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Examples

[Graphics:../Images/index_gr_122.gif]

Figure 5. The zonohedrification of the rhombic triacontahedron.

To generate the rhombic triacontahedron (Figure 2a) and then its zonohedrification (Figure 5), we can type the following.

[Graphics:../Images/index_gr_123.gif]

Using Nest, we can construct [Graphics:../Images/index_gr_124.gif], the truncated rhombic dodecahedron of Figure 1g. The result is composed of squares and hexagons, and is easy to confuse with the truncated octahedron (Figure 1b) if one doesn't look carefully.

[Graphics:../Images/index_gr_125.gif]

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:

[Graphics:../Images/index_gr_126.gif]

A polar zonohedron, illustrated in [1], 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.

[Graphics:../Images/index_gr_127.gif]
[Graphics:../Images/index_gr_128.gif]
[Graphics:../Images/index_gr_129.gif]
[Graphics:../Images/index_gr_130.gif]

[Graphics:../Images/index_gr_131.gif]

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.

[Graphics:../Images/index_gr_132.gif]

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 unit provides a unit-length vector in the given direction, so the result is an equilateral zonohedron.

[Graphics:../Images/index_gr_133.gif]
[Graphics:../Images/index_gr_134.gif]
[Graphics:../Images/index_gr_135.gif]
[Graphics:../Images/index_gr_136.gif]
[Graphics:../Images/index_gr_137.gif]
[Graphics:../Images/index_gr_138.gif]
[Graphics:../Images/index_gr_139.gif]

[Graphics:../Images/index_gr_140.gif]

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 [6] 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 [3]. 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.

[Graphics:../Images/index_gr_141.gif]
[Graphics:../Images/index_gr_142.gif]
[Graphics:../Images/index_gr_143.gif]
[Graphics:../Images/index_gr_144.gif]
[Graphics:../Images/index_gr_145.gif]
[Graphics:../Images/index_gr_146.gif]
[Graphics:../Images/index_gr_147.gif]
[Graphics:../Images/index_gr_148.gif]
[Graphics:../Images/index_gr_149.gif]

[Graphics:../Images/index_gr_150.gif]

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.


Converted by Mathematica      September 30, 1999

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