What happens if we truncate both members of a pair of dual polyhedra? The octahedron and cube provide a representative pair for Loeb's elegant experiment. Chop off the corners of the cube, creating eight new triangle faces, while changing the six squares into octagons (Fig. 4-l0a). The truncated octahedron (also called tetrakaidecahedron for its fourteen faces) (Fig. 4-10b) is a space-filling shape—a subject that we shall explore more fully in Chapter 12.

Fig. 4-10. (a), (b) Truncation of cube and octahedron. (c), (d) Degenerate truncation of cube produces the same polyhedron as degenerate truncation of octahedron: cuboctahedron. Click on thumbnail for larger image.

      Now, notice what happens if we allow all truncation planes to expand. The truncated cube's triangles and octahedron's squares independently spread to the "degenerate case," with truncation planes meeting at mid-edge points (Fig. 4-l0c, d). Suddenly octagons and hexagons are phased out, becoming squares and triangles respectively. The two different systems turn into the same polyhedron. Conventionally called the cuboctahedron for reasons now apparent (and renamed "vector equilibrium" by Fuller, for a reason explored in Chapter 7), this polyhedron plays a crucial role in Synergetics. In the present context it merely provides an exemplary illustration of the interactions of duality and truncation.

Loeb's further investigation reveals that degenerate truncation of any dual pair leads to the same polyhedron. For instance, degenerate truncation of both the pentagonal dodecahedron and the icosahedron creates the icosadodecahedron, named for its twenty triangles and twelve pentagons. The pentagons, created by slicing the twelve five-valent vertices, alternate with the twenty icosahedral triangles. Looked at another way, twenty triangles have resulted from chopping off all the three-valent vertices of the pentagonal dodecahedron (Fig. 4-11). With either outlook, we begin to see the effect of duality. (Take special note of the system's twelve fivefold elements; their presence will soon be interpreted as a fundamental law under certain conditions.)

Fig. 4-11. Icosadodecahedron. Click on thumbnail for larger image.

      We now need a new category as we uncover new polyhedra that are not regular, but certainly far from random or irregular. The cuboctahedron and the icosadodecahedron are alike in having only one kind of vertex but two different kinds of faces. Such polyhedra are called semiregular. It will not come as a surprise that their duals have one kind of face and two kinds of vertices. You can begin to imagine the potential for generating new systems, as reflected in the term "intertransformabilities."

      The elegant results of degenerately truncating dual polyhedra inspire further questions. Does it follow that degenerate stellation of a dual pair will create the same polyhedron? And if so, how do we define degenerate stellation? Experimentation answers both questions.

      To check the hypothesis with our reliable octahedron-cube pair, we stellate both systems independently. Try to imagine the transition: six shallow square pyramids are superimposed on the cube, while eight triangular pyramids are added to the octahedron, as seen in Figure 4-9. We then increase the altitude of all pyramids, until triangles of adjacent pyramids just become coplanar. In both cases, twenty-four individual triangular facets suddenly merge into twelve rhombic (or diamond) shapes, thereby creating the rhombic dodecahedron, named for its twelve rhombic faces. The original edges of the cube form the short diagonals of the twelve faces, while the octahedral edges turn into the twelve long diagonals (Fig. 4-12). We thereby have illustrated degenerate stellation.

Fig. 4-12. Degenerate stellation of cube produces the same polyhedron as stellation of octahedron: rhombic dodecahedron. Click on thumbnail for larger image.

      Notice that both members of the original dual pair have the same number of edges. This turns out to be a necessary condition of duality, which follows logically from the nature of the geometric correspondence. In order for each vertex to line up with a face, the edges of two dual polyhedra must cross each other, as can be seen in Figure 4-6. We further observe that whereas degenerate truncation produced quadrivalent vertices (cuboctahedron), degenerate stellation produced quadrilateral faces (rhombic dodecahedron). A generalized principle emerges.