathematical perspective, zonohedra are particularly notable for the way in which they intertwine combinatorial and geometric properties. One relevant theorem [3, p. 28] is that if each face of a convex polyhedron is centrally symmetric, then the entire solid is centrally symmetric. We only consider convex zonohedra here; two interesting nonconvex zonohedra are illustrated in [3, p. 103].

The zonohedra illustrated in Figure 1 are zonohedrifications of other polyhedra listed in the caption. The geometric relation between a polyhedron, [Graphics:../Images/index_gr_2.gif], and its zonohedrification, [Graphics:../Images/index_gr_3.gif], is the following: For every plane defined by the center, [Graphics:../Images/index_gr_4.gif], of [Graphics:../Images/index_gr_5.gif] and (at least) two vertices, [Graphics:../Images/index_gr_6.gif], [Graphics:../Images/index_gr_7.gif], of [Graphics:../Images/index_gr_8.gif], there are two equal opposite faces of [Graphics:../Images/index_gr_9.gif] parallel to that plane. Furthermore, these two opposite faces of [Graphics:../Images/index_gr_10.gif] have edges that are parallel and equal to the lines [Graphics:../Images/index_gr_11.gif] and [Graphics:../Images/index_gr_12.gif]. Furthermore, every edge of [Graphics:../Images/index_gr_13.gif] comes about in that manner, (i.e., every edge is congruent and parallel to a line [Graphics:../Images/index_gr_14.gif] for some vertex [Graphics:../Images/index_gr_15.gif] of [Graphics:../Images/index_gr_16.gif]).

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 [Graphics:../Images/index_gr_17.gif], where [Graphics:../Images/index_gr_18.gif] indicates the zonohedrification of [Graphics:../Images/index_gr_19.gif]. (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 [Graphics:../Images/index_gr_20.gif], i.e., [Graphics:../Images/index_gr_21.gif], is the rhombic dodecahedron. Observe in Figure 1e that the edges of the rhombic dodecahedron are each in one of four directions, corresponding to the four diagonals of the cube. Continuing, [Graphics:../Images/index_gr_22.gif], i.e., [Graphics:../Images/index_gr_23.gif], is a form of truncated rhombic dodecahedron (Figure 1g). The process can be nested indefinitely, producing a sequence of new polyhedra starting with any polyhedron.

The definition above assumes that [Graphics:../Images/index_gr_24.gif] (the center of [Graphics:../Images/index_gr_25.gif]) is well defined. If [Graphics:../Images/index_gr_26.gif] has a center of symmetry, it is a natural choice for [Graphics:../Images/index_gr_27.gif], but the zonohedrification of an arbitrary irregular polyhedron will depend on what point we choose to call [Graphics:../Images/index_gr_28.gif]. The software below takes the origin as [Graphics:../Images/index_gr_29.gif]. Starting with a square pyramid (half an octahedron), the zonohedrification is still a cube if [Graphics:../Images/index_gr_30.gif] is chosen to be the center of the base, but an elongated dodecahedron (Figure 1h) if [Graphics:../Images/index_gr_31.gif] 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 zones. Given one edge of any zonohedron, there exists a set of equal parallel edges (e.g., four vertical edges of a cube). A zone is the set of faces that contain edges from that set (e.g., four vertical faces of the cube). Such a set must form a band, which encircles the zonohedron like a belt. This is because, by definition, all the faces are zonogons. Given an edge, a face incident with that edge has an equal opposite edge, and we can hop from edge to edge moving across a face with each hop and stepping only on equal parallel edges. As there are only a finite number of such edges, eventually the sequence of hops must return to the starting edge.

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 star of [Graphics:../Images/index_gr_32.gif] vectors. When we zonohedrify a polyhedron, we will use its vertices to determine the star and then the star will determine the generated polyhedron. If the vectors in the star are of equal length, the edges of the resulting zonohedron are equal, and it is called an equilateral zonohedron. For example, Figure 1h is not equilateral; by shortening its vertical edges an equilateral zonohedron can be obtained. Similarly, any zonohedron can be made equilateral (i.e., unit length vectors can be chosen for the star).

[Graphics:../Images/index_gr_33.gif]

Figure 2. By a series of zone removals, any zonohedron bounded by parallelograms is reducible to a parallelepiped. In each step, a zone is removed and the two remaining "hemispheres" are brought together. Starting with (a) the rhombic triacontahedron, we get (b) a rhombic icosahedron, (c) a rhombic dodecahedron of the second kind, and (d) a parallelepiped. The reverse process can be used in a zonohedra construction algorithm.

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 [Graphics:../Images/index_gr_34.gif]; 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, [Graphics:../Images/index_gr_35.gif], 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 [Graphics:../Images/index_gr_36.gif]-zone zonohedra have [Graphics:../Images/index_gr_37.gif] 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 [Graphics:../Images/index_gr_38.gif] choose [Graphics:../Images/index_gr_39.gif] possible pairs of zones, it follows that there are [Graphics:../Images/index_gr_40.gif] 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.

[Graphics:../Images/index_gr_41.gif]

Figure 3. Projecting a 4-dimensional hypercube into a 3-space and looking at its exterior typically gives (a) an object whose exterior is a skewed rhombic dodecahedron. In the special case where the projections of three hypercube edges happen to be coplanar, the result can be (b) an irregular hexagonal prism.

If an n-dimensional hypercube is linearly projected into 3-space, the convex hull of the projection is an [Graphics:../Images/index_gr_42.gif]-zone zonohedron. Figure 3 shows how a four-dimensional hypercube projects into 3-space giving a 4-zone solid, which can be either (a) a skewed version of the rhombic dodecahedron or (b) a hexagonal prism. Hexagonal faces arise if three of the hypercube's edge directions are projected into coplanar vectors. A relatively simple program can carry out these steps, but is prohibitively slow because the hypercube has [Graphics:../Images/index_gr_43.gif] vertices. Although the final zonohedron (after taking the convex hull) has a number of components, which is polynomial in [Graphics:../Images/index_gr_44.gif], the intermediate steps require time, which is exponential in [Graphics:../Images/index_gr_45.gif]. So the method is unsuitable for the larger examples below, but it leads to a useful data structure for zonohedral components.


Converted by Mathematica      September 30, 1999

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