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Table I reveals a curious pattern. Notice the relationship between cube and octahedron, along with the similar pairing of pentagonal dodecahedron and icosahedron. Polyhedra thus related, each with the same number of vertices as the other has faces, are called each other's dual. (6) We are thereby introduced to duality as a numerical relationship; the vertex and face tallies are simply switched. Reassuringly, the significance extends: Loeb observes the geometric manifestation of duality in precise matching of vertex to face. Two dual polyhedra line up with every corner of each meeting the center of a window of the other, as the correspondence implies. (Figure 4-7a shows the dual relationship of the cube and octahedron.)
We have conspicuously ignored one member of Plato's polyhedral family. What is the tetrahedron's dual? Following Loeb's example, we count elements to predict the answer, and find the same number of vertices as faces. Therefore, by interchanging the two elements to find the tetrahedron's dual, we generate another tetrahedron (Fig.4-7b). Once again, the minimum system stands out: only the tetrahedron is its own dual. Put two of them together to check: the four windows and corners line up, and curiously, the combined eight vertices of two same-size tetrahedra outline the corners of a cube, as Fuller never tired of explaining. ["Two equal tetrahedra (positive and negative) joined at their common centers define the cube" (462.00, figure; Fig. 4-6c).]
Truncation and Stellation
The introduction of two operations explored by Arthur Loeb will further elucidate the relationships between polyhedra. By altering and combining the five regular polyhedra, we can generate new shapes-one of the geometric phenomena that inspired Fuller's coinage "intertransformabilities." Loeb's observations will help to clarify the meaning of this polysyllabic Fullerism.
"Truncation" involves chopping off corners so that they are replaced by surfaces (Fig. 4-8a). (7) Truncation of a three-valent vertex will generate a triangle; four-valent vertices become squares, and so on. The number of edges determines the number of sides of the new polygon. Notice that the definition does not specify how much of the corner is sliced away in truncation. Loeb's work reinforces our emphasis thus far on topology (studying numbers of elements, or valency) rather than size. The location of slicing is therefore unimportant, until the truncation planes move inward far enough to touch each other. At that point the edges between the new planes disappear, and so the topology changes. Figure 4-8 shows various possibilities, including that final chop at the mid-edge point. This special limit case, called by Loeb degenerate, yields some interesting results, as we shall see. (For example, the "degenerate truncation" of a tetrahedron unexpectedly turns out to be another member of the Platonic Family, the octahedron, as revealed by Fig. 4-8d.)
Readers interested in learning more about these concepts and the mathematical analyses involved should read Space Structures by Arthur Loeb. Although we introduce only "vertex truncation," Loeb's studies extend to "edge truncation" as well. For our purpose of becoming familiar with the interconnectedness of basic polyhedra, we explore just one of Loeb's discoveries in the following pages. Then when we see these shapes again in the context of synergetics, some of their important relationships can be anticipated.
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