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Locating New Polyhedral Systems
The first new polyhedron consists simply of one octahedron with a tetrahedron on two opposite sides (Fig. 9-8). The result, a rhombohedron, can be seen as a partially flattened cube. A toothpick-marshmallow model demonstrates the transition effectively, because the marshmallow joints have sufficient stiffness to hold either inherently unstable shape. The rhombohedron's direct relationship to the cube suggests a space-filling capability, which we shall explore in greater depth in Chapter 12.
The next candidate, the VE, is too familiar to warrant further description at this point. Twelve cuboctahedral vertices can be located around every point in the IVM, thereby embracing eight tetrahedra and six half octahedra.
Furthermore, higher-frequency versions of any of the above polyhedratetrahedron, octahedron, rhombohedron, and VEcan be easily located within the matrix, thus establishing the foundation for truncated polyhedra. Subtract a half octahedron from each of the six corners of a three-frequency octahedron to yield a symmetrical "truncated octahedron" with fourteen faces: six squares and eight regular hexagons (Fig. 9-9b). A "truncated tetrahedron," with four hexagons and four triangles, is left after a single-frequency tetrahedron is removed from each corner of a three-frequency tetrahedron (Fig. 9-9a). The same procedure applies to higher-frequency versions of any of the above shapes, as well as further truncations of truncated shapes. Such transformations can be plotted indefinitely.
Duality and the IVM
We now introduce a new level of flexibility with the addition of a new set of verticesin the exact center of each tetrahedron and octahedron. These new vertices are connected to the original IVM vertices, thereby introducing radial vectors into each of the original cells. Figure 9-10 shows a single octahedron and tetrahedron with these central nodes.
The central angles of a tetrahedron is approximately 109° 28', exactly equal to the octahedron's dihedral angle. No longer surprised by such relationships, we go on to look inside the octahedron and note its central angle of 90 degrees, which is the surface angle of a cube. The octahedron's three body diagonals, or six radil, thus form the XYZ axes. Figure 9-10 highlights these central angles by showing these two polyhedra with central nodes and radli. Right angles are thus integrated into the IVM system as by-products of the (stable) triangulated octahedron, rather than by an arbitrary initial choice of a network of unstable cubes. Since the IVM complex of octahedra and tetrahedra emerges automatically as a consequence of its unique property of spatial omnisymmetry, the array is not the product of an arbitrary choice.
We now observe considerable expansion of our inventory of generated shapes. Starting with the most familiar, we isolate the minimum cube. Formed by a single tetrahedron embraced by four neighboring eighth-octahedral pyramids, or octants, the cube is once again based on the tetrahedron. We first encountered this relationship in "Structure and Pattern Integrity" (using the tetrahedron to establish the minimum stable cube), and now we have determined the exact shape of the leftover space: four eighth-octahedra. This observation indicates that "degenerate stellation" of the tetrahedron forms a cube. The four vertices of the tetrahedron, together with the four centers of neighboring octahedra, provide the eight comers of this basic building block. Its six square faces are created by two adjacent quarters of the square cross-sections of single-frequency octahedra (Fig. 9-11). As with other IVM systems, larger and larger cubes will be outlined by more remote octahedron centers.
Next, we embrace a single octahedron by eight quarter tetrahedra, thereby outlining the rhombic dodecahedron, whose twelve diamond faces have obtuse angles of 109° 28' and acute angles of 70° 32'generated by the tetrahedral central angle and two adjacent axial angles, respectively. Its eight three-valent vertices are the centers of embracing tetrahedra, while its six four-valent vertices are the original octahedron vertices (Fig. 9-12). Once again, we observe the relationship of duality between the VE and rhombic dodecahedron. The former has fourteen faces (six four-sided and eight three-sided) corresponding to the four-valent vertices and three-valent vertices of the latter. Likewise, the twelve four-valent VE vertices line up with the twelve rhombic faces. (Refer to Fig. 4-12.)
The duality between VE and rhombic dodecahedron illustrates the relationship of duality and domain. Having already seen that spheres in closest packing outline the vertices of the VE, we now turn our attention to the domain of individual spheres. (2) The domain of a sphere is defined as the region closer to a given sphere's center than to the center of any other sphere. This necessarily includes the sphere itself, as well as the portion of its surrounding gap that is closer to that sphere than to any other. Imagine a point at the exact center of an interstitial gap; this will be the dividing point between neighboring domains, that is, a vertex of the polyhedron outlined by the sphere's domain. This domain polyhedron happens to be the rhombic dodecahedron. As each sphere in cubic packing is by definition identically situated, each domain must be the same. Therefore, the shape of this region consistently fits together to fill space. Fuller's term for the rhombic dodecahedron is "spheric" because of this relationship to spheres in closest packing.
We now have an experiential basis for the VErhombic-dodecahedron duality. Twelve vectors emanate from any point in the IVM, locating the vertices of the VE, while poking through the center of the twelve diamond faces which frame the point's domain. We were introduced to duality as exact face-to-vertex correspondence, and now we see how duals can be instrumental in locating a system's domain. Our investigation of space-filling in Chapter 12 will explore this relationship more fully.
Returning to the IVM, we observe that four rhombic dodecahedra, or "spherics," come together at the center of each tetrahedron, such that the tetrahedron central angle becomes the obtuse surface angle of the spheric. In the same way, eight cubes meet at the center of each octahedron, as allowed by the shared 90-degree surface and central angles, respectively.
For clarity, we shall refer to the new network, interconnecting the centers of all octahedral and tetrahedral cells, as IVM', and we can draw the following conclusion. If the vertices of a given polyhedron are located in the IVM, then that system's dual will be outlined by the IVM', and vice versa. Similarly, if a polyhedron is centered on a vertex of the IVM, its dual will be centered on a vertex in IVM'. For example, we recall our first case of duality: the vertices of the octahedron's dual, the cube, are supplied by octahedron centers, which are nodes of IVM'.
This discovery leads us to another assumption. As truncation of our familiar polyhedra yields shapes contained within the IVM, the dual operation, stellation, should produce polyhedra outlined by IVM'. The assumption is valid: the additional IVM' vertices provide the loci for the vertices of stellated versions of these basic shapes. Actually, this observation is not new, for we have already seen that quarter-tetrahedral pyramids affixed to octahedron faces produce Fuller's spheric, or, in other words, that a degenerately stellated octahedron becomes a rhombic dodecahedron. The three-valent vertices of this diamond faceted shape are tetrahedron centers, by definition nodes of IVM'.
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