As Fuller so often observed, nature consists exclusively of endlessly transforming energy. Atoms gather in "high-frequency" clusters, disassociate periodically, and regroup elsewhere—new patterns, different substances. We interpret these transient events as solids and liquids because of the limitations of our five senses. To Fuller, "exquisitely transformable" polyhedra were highly logical models with which to elucidate nature's behavior. "Intertransformability" thus applies to both nature and her models.

      We now consider a final experiment from Loeb's research, to bring the subject to a close. Start by truncating a representative polyhedron such as the cube. Continue to the degenerate polyhedron (in this case the cuboctahedron, as illustrated in Fig. 4-l0c), and put it aside for a moment. Now take that same cube, the degenerate stellation of which yields the rhombic dodecahedron (as seen in Fig. 4-12) with its twelve rhombic faces.

      Twelve identical faces? Have we accidentally overlooked a regular polyhedron? No, for the other requirement, of identical vertices, is not satisfied. The diamonds' obtuse angles ⁽⁸⁾ come together at eight three-valent vertices, and their acute angles ⁽⁸⁾ at six four-valent vertices. (Refer to Fig. 4-12.)

      Those numbers are familiar. Twelve identical faces, along with six four-valent and eight three-valent vertices, correspond to the twelve identical vertices, six four-valent faces (squares), and eight three-valent faces (triangles) of the cuboctahedron (Fig. 4-13). Loeb thus shows that duality extends to semiregular polyhedra. This fact would have enabled us to predict the rhombic dodecahedron's existence, by specifying twelve similar quadrilateral faces to correspond to twelve four-valent vertices, and so on. Now we can examine the results of the experiment. A pair of dual polyhedra was created by applying the two inverse operations to the same initial shape.

Fig. 4-13. Cuboctahedron and rhombic dodecahedron are dual polyhedra. Click on thumbnail for larger image.

      Had we started with an octahedron, the cube's dual, the results would have been the same. Degenerate stellation creates the rhombic dodecahedron (Fig. 4-8), and degenerate truncation, the cuboctahedron, as described above (Fig. 4-9). Thus, the two operations generate dual polyhedra from one starting point. Loeb concludes that degenerate truncation and stellation are dual operations.

      To generalize, Loeb discovers that if both members of a pair of dual polyhedra are truncated (or both stellated) to the degenerate case, they will lead to the same result. Conversely, separate degenerate truncation and stellation of the same shape create a new dual pair. Finally, regular polyhedra usually generate semiregular ones.

The foundation is almost in place. A final tool to pick up is an understanding of symmetry: "exact correspondence of form or constituent configuration on opposite sides of a dividing line or plane or about a center or axis." This somewhat abstruse definition from The American Heritage Dictionary introduces the two types of symmetry.

Fig. 4-14 Click on thumbnail for larger image.

Mirror symmetry is the more familiar, involving the exact reflection of a pattern on either side of a "mirror line" (or plane). The letter "M" exhibits mirror symmetry; "R" does not (Fig. 4-14b).       Rotational symmetry specifies that a configuration can be rotated some fraction of 360 degrees (depending on the numerical type of rotational symmetry) without changing the pattern. For example, a square, exhibiting fourfold rotational symmetry about its center, can be rotated 90, 180, or 270 degrees without detectable change. Similarly, an icosahedron has fivefold rotational symmetry about an axis through a pair of opposite vertices (Fig. 4-14a). The letter "S" exhibits twofold rotational symmetry; it looks the same after a 180-degree turn (Fig. 4-14b). In other words, in x-fold rotational symmetry, constituents of a pattern are repeated x times about a common center.

      These concepts will prove useful as we proceed through Fuller's discoveries.