Figure 5. The zonohedrification of the rhombic triacontahedron.
The zonohedrify procedure can also create a zonohedron from any arbitrary star, if we embed the star in a list (so it appears like the vertices of one face of a polyhedron). For example, to generate the hexagonal prism of Figure 3b using the star defined above, do the following:
A polar zonohedron, illustrated in , is the zonohedrification of a prism or antiprism. Although prisms and antiprisms are not present in the built-in polyhedra package, their zonohedrifications can be created with a star of vectors arranged like the ribs of an umbrella.
Particularly interesting zonohedra result if we use the symmetry axes of a regular polyhedron as the star. These axes pass through the vertices, face centers, and edge midpoints of a regular polyhedron, so the following three auxiliary routines are useful.
With them, we can create a list of the 31 axes of symmetry of an icosahedron (six five-fold axes through the vertices, 10 three-fold axes through the face centers, and 15 two-fold axes through the edge midpoints) to use as a star. The function
Figure 6. 31-zone zonohedron based on the 31 axes of symmetry of the icosahedron (242 faces).
The resulting 31-zone, 242-face polyhedron is shown as Figure 6. It has been proposed  as the basis for an architectural system, akin to geodesic domes but with a smaller inventory of parts and a greater variety of forms. One final example is the zonohedrification of the truncated cuboctahedron. This is the largest of 15 paper zonohedra models shown in Plate II of . It is easy to generate if we first create a truncated cuboctahedron (Figure 1c) as the zonohedrification of a star of the three four-fold axes and six two-fold axes of a cube.
Figure 7. The zonohedrification of the truncated cuboctahedron (24-zones, 552 faces).
The result, shown here as Figure 7, has 24 zones and 552 parallelogram faces.
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